File:Eu funded en.jpg

2025-10-29T07:36:09Z

Miguel Caro:

TurboGAP

2025-06-11T12:48:05Z

Miguel Caro:

{| class="wikitable" style="float:right; margin-left: 10px; width: 45%; margin-top: 0px;"
| style="text-align: center;"|
[[File:turbogap_logo.png|320px]]
|-
|
'''What's new?'''
|-
|
<div style = "height: 200px; overflow-y: scroll;" >
'''11/06/2025''': There are new [[potentials]] available for P, CH, CO and AuPt:H.
-----
'''10/06/2023''': There are new [[potentials]] available for Fe and Pt.
-----
'''01/04/2022''': There is a new [[tutorials|tutorial]] available: [[Simulating icosahedral gold clusters]].
-----
'''28/02/2022''': We have added a [[potential]] for elemental gold. Potentials for C, Si and Au are now publicly available for use with TurboGAP, and many more are under development.
-----
'''28/11/2021''': We have added a [[tutorials|tutorial]] on how to use barostating and van der Waals corrections in '''TurboGAP'''.
-----
'''10/05/2021''': The '''TurboGAP''' repo has been made public. Expansion of this website will now take place rapidly.
</div>
|}

Welcome to the official '''TurboGAP''' website!

'''TurboGAP''' is a Fortran code that offers different levels of functionality for machine-learning driven atomistic simulation. The main functionality is provided via the [http://github.com/libAtoms/soap_turbo soap_turbo] library for efficient construction of accurate SOAP-type many-body atomic descriptors. These can also be used in combination with the [http://libatoms.github.io QUIP/GAP code] or with '''TurboGAP''''s native interface. Thanks to QUIP/GAP, '''TurboGAP''''s <code>soap_turbo</code> descriptors can be used to run MD simulations with LAMMPS. Together with the libraries and the native interface, we also provide a list of interatomic [[potentials]] to be used with the code.

'''TurboGAP''' is developed in [https://miguelcaro.org Miguel Caro's group] at Aalto University.

'''TurboGAP''' is free to use for non-commercial purposes, provided that you adhere to its [https://github.com/mcaroba/turbogap/blob/master/LICENSE.md license]. If you want to use '''TurboGAP''' for commercial purposes, or purposes not covered by the '''TurboGAP''' non-commercial license, please contact the [[TurboGAP team]].

{| style="width: 80%" align="center"
| [[File:Aalto_logo.png|250px]]
| [[File:AKA_logo.jpg|200px]]
| [[File:CSC_logo.png|200px]]
|}

Core pot buffer

2025-04-20T07:21:59Z

Miguel Caro: Created page with "<code>core_pot_buffer</code> defines the buffer zone of the cosine cutoff function, in Angstrom, for use with <code>core_pot_cutoff</code>. The value of <code>core_p..."

<code>[[core_pot_buffer]]</code> defines the buffer zone of the cosine cutoff function, in Angstrom, for use with <code>[[core_pot_cutoff]]</code>. The value of <code>[[core_pot_buffer]]</code> is only used if the value of <code>[[core_pot_cutoff]]</code> is smaller than the default cutoff for the tabulated [[core potential]].

=== Summary ===

{| class="wikitable"
|+ Summary for <code>core_pot_buffer</code> keyword
! style="text-align:left;"| Required/optional
! style="text-align:left;"| Type
! style="text-align:left;"| Accepted values
! style="text-align:left;"| Default
! style="text-align:left;"| See also
|-
| Optional
| Real
| Any positive real
| 1. (only used if [[core_pot_cutoff]] is smaller than the tabulated [[core potential]] cutoff)
| <code>[[core potential]]</code>, <code>[[core_pot_cutoff]]</code>
|}

=== Example ===

[[core_pot_cutoff]] = 10.
[[core_pot_buffer]] = 1.

Core pot cutoff

2025-04-20T07:20:32Z

Miguel Caro:

<code>[[core_pot_cutoff]]</code> defines an outer cutoff, in Angstrom, for the tabulated [[core potential]]. This is useful when the tabulated [[core potential]] has a long tail (20 or 25 Angstrom) and the user wants to reduce this value to avoid computing large neighbor lists, which may slow down the calculations. <code>core_pot_cutoff</code> multiplies the tabulated potential(s) by a cosine cutoff function that goes to zero at <code>core_pot_cutoff</code>, switching from 1 to 0 over a distance defined by <code>[[core_pot_buffer]]</code>.

=== Summary ===

{| class="wikitable"
|+ Summary for <code>core_pot_cutoff</code> keyword
! style="text-align:left;"| Required/optional
! style="text-align:left;"| Type
! style="text-align:left;"| Accepted values
! style="text-align:left;"| Default
! style="text-align:left;"| See also
|-
| Optional
| Real
| Any positive real
| None (the default [[core potential]] cutoff is used if [[core_pot_cutoff]] is not defined)
| <code>[[core potential]]</code>, <code>[[core_pot_buffer]]</code>
|}

=== Example ===

[[core_pot_cutoff]] = 10.
[[core_pot_buffer]] = 1.

Energetic and structural analysis of a database of PtAu nanoclusters

2024-05-15T07:17:34Z

Miguel Caro: /* Generating the nanoparticles */

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': to be able to generate the nanoparticles with '''TurboGAP''', you'll need to download the gap_files from [https://zenodo.org/records/11184039 Zenodo]. If you want to skip this step (e.g., to speed up the process of going through the tutorial), you can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found in Ref.<ref name="kloppenburg_2023" />), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2016" />. First, we use Quippy to compute the SOAP descriptors (using the <code>soap_turbo</code> implementation) of our NPs. With these descriptors, which are a set of vectors (one per atom) for each NP, <math>\{ \textbf{q}_i \}</math>, we construct the kernels. These kernels constitute a measure of similarity between individual atomic environments, and are given as dot products, <math>k(i,j) = (\textbf{q}_i \cdot \textbf{q}_j)^\zeta</math>, where <math>k(i,j) \in [0,1]</math> (0 means nothing alike and 1 identical up to symmetry operations). With these individual kernels we compute the global kernels, which allow us to compare whole NPs, <math>K(I,J) = (\sum_{i \in I} \sum_{j \in J} k(i,j)) / \sqrt{K(I,I) K(J,J)}</math>. From the kernels, we construct a distance metric, <math>d(I,J) = \sqrt{1-K(I,J)}</math>. FInally, we use this distance to build a dissimilarity matrix which is passed to the [https://github.com/mcaroba/cl-MDS cl-MDS] code, which performs k-medoids clustering and hierarchical MDS-based embedding of the data points.<ref name="hernandez-leon_2023" />

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

The code will select a series of "medoids" or representative NPs, together with the embedding in two dimensions of the data. Each point in the MDS map represents a NP, and distances on the plot represent how similar two NPs are: the closer they are on the plot the more similar they are in the significantly higher-dimensional configuration space (as goven by the kernel similarities). The initial database <code>db0.xyz</code> has been updated with information related to the clustering and embedding. You can download a sample updated file <code>db1.xyz</code> from [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db1.xyz here]. In the "info" line, the tags "clmds_coordinates", "clmds_cluster" and "clmds_medoid" will appear, respectively indicating the 2D MDS embedding coordinates, cluster it belongs to and whether it's a medoid or not of the corresponding NP.

Now, let's make a <code>plot_nps</code> directory and run this to get the cartoons for the medoid NPs:

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):
atoms = db[k].copy()
try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

We can see how the most representative NPs span all alloy compositions encountered in the database. Here we chose 22 clusters/medoids for the k-medoids algorithm. In k-medoids this number is chosen by the user, althugh one can also sweep over different values and monitor the evolution of some "incoherence" measure, such as average intracluster distances, to select the optimal number of clusters.

[[File:All_np.png|800px]]

Now, we can also plot the MDS maps to visualize the global dissimilarities at a glance:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

Here we are plotting the similarities on the XY coordinates and choose three different color codings to display further information: to which data cluster the NP corresponds to, what is the composition of each NP, and what is the energy (per atom) of each NP. Unsurprisingly, there is a very strong correlation between these three quantities, because for these NP the most differentiating property is their chemical composition (i.e., their Pt fraction).

[[File:Mds_np.png|800px]]

== Analyzing the individual atomic sites ==

After using the global SOAP kernel <math>K(I,J)</math> to compare and analyze the structure of whole NPs, we are going to use the regular (atom- or site-wise) SOAP kernel <math>k(i,j)</math> to gain insight into the site distribution in our database. We are further going to use some of the <code>ase_tools</code> functionality to identify the surface atoms. This might be relevant, e.g., for applications in catalysis. We now use the sparsification features of <code>cl-MDS</code> because the number of atomic sites is too large compared to the number of NPs, and this would significantly slow down the clustering+embedding procedure if we were using the entire set of data points. The following script takes care of all the different steps:

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

We have again updated the database, this time with per-atom information. The new <code>db2.xyz</code> file (an example of which you can download [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db2.xyz here]) now contains extra columns with the cluster and embedding information for all atoms, as well as a column indicating whether the atom is a surface atom or not. Let us now create a new directory, <code>plot_sites</code>, to run the plotting script within, again aiming to keep our top directory as clean as possible.

Note how our Ovito code is now significantly more complex, since we are doing two things: 1) we are aligning the line of view with the center of mass of the NP and the position of the medoids; 2) when the medoid is not a surface atom, we are adding transparency to all the atoms between the medoid and the viewer. Besides, we are doing the less sophisticated operation of mixing the color of the Pt/Au atom (gray/yellow, respectively) with red, to highlight the medoid.

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []
cluster = []
for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)
cluster.append(atoms2.arrays["clmds_cluster"][i])

for k in range(0, len(db)):
atoms = db[k].copy()
medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective

vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (cluster[k]+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()
</syntaxhighlight>

This is what the Ovito rendering of the medoid sites looks like:

[[File:All_sites.png|800px]]

Finally, let's plot the MDS maps and color code according to useful information:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

We have now chosen to color code that data points (of which we have many more than before!) by cluster, species+bulk/surface character, and local GAP energy. We can see that the clustering and embedding algorithms separate preferentially according to species, but also by surface/bulk character. It is also interesting to see how the GAP local energy clearly establishes bulk Pt atoms to be more stable than surface Pt atoms. However, take this with a grain of salt: due to the very small size of these NPs, there is no proper bulk. Nevertheless, the stabilizing effect of the increased atomic coordination is quite evident going from the interior sites to the surface sites.

[[File:Mds_sites.png|800px]]

{{Reference list}}

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-15T12:08:19Z

Miguel Caro:

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found in Ref.<ref name="kloppenburg_2023" />), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2016" />. First, we use Quippy to compute the SOAP descriptors (using the <code>soap_turbo</code> implementation) of our NPs. With these descriptors, which are a set of vectors (one per atom) for each NP, <math>\{ \textbf{q}_i \}</math>, we construct the kernels. These kernels constitute a measure of similarity between individual atomic environments, and are given as dot products, <math>k(i,j) = (\textbf{q}_i \cdot \textbf{q}_j)^\zeta</math>, where <math>k(i,j) \in [0,1]</math> (0 means nothing alike and 1 identical up to symmetry operations). With these individual kernels we compute the global kernels, which allow us to compare whole NPs, <math>K(I,J) = (\sum_{i \in I} \sum_{j \in J} k(i,j)) / \sqrt{K(I,I) K(J,J)}</math>. From the kernels, we construct a distance metric, <math>d(I,J) = \sqrt{1-K(I,J)}</math>. FInally, we use this distance to build a dissimilarity matrix which is passed to the [https://github.com/mcaroba/cl-MDS cl-MDS] code, which performs k-medoids clustering and hierarchical MDS-based embedding of the data points.<ref name="hernandez-leon_2023" />

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

The code will select a series of "medoids" or representative NPs, together with the embedding in two dimensions of the data. Each point in the MDS map represents a NP, and distances on the plot represent how similar two NPs are: the closer they are on the plot the more similar they are in the significantly higher-dimensional configuration space (as goven by the kernel similarities). The initial database <code>db0.xyz</code> has been updated with information related to the clustering and embedding. You can download a sample updated file <code>db1.xyz</code> from [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db1.xyz here]. In the "info" line, the tags "clmds_coordinates", "clmds_cluster" and "clmds_medoid" will appear, respectively indicating the 2D MDS embedding coordinates, cluster it belongs to and whether it's a medoid or not of the corresponding NP.

Now, let's make a <code>plot_nps</code> directory and run this to get the cartoons for the medoid NPs:

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):
atoms = db[k].copy()
try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

We can see how the most representative NPs span all alloy compositions encountered in the database. Here we chose 22 clusters/medoids for the k-medoids algorithm. In k-medoids this number is chosen by the user, althugh one can also sweep over different values and monitor the evolution of some "incoherence" measure, such as average intracluster distances, to select the optimal number of clusters.

[[File:All_np.png|800px]]

Now, we can also plot the MDS maps to visualize the global dissimilarities at a glance:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

Here we are plotting the similarities on the XY coordinates and choose three different color codings to display further information: to which data cluster the NP corresponds to, what is the composition of each NP, and what is the energy (per atom) of each NP. Unsurprisingly, there is a very strong correlation between these three quantities, because for these NP the most differentiating property is their chemical composition (i.e., their Pt fraction).

[[File:Mds_np.png|800px]]

== Analyzing the individual atomic sites ==

After using the global SOAP kernel <math>K(I,J)</math> to compare and analyze the structure of whole NPs, we are going to use the regular (atom- or site-wise) SOAP kernel <math>k(i,j)</math> to gain insight into the site distribution in our database. We are further going to use some of the <code>ase_tools</code> functionality to identify the surface atoms. This might be relevant, e.g., for applications in catalysis. We now use the sparsification features of <code>cl-MDS</code> because the number of atomic sites is too large compared to the number of NPs, and this would significantly slow down the clustering+embedding procedure if we were using the entire set of data points. The following script takes care of all the different steps:

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

We have again updated the database, this time with per-atom information. The new <code>db2.xyz</code> file (an example of which you can download [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db2.xyz here]) now contains extra columns with the cluster and embedding information for all atoms, as well as a column indicating whether the atom is a surface atom or not. Let us now create a new directory, <code>plot_sites</code>, to run the plotting script within, again aiming to keep our top directory as clean as possible.

Note how our Ovito code is now significantly more complex, since we are doing two things: 1) we are aligning the line of view with the center of mass of the NP and the position of the medoids; 2) when the medoid is not a surface atom, we are adding transparency to all the atoms between the medoid and the viewer. Besides, we are doing the less sophisticated operation of mixing the color of the Pt/Au atom (gray/yellow, respectively) with red, to highlight the medoid.

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []
cluster = []
for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)
cluster.append(atoms2.arrays["clmds_cluster"][i])

for k in range(0, len(db)):
atoms = db[k].copy()
medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective

vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (cluster[k]+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()
</syntaxhighlight>

This is what the Ovito rendering of the medoid sites looks like:

[[File:All_sites.png|800px]]

Finally, let's plot the MDS maps and color code according to useful information:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

We have now chosen to color code that data points (of which we have many more than before!) by cluster, species+bulk/surface character, and local GAP energy. We can see that the clustering and embedding algorithms separate preferentially according to species, but also by surface/bulk character. It is also interesting to see how the GAP local energy clearly establishes bulk Pt atoms to be more stable than surface Pt atoms. However, take this with a grain of salt: due to the very small size of these NPs, there is no proper bulk. Nevertheless, the stabilizing effect of the increased atomic coordination is quite evident going from the interior sites to the surface sites.

[[File:Mds_sites.png|800px]]

{{Reference list}}

Template:Reference list

2023-08-15T12:08:05Z

Miguel Caro: /* Reference list */

== Reference list ==

<references>
<ref name="bartok_2010">
'''A.P. Bartók, M.C. Payne, R. Kondor, and G. Csányi'''. ''Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons''.
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.104.136403 Phys. Rev. Lett. '''104''', 136403 (2010)].
</ref>

<ref name="muhli_2021">
'''H. Muhli and M.A. Caro'''. ''GAP interatomic potential for C<sub>60</sub>''. [https://doi.org/10.5281/zenodo.4616343 Zenodo: 10.5281/zenodo.4616343 (2021)].
</ref>

<ref name="muhli_2021b">
'''H. Muhli, X. Chen, A.P. Bartók, P. Hernández-León, G. Csányi, T. Ala-Nissila, and M.A. Caro'''. ''Machine learning force fields based on local parametrization of dispersion interactions: Application to the phase diagram of C<sub>60</sub>''. [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.054106 Phys. Rev. B '''104''', 054106 (2021)].
</ref>

<ref name="bartok_2013">
'''AP Bartók, R Kondor, G Csányi'''. ''On representing chemical environments''. [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.184115 Phys. Rev. B '''87''', 184115 (2013)].
</ref>

<ref name="bartok_2015">
'''A.P. Bartók and G. Csányi'''. ''Gaussian Approximation Potentials: a brief tutorial introduction''. [https://doi.org/10.1002/qua.24927 Int. J. Quantum Chem. '''115''', 1051 (2015)].
</ref>

<ref name="caro_2019">
'''M.A. Caro'''. ''Optimizing many-body atomic descriptors for enhanced computational performance of machine learning based interatomic potentials''. [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.100.024112 Phys. Rev. B '''100''', 024112 (2019)].
</ref>

<ref name="tkatchenko_2009">
'''A. Tkatchenko and M. Scheffler'''. ''Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data''.
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.102.073005 Phys. Rev. Lett. '''102''', 073005 (2009)].
</ref>

<ref name="tkatchenko_2012">
'''A. Tkatchenko, R.A. DiStasio Jr, R. Car, and M. Scheffler'''. ''Accurate and efficient method for many-body van der Waals interactions''.
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.236402 Phys. Rev. Lett. '''108''', 236402 (2012)].
</ref>

<ref name="bartok_2018">
'''A.P. Bartók, J. Kermode, N. Bernstein, and G. Csányi'''. ''Machine learning a general-purpose interatomic potential for silicon''. [https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.041048 Phys. Rev. X '''8''', 041048 (2018)].
</ref>

<ref name="bartok_2017">
'''A.P. Bartók, S. De, C. Poelking, N. Bernstein, J.R. Kermode, G. Csányi, and M. Ceriotti'''. ''Machine learning unifies the modeling of materials and molecules''. [https://www.science.org/doi/10.1126/sciadv.1701816 Sci. Adv. '''3''', e1701816 (2017)].
</ref>

<ref name="de_2016">
'''S. De, A.P. Bartók, G. Csányi, and M. Ceriotti'''. ''Comparing molecules and solids across structural and alchemical space''. [https://pubs.rsc.org/en/content/articlelanding/2016/cp/c6cp00415f Phys. Chem. Chem. Phys. '''18''', 13754 (2016)].
</ref>

<ref name="hernandez-leon_2023">
'''P. Hernández-León and M.A. Caro'''. ''Cluster-based multidimensional scaling embedding tool for data visualization''. [https://arxiv.org/abs/2209.06614 arXiv:2209.06614 (2023)].
</ref>

<ref name="kloppenburg_2023">
'''J. Kloppenburg, L.B. Pártay, H. Jónsson, and M.A. Caro'''. ''A general-purpose machine learning Pt interatomic potential for an accurate description of bulk, surfaces, and nanoparticles''. [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 J. Chem. Phys. '''158''', 134704 (2023)].
</ref>

</references>

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-15T12:07:53Z

Miguel Caro: /* Generating the nanoparticles */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found in Ref.<ref name="kloppenburg_2023" />), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2016" />. First, we use Quippy to compute the SOAP descriptors (using the <code>soap_turbo</code> implementation) of our NPs. With these descriptors, which are a set of vectors (one per atom) for each NP, <math>\{ \textbf{q}_i \}</math>, we construct the kernels. These kernels constitute a measure of similarity between individual atomic environments, and are given as dot products, <math>k(i,j) = (\textbf{q}_i \cdot \textbf{q}_j)^\zeta</math>, where <math>k(i,j) \in [0,1]</math> (0 means nothing alike and 1 identical up to symmetry operations). With these individual kernels we compute the global kernels, which allow us to compare whole NPs, <math>K(I,J) = (\sum_{i \in I} \sum_{j \in J} k(i,j)) / \sqrt{K(I,I) K(J,J)}</math>. From the kernels, we construct a distance metric, <math>d(I,J) = \sqrt{1-K(I,J)}</math>. FInally, we use this distance to build a dissimilarity matrix which is passed to the [https://github.com/mcaroba/cl-MDS cl-MDS] code, which performs k-medoids clustering and hierarchical MDS-based embedding of the data points.<ref name="hernandez-leon_2023" />

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

The code will select a series of "medoids" or representative NPs, together with the embedding in two dimensions of the data. Each point in the MDS map represents a NP, and distances on the plot represent how similar two NPs are: the closer they are on the plot the more similar they are in the significantly higher-dimensional configuration space (as goven by the kernel similarities). The initial database <code>db0.xyz</code> has been updated with information related to the clustering and embedding. You can download a sample updated file <code>db1.xyz</code> from [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db1.xyz here]. In the "info" line, the tags "clmds_coordinates", "clmds_cluster" and "clmds_medoid" will appear, respectively indicating the 2D MDS embedding coordinates, cluster it belongs to and whether it's a medoid or not of the corresponding NP.

Now, let's make a <code>plot_nps</code> directory and run this to get the cartoons for the medoid NPs:

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):
atoms = db[k].copy()
try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

We can see how the most representative NPs span all alloy compositions encountered in the database. Here we chose 22 clusters/medoids for the k-medoids algorithm. In k-medoids this number is chosen by the user, althugh one can also sweep over different values and monitor the evolution of some "incoherence" measure, such as average intracluster distances, to select the optimal number of clusters.

[[File:All_np.png|800px]]

Now, we can also plot the MDS maps to visualize the global dissimilarities at a glance:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

Here we are plotting the similarities on the XY coordinates and choose three different color codings to display further information: to which data cluster the NP corresponds to, what is the composition of each NP, and what is the energy (per atom) of each NP. Unsurprisingly, there is a very strong correlation between these three quantities, because for these NP the most differentiating property is their chemical composition (i.e., their Pt fraction).

[[File:Mds_np.png|800px]]

== Analyzing the individual atomic sites ==

After using the global SOAP kernel <math>K(I,J)</math> to compare and analyze the structure of whole NPs, we are going to use the regular (atom- or site-wise) SOAP kernel <math>k(i,j)</math> to gain insight into the site distribution in our database. We are further going to use some of the <code>ase_tools</code> functionality to identify the surface atoms. This might be relevant, e.g., for applications in catalysis. We now use the sparsification features of <code>cl-MDS</code> because the number of atomic sites is too large compared to the number of NPs, and this would significantly slow down the clustering+embedding procedure if we were using the entire set of data points. The following script takes care of all the different steps:

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

We have again updated the database, this time with per-atom information. The new <code>db2.xyz</code> file (an example of which you can download [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db2.xyz here]) now contains extra columns with the cluster and embedding information for all atoms, as well as a column indicating whether the atom is a surface atom or not. Let us now create a new directory, <code>plot_sites</code>, to run the plotting script within, again aiming to keep our top directory as clean as possible.

Note how our Ovito code is now significantly more complex, since we are doing two things: 1) we are aligning the line of view with the center of mass of the NP and the position of the medoids; 2) when the medoid is not a surface atom, we are adding transparency to all the atoms between the medoid and the viewer. Besides, we are doing the less sophisticated operation of mixing the color of the Pt/Au atom (gray/yellow, respectively) with red, to highlight the medoid.

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []
cluster = []
for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)
cluster.append(atoms2.arrays["clmds_cluster"][i])

for k in range(0, len(db)):
atoms = db[k].copy()
medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective

vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (cluster[k]+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()
</syntaxhighlight>

This is what the Ovito rendering of the medoid sites looks like:

[[File:All_sites.png|800px]]

Finally, let's plot the MDS maps and color code according to useful information:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

We have now chosen to color code that data points (of which we have many more than before!) by cluster, species+bulk/surface character, and local GAP energy. We can see that the clustering and embedding algorithms separate preferentially according to species, but also by surface/bulk character. It is also interesting to see how the GAP local energy clearly establishes bulk Pt atoms to be more stable than surface Pt atoms. However, take this with a grain of salt: due to the very small size of these NPs, there is no proper bulk. Nevertheless, the stabilizing effect of the increased atomic coordination is quite evident going from the interior sites to the surface sites.

[[File:Mds_sites.png|800px]]

{{Reference list}}

Template:Reference list

2023-08-15T12:07:15Z

Miguel Caro: /* Reference list */

== Reference list ==

<references>
<ref name="bartok_2010">
'''A.P. Bartók, M.C. Payne, R. Kondor, and G. Csányi'''. ''Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons''.
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.104.136403 Phys. Rev. Lett. '''104''', 136403 (2010)].
</ref>

<ref name="muhli_2021">
'''H. Muhli and M.A. Caro'''. ''GAP interatomic potential for C<sub>60</sub>''. [https://doi.org/10.5281/zenodo.4616343 Zenodo: 10.5281/zenodo.4616343 (2021)].
</ref>

<ref name="muhli_2021b">
'''H. Muhli, X. Chen, A.P. Bartók, P. Hernández-León, G. Csányi, T. Ala-Nissila, and M.A. Caro'''. ''Machine learning force fields based on local parametrization of dispersion interactions: Application to the phase diagram of C<sub>60</sub>''. [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.054106 Phys. Rev. B '''104''', 054106 (2021)].
</ref>

<ref name="bartok_2013">
'''AP Bartók, R Kondor, G Csányi'''. ''On representing chemical environments''. [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.184115 Phys. Rev. B '''87''', 184115 (2013)].
</ref>

<ref name="bartok_2015">
'''A.P. Bartók and G. Csányi'''. ''Gaussian Approximation Potentials: a brief tutorial introduction''. [https://doi.org/10.1002/qua.24927 Int. J. Quantum Chem. '''115''', 1051 (2015)].
</ref>

<ref name="caro_2019">
'''M.A. Caro'''. ''Optimizing many-body atomic descriptors for enhanced computational performance of machine learning based interatomic potentials''. [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.100.024112 Phys. Rev. B '''100''', 024112 (2019)].
</ref>

<ref name="tkatchenko_2009">
'''A. Tkatchenko and M. Scheffler'''. ''Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data''.
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.102.073005 Phys. Rev. Lett. '''102''', 073005 (2009)].
</ref>

<ref name="tkatchenko_2012">
'''A. Tkatchenko, R.A. DiStasio Jr, R. Car, and M. Scheffler'''. ''Accurate and efficient method for many-body van der Waals interactions''.
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.236402 Phys. Rev. Lett. '''108''', 236402 (2012)].
</ref>

<ref name="bartok_2018">
'''A.P. Bartók, J. Kermode, N. Bernstein, and G. Csányi'''. ''Machine learning a general-purpose interatomic potential for silicon''. [https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.041048 Phys. Rev. X '''8''', 041048 (2018)].
</ref>

<ref name="bartok_2017">
'''A.P. Bartók, S. De, C. Poelking, N. Bernstein, J.R. Kermode, G. Csányi, and M. Ceriotti'''. ''Machine learning unifies the modeling of materials and molecules''. [https://www.science.org/doi/10.1126/sciadv.1701816 Sci. Adv. '''3''', e1701816 (2017)].
</ref>

<ref name="de_2016">
'''S. De, A.P. Bartók, G. Csányi, and M. Ceriotti'''. ''Comparing molecules and solids across structural and alchemical space''. [https://pubs.rsc.org/en/content/articlelanding/2016/cp/c6cp00415f Phys. Chem. Chem. Phys. '''18''', 13754 (2016)].
</ref>

<ref name="hernandez-leon_2023">
'''P. Hernández-León and M.A. Caro'''. ''Cluster-based multidimensional scaling embedding tool for data visualization''. [https://arxiv.org/abs/2209.06614 arXiv:2209.06614 (2023)].
</ref>

<ref name="kloppenburg_2023">
'''J. Kloppenburg, L.B. Pártay, H. Jónsson, and M.A. Caro'''. ''A general-purpose machine learning Pt interatomic potential for an accurate description of bulk, surfaces, and nanoparticles''. J. Chem. Phys. '''158''', 134704 (2023).
</ref>

</references>

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-15T12:03:10Z

Miguel Caro: /* Analyzing the individual atomic sites */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2016" />. First, we use Quippy to compute the SOAP descriptors (using the <code>soap_turbo</code> implementation) of our NPs. With these descriptors, which are a set of vectors (one per atom) for each NP, <math>\{ \textbf{q}_i \}</math>, we construct the kernels. These kernels constitute a measure of similarity between individual atomic environments, and are given as dot products, <math>k(i,j) = (\textbf{q}_i \cdot \textbf{q}_j)^\zeta</math>, where <math>k(i,j) \in [0,1]</math> (0 means nothing alike and 1 identical up to symmetry operations). With these individual kernels we compute the global kernels, which allow us to compare whole NPs, <math>K(I,J) = (\sum_{i \in I} \sum_{j \in J} k(i,j)) / \sqrt{K(I,I) K(J,J)}</math>. From the kernels, we construct a distance metric, <math>d(I,J) = \sqrt{1-K(I,J)}</math>. FInally, we use this distance to build a dissimilarity matrix which is passed to the [https://github.com/mcaroba/cl-MDS cl-MDS] code, which performs k-medoids clustering and hierarchical MDS-based embedding of the data points.<ref name="hernandez-leon_2023" />

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

The code will select a series of "medoids" or representative NPs, together with the embedding in two dimensions of the data. Each point in the MDS map represents a NP, and distances on the plot represent how similar two NPs are: the closer they are on the plot the more similar they are in the significantly higher-dimensional configuration space (as goven by the kernel similarities). The initial database <code>db0.xyz</code> has been updated with information related to the clustering and embedding. You can download a sample updated file <code>db1.xyz</code> from [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db1.xyz here]. In the "info" line, the tags "clmds_coordinates", "clmds_cluster" and "clmds_medoid" will appear, respectively indicating the 2D MDS embedding coordinates, cluster it belongs to and whether it's a medoid or not of the corresponding NP.

Now, let's make a <code>plot_nps</code> directory and run this to get the cartoons for the medoid NPs:

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):
atoms = db[k].copy()
try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

We can see how the most representative NPs span all alloy compositions encountered in the database. Here we chose 22 clusters/medoids for the k-medoids algorithm. In k-medoids this number is chosen by the user, althugh one can also sweep over different values and monitor the evolution of some "incoherence" measure, such as average intracluster distances, to select the optimal number of clusters.

[[File:All_np.png|800px]]

Now, we can also plot the MDS maps to visualize the global dissimilarities at a glance:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

Here we are plotting the similarities on the XY coordinates and choose three different color codings to display further information: to which data cluster the NP corresponds to, what is the composition of each NP, and what is the energy (per atom) of each NP. Unsurprisingly, there is a very strong correlation between these three quantities, because for these NP the most differentiating property is their chemical composition (i.e., their Pt fraction).

[[File:Mds_np.png|800px]]

== Analyzing the individual atomic sites ==

After using the global SOAP kernel <math>K(I,J)</math> to compare and analyze the structure of whole NPs, we are going to use the regular (atom- or site-wise) SOAP kernel <math>k(i,j)</math> to gain insight into the site distribution in our database. We are further going to use some of the <code>ase_tools</code> functionality to identify the surface atoms. This might be relevant, e.g., for applications in catalysis. We now use the sparsification features of <code>cl-MDS</code> because the number of atomic sites is too large compared to the number of NPs, and this would significantly slow down the clustering+embedding procedure if we were using the entire set of data points. The following script takes care of all the different steps:

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

We have again updated the database, this time with per-atom information. The new <code>db2.xyz</code> file (an example of which you can download [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db2.xyz here]) now contains extra columns with the cluster and embedding information for all atoms, as well as a column indicating whether the atom is a surface atom or not. Let us now create a new directory, <code>plot_sites</code>, to run the plotting script within, again aiming to keep our top directory as clean as possible.

Note how our Ovito code is now significantly more complex, since we are doing two things: 1) we are aligning the line of view with the center of mass of the NP and the position of the medoids; 2) when the medoid is not a surface atom, we are adding transparency to all the atoms between the medoid and the viewer. Besides, we are doing the less sophisticated operation of mixing the color of the Pt/Au atom (gray/yellow, respectively) with red, to highlight the medoid.

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []
cluster = []
for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)
cluster.append(atoms2.arrays["clmds_cluster"][i])

for k in range(0, len(db)):
atoms = db[k].copy()
medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective

vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (cluster[k]+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()
</syntaxhighlight>

This is what the Ovito rendering of the medoid sites looks like:

[[File:All_sites.png|800px]]

Finally, let's plot the MDS maps and color code according to useful information:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

We have now chosen to color code that data points (of which we have many more than before!) by cluster, species+bulk/surface character, and local GAP energy. We can see that the clustering and embedding algorithms separate preferentially according to species, but also by surface/bulk character. It is also interesting to see how the GAP local energy clearly establishes bulk Pt atoms to be more stable than surface Pt atoms. However, take this with a grain of salt: due to the very small size of these NPs, there is no proper bulk. Nevertheless, the stabilizing effect of the increased atomic coordination is quite evident going from the interior sites to the surface sites.

[[File:Mds_sites.png|800px]]

{{Reference list}}

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-15T11:46:35Z

Miguel Caro: /* Analyzing the individual atomic sites */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2016" />. First, we use Quippy to compute the SOAP descriptors (using the <code>soap_turbo</code> implementation) of our NPs. With these descriptors, which are a set of vectors (one per atom) for each NP, <math>\{ \textbf{q}_i \}</math>, we construct the kernels. These kernels constitute a measure of similarity between individual atomic environments, and are given as dot products, <math>k(i,j) = (\textbf{q}_i \cdot \textbf{q}_j)^\zeta</math>, where <math>k(i,j) \in [0,1]</math> (0 means nothing alike and 1 identical up to symmetry operations). With these individual kernels we compute the global kernels, which allow us to compare whole NPs, <math>K(I,J) = (\sum_{i \in I} \sum_{j \in J} k(i,j)) / \sqrt{K(I,I) K(J,J)}</math>. From the kernels, we construct a distance metric, <math>d(I,J) = \sqrt{1-K(I,J)}</math>. FInally, we use this distance to build a dissimilarity matrix which is passed to the [https://github.com/mcaroba/cl-MDS cl-MDS] code, which performs k-medoids clustering and hierarchical MDS-based embedding of the data points.<ref name="hernandez-leon_2023" />

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

The code will select a series of "medoids" or representative NPs, together with the embedding in two dimensions of the data. Each point in the MDS map represents a NP, and distances on the plot represent how similar two NPs are: the closer they are on the plot the more similar they are in the significantly higher-dimensional configuration space (as goven by the kernel similarities). The initial database <code>db0.xyz</code> has been updated with information related to the clustering and embedding. You can download a sample updated file <code>db1.xyz</code> from [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db1.xyz here]. In the "info" line, the tags "clmds_coordinates", "clmds_cluster" and "clmds_medoid" will appear, respectively indicating the 2D MDS embedding coordinates, cluster it belongs to and whether it's a medoid or not of the corresponding NP.

Now, let's make a <code>plot_nps</code> directory and run this to get the cartoons for the medoid NPs:

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):
atoms = db[k].copy()
try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

We can see how the most representative NPs span all alloy compositions encountered in the database. Here we chose 22 clusters/medoids for the k-medoids algorithm. In k-medoids this number is chosen by the user, althugh one can also sweep over different values and monitor the evolution of some "incoherence" measure, such as average intracluster distances, to select the optimal number of clusters.

[[File:All_np.png|800px]]

Now, we can also plot the MDS maps to visualize the global dissimilarities at a glance:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

Here we are plotting the similarities on the XY coordinates and choose three different color codings to display further information: to which data cluster the NP corresponds to, what is the composition of each NP, and what is the energy (per atom) of each NP. Unsurprisingly, there is a very strong correlation between these three quantities, because for these NP the most differentiating property is their chemical composition (i.e., their Pt fraction).

[[File:Mds_np.png|800px]]

== Analyzing the individual atomic sites ==

After using the global SOAP kernel <math>K(I,J)</math> to compare and analyze the structure of whole NPs, we are going to use the regular (atom- or site-wise) SOAP kernel <math>k(i,j)</math> to gain insight into the site distribution in our database. We are further going to use some of the <code>ase_tools</code> functionality to identify the surface atoms. This might be relevant, e.g., for applications in catalysis. We now use the sparsification features of <code>cl-MDS</code> because the number of atomic sites is too large compared to the number of NPs, and this would significantly slow down the clustering+embedding procedure if we were using the entire set of data points. The following script takes care of all the different steps:

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

We have again updated the database, this time with per-atom information. The new <code>db2.xyz</code> file (an example of which you can download [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db2.xyz here]) now contains extra columns with the cluster and embedding information for all atoms, as well as a column indicating whether the atom is a surface atom or not. Let us now create a new directory, <code>plot_sites</code>, to run the plotting script within, again aiming to keep our top directory as clean as possible.

Note how our Ovito code is now significantly more complex, since we are doing two things: 1) we are aligning the line of view with the center of mass of the NP and the position of the medoids; 2) when the medoid is not a surface atom, we are adding transparency to all the atoms between the medoid and the viewer. Besides, we are doing the less sophisticated operation of mixing the color of the Pt/Au atom (gray/yellow, respectively) with red, to highlight the medoid.

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):
atoms = db[k].copy()
medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective

vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()
</syntaxhighlight>

This is what the Ovito rendering of the medoid sites looks like:

[[File:All_sites.png|800px]]

Finally, let's plot the MDS maps and color code according to useful information:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

We have now chosen to color code that data points (of which we have many more than before!) by cluster, species+bulk/surface character, and local GAP energy. We can see that the clustering and embedding algorithms separate preferentially according to species, but also by surface/bulk character. It is also interesting to see how the GAP local energy clearly establishes bulk Pt atoms to be more stable than surface Pt atoms. However, take this with a grain of salt: due to the very small size of these NPs, there is no proper bulk. Nevertheless, the stabilizing effect of the increased atomic coordination is quite evident going from the interior sites to the surface sites.

[[File:Mds_sites.png|800px]]

{{Reference list}}

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-15T08:00:11Z

Miguel Caro: /* Analyze the structure */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2016" />. First, we use Quippy to compute the SOAP descriptors (using the <code>soap_turbo</code> implementation) of our NPs. With these descriptors, which are a set of vectors (one per atom) for each NP, <math>\{ \textbf{q}_i \}</math>, we construct the kernels. These kernels constitute a measure of similarity between individual atomic environments, and are given as dot products, <math>k(i,j) = (\textbf{q}_i \cdot \textbf{q}_j)^\zeta</math>, where <math>k(i,j) \in [0,1]</math> (0 means nothing alike and 1 identical up to symmetry operations). With these individual kernels we compute the global kernels, which allow us to compare whole NPs, <math>K(I,J) = (\sum_{i \in I} \sum_{j \in J} k(i,j)) / \sqrt{K(I,I) K(J,J)}</math>. From the kernels, we construct a distance metric, <math>d(I,J) = \sqrt{1-K(I,J)}</math>. FInally, we use this distance to build a dissimilarity matrix which is passed to the [https://github.com/mcaroba/cl-MDS cl-MDS] code, which performs k-medoids clustering and hierarchical MDS-based embedding of the data points.<ref name="hernandez-leon_2023" />

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

The code will select a series of "medoids" or representative NPs, together with the embedding in two dimensions of the data. Each point in the MDS map represents a NP, and distances on the plot represent how similar two NPs are: the closer they are on the plot the more similar they are in the significantly higher-dimensional configuration space (as goven by the kernel similarities). The initial database <code>db0.xyz</code> has been updated with information related to the clustering and embedding. You can download a sample updated file <code>db1.xyz</code> from [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db1.xyz here]. In the "info" line, the tags "clmds_coordinates", "clmds_cluster" and "clmds_medoid" will appear, respectively indicating the 2D MDS embedding coordinates, cluster it belongs to and whether it's a medoid or not of the corresponding NP.

Now, let's make a <code>plot_nps</code> directory and run this to get the cartoons for the medoid NPs:

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):
atoms = db[k].copy()
try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

We can see how the most representative NPs span all alloy compositions encountered in the database. Here we chose 22 clusters/medoids for the k-medoids algorithm. In k-medoids this number is chosen by the user, althugh one can also sweep over different values and monitor the evolution of some "incoherence" measure, such as average intracluster distances, to select the optimal number of clusters.

[[File:All_np.png|800px]]

Now, we can also plot the MDS maps to visualize the global dissimilarities at a glance:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

Here we are plotting the similarities on the XY coordinates and choose three different color codings to display further information: to which data cluster the NP corresponds to, what is the composition of each NP, and what is the energy (per atom) of each NP. Unsurprisingly, there is a very strong correlation between these three quantities, because for these NP the most differentiating property is their chemical composition (i.e., their Pt fraction).

[[File:Mds_np.png|800px]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

{{Reference list}}

Gap files

2023-08-15T06:08:09Z

Miguel Caro: Redirected page to Documentation#Potential directory (gap files/)

#REDIRECT [[Documentation#Potential directory (gap_files/)]]

XYZ file

2023-08-15T06:06:46Z

Miguel Caro: Redirected page to Documentation#Atoms file (*.xyz)

#REDIRECT [[Documentation#Atoms file (*.xyz)]]

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-15T05:17:43Z

Miguel Caro: /* Analyze the structure */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2016" />. First, we use Quippy to compute the SOAP descriptors (using the <code>soap_turbo</code> implementation) of our NPs. With these descriptors, which are a set of vectors (one per atom) for each NP, <math>\{ \textbf{q}_i \}</math>, we construct the kernels. These kernels constitute a measure of similarity between individual atomic environments, and are given as dot products, <math>k(i,j) = (\textbf{q}_i \cdot \textbf{q}_j)^\zeta</math>, where <math>k(i,j) \in [0,1]</math> (0 means nothing alike and 1 identical up to symmetry operations). With these individual kernels we compute the global kernels, which allow us to compare whole NPs, <math>K(I,J) = (\sum_{i \in I} \sum_{j \in J} k(i,j)) / \sqrt{K(I,I) K(J,J)}</math>. From the kernels, we construct a distance metric, <math>d(I,J) = \sqrt{1-K(I,J)}</math>. FInally, we use this distance to build a dissimilarity matrix which is passed to the [https://github.com/mcaroba/cl-MDS cl-MDS] code, which performs k-medoids clustering and hierarchical MDS-based embedding of the data points.<ref name="hernandez-leon_2023" />

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

The code will select a series of "medoids" or representative NPs, together with the embedding in two dimensions of the data. Each point in the MDS map represents a NP, and distances on the plot represent how similar two NPs are: the closer they are on the plot the more similar they are in the significantly higher-dimensional configuration space (as goven by the kernel similarities). The initial database <code>db0.xyz</code> has been updated with information related to the clustering and embedding. In the "info" line, the tags "clmds_coordinates", "clmds_cluster" and "clmds_medoid" will appear, respectively indicating the 2D MDS embedding coordinates, cluster it belongs to and whether it's a medoid or not of the corresponding NP.

Now, let's make a <code>plot_nps</code> directory and run this to get the cartoons for the medoid NPs:

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):
atoms = db[k].copy()
try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

We can see how the most representative NPs span all alloy compositions encountered in the database. Here we chose 22 clusters/medoids for the k-medoids algorithm. In k-medoids this number is chosen by the user, althugh one can also sweep over different values and monitor the evolution of some "incoherence" measure, such as average intracluster distances, to select the optimal number of clusters.

[[File:All_np.png|800px]]

Now, we can also plot the MDS maps to visualize the global dissimilarities at a glance:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

Here we are plotting the similarities on the XY coordinates and choose three different color codings to display further information: to which data cluster the NP corresponds to, what is the composition of each NP, and what is the energy (per atom) of each NP. Unsurprisingly, there is a very strong correlation between these three quantities, because for these NP the most differentiating property is their chemical composition (i.e., their Pt fraction).

[[File:Mds_np.png|800px]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

{{Reference list}}

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-14T18:36:15Z

Miguel Caro:

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2016" />. First, we use Quippy to compute the SOAP descriptors (using the <code>soap_turbo</code> implementation) of our NPs. With these descriptors, which are a set of vectors (one per atom) for each NP, <math>\{ \textbf{q}_i \}</math>, we construct the kernels. These kernels constitute a measure of similarity between individual atomic environments, and are given as dot products, <math>k(i,j) = (\textbf{q}_i \cdot \textbf{q}_j)^\zeta</math>, where <math>k(i,j) \in [0,1]</math> (0 means nothing alike and 1 identical up to symmetry operations). With these individual kernels we compute the global kernels, which allow us to compare whole NPs, <math>K(I,J) = (\sum_{i \in I} \sum_{j \in J} k(i,j)) / \sqrt{K(I,I) K(J,J)}</math>. From the kernels, we construct a distance metric, <math>d(I,J) = \sqrt{1-K(I,J)}</math>. FInally, we use this distance to build a dissimilarity matrix which is passed to the [https://github.com/mcaroba/cl-MDS cl-MDS] code, which performs k-medoids clustering and hierarchical MDS-based embedding of the data points.<ref name="hernandez-leon_2023" />

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

The code will select a series of "medoids" or representative NPs, together with the embedding in two dimensions of the data. Each point in the MDS map represents a NP, and distances on the plot represent how similar two NPs are: the closer they are on the plot the more similar they are in the significantly higher-dimensional configuration space (as goven by the kernel similarities).

First, let's make a <code>plot_nps</code> directory and run this to get the medoid NPs:

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):
atoms = db[k].copy()
try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

We can see how the most representative NPs span all alloy compositions encountered in the database. Here we chose 22 clusters/medoids for the k-medoids algorithm. In k-medoids this number is chosen by the user, althugh one can also sweep over different values and monitor the evolution of some "incoherence" measure, such as average intracluster distances, to select the optimal number of clusters.

[[File:All_np.png|800px]]

Now, we can also plot the MDS maps to visualize the global dissimilarities are a glance:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png|800px]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

{{Reference list}}

File:Mds sites.png

2023-08-14T18:34:18Z

Miguel Caro:

File:Mds np.png

2023-08-14T18:33:46Z

Miguel Caro:

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-14T18:26:10Z

Miguel Caro: /* Analyze the structure */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2019" />. First, we use Quippy to compute the SOAP descriptors (using the <code>soap_turbo</code> implementation) of our NPs. With these descriptors, which are a set of vectors (one per atom) for each NP, <math>\{ \textbf{q}_i \}</math>, we construct the kernels. These kernels constitute a measure of similarity between individual atomic environments, and are given as dot products, <math>k(i,j) = (\textbf{q}_i \cdot \textbf{q}_j)^\zeta</math>, where <math>k(i,j) \in [0,1]</math> (0 means nothing alike and 1 identical up to symmetry operations). With these individual kernels we compute the global kernels, which allow us to compare whole NPs, <math>K(I,J) = (\sum_{i \in I} \sum_{j \in J} k(i,j)) / \sqrt{K(I,I) K(J,J)}</math>. From the kernels, we construct a distance metric, <math>d(I,J) = \sqrt{1-K(I,J)}</math>. FInally, we use this distance to build a dissimilarity matrix which is passed to the [https://github.com/mcaroba/cl-MDS cl-MDS] code, which performs k-medoids clustering and hierarchical MDS-based embedding of the data points.<ref name="hernandez-leon_2023" />

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

The code will select a series of "medoids" or representative NPs, together with the embedding in two dimensions of the data. Each point in the MDS map represents a NP, and distances on the plot represent how similar two NPs are: the closer they are on the plot the more similar they are in the significantly higher-dimensional configuration space (as goven by the kernel similarities).

First, let's make a <code>plot_nps</code> directory and run this to get the medoid NPs:

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):
atoms = db[k].copy()
try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

We can see how the most representative NPs span all alloy compositions encountered in the database. Here we chose 22 clusters/medoids for the k-medoids algorithm. In k-medoids this number is chosen by the user, althugh one can also sweep over different values and monitor the evolution of some "incoherence" measure, such as average intracluster distances, to select the optimal number of clusters.

[[File:All_np.png|800px]]

Now, we can also plot the MDS maps to visualize the global dissimilarities are a glance:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png|800px]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

{{Reference list}}

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-14T18:25:42Z

Miguel Caro: /* Analyze the structure */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2019" />. First, we use Quippy to compute the SOAP descriptors (using the <code>soap_turbo</code> implementation) of our NPs. With these descriptors, which are a set of vectors (one per atom) for each NP, <math>\{ \textbf{q}_i \}</math>, we construct the kernels. These kernels constitute a measure of similarity between individual atomic environments, and are given as dot products, <math>k(i,j) = (\textbf{q}_i \cdot \textbf{q}_j)^zeta</math>, where <math>k(i,j) \in [0,1]</math> (0 means nothing alike and 1 identical up to symmetry operations). With these individual kernels we compute the global kernels, which allow us to compare whole NPs, <math>K(I,J) = (\sum_{i \in I} \sum_{j \in J} k(i,j)) / \sqrt{K(I,I) K(J,J)}</math>. From the kernels, we construct a distance metric, <math>d(I,J) = \sqrt{1-K(I,J)}</math>. FInally, we use this distance to build a dissimilarity matrix which is passed to the [https://github.com/mcaroba/cl-MDS cl-MDS] code, which performs k-medoids clustering and hierarchical MDS-based embedding of the data points.<ref name="hernandez-leon_2023" />

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

The code will select a series of "medoids" or representative NPs, together with the embedding in two dimensions of the data. Each point in the MDS map represents a NP, and distances on the plot represent how similar two NPs are: the closer they are on the plot the more similar they are in the significantly higher-dimensional configuration space (as goven by the kernel similarities).

First, let's make a <code>plot_nps</code> directory and run this to get the medoid NPs:

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):
atoms = db[k].copy()
try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

We can see how the most representative NPs span all alloy compositions encountered in the database. Here we chose 22 clusters/medoids for the k-medoids algorithm. In k-medoids this number is chosen by the user, althugh one can also sweep over different values and monitor the evolution of some "incoherence" measure, such as average intracluster distances, to select the optimal number of clusters.

[[File:All_np.png|800px]]

Now, we can also plot the MDS maps to visualize the global dissimilarities are a glance:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png|800px]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

{{Reference list}}

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-14T18:24:51Z

Miguel Caro: /* Analyze the structure */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2019" />. First, we use Quippy to compute the SOAP descriptors (using the <code>soap_turbo</code> implementation) of our NPs. With these descriptors, which are a set of vectors (one per atom) for each NP, <math>\{ \textbf{q}_i \}</math>, we construct the kernels. These kernels constitute a measure of similarity between individual atomic environments, and are given as dot products, <math>k(i,j) = (\textbf{q}_i \cdot \texbf{q}_j)^zeta</math>, where <math>k(i,j) \in [0,1]</math> (0 means nothing alike and 1 identical up to symmetry operations). With these individual kernels we compute the global kernels, which allow us to compare whole NPs, <math>K(I,J) = (\sum_{i \in I} \sum_{j \in J} k(i,j)) / \sqrt{K(I,I) K(J,J)}</math>. From the kernels, we construct a distance metric, <math>d(I,J) = sqrt{1-K(I,J)}</math>. FInally, we use this distance to build a dissimilarity matrix which is passed to the [https://github.com/mcaroba/cl-MDS cl-MDS] code, which performs k-medoids clustering and hierarchical MDS-based embedding of the data points.<ref name="hernandez-leon_2023" />

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f \
atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} \
radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} \
n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

The code will select a series of "medoids" or representative NPs, together with the embedding in two dimensions of the data. Each point in the MDS map represents a NP, and distances on the plot represent how similar two NPs are: the closer they are on the plot the more similar they are in the significantly higher-dimensional configuration space (as goven by the kernel similarities).

First, let's make a <code>plot_nps</code> directory and run this to get the medoid NPs:

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):
atoms = db[k].copy()
try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

We can see how the most representative NPs span all alloy compositions encountered in the database. Here we chose 22 clusters/medoids for the k-medoids algorithm. In k-medoids this number is chosen by the user, althugh one can also sweep over different values and monitor the evolution of some "incoherence" measure, such as average intracluster distances, to select the optimal number of clusters.

[[File:All_np.png|800px]]

Now, we can also plot the MDS maps to visualize the global dissimilarities are a glance:

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset xtics
unset ytics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png|800px]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

{{Reference list}}

Template:Reference list

2023-08-14T18:10:18Z

Miguel Caro: /* Reference list */

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-14T16:14:59Z

Miguel Caro:

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2019" />.

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

[[File:All_np.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

{{Reference list}}

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-14T16:14:25Z

Miguel Caro: /* Analyze the structure */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

To grasp the overall structural features of our NPs, we are going to ''global'' similarity kernels based on SOAP descriptors, as described in Ref.<ref name="bartok_2017" /> and Ref.<ref name="de_2019" />.

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

[[File:All_np.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

Template:Reference list

2023-08-14T16:10:43Z

Miguel Caro: /* Reference list */

== Reference list ==

<references>
<ref name="bartok_2010">
'''A.P. Bartók, M.C. Payne, R. Kondor, and G. Csányi'''. ''Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons''.
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.104.136403 Phys. Rev. Lett. '''104''', 136403 (2010)].
</ref>

<ref name="muhli_2021">
'''H. Muhli and M.A. Caro'''. ''GAP interatomic potential for C<sub>60</sub>''. [https://doi.org/10.5281/zenodo.4616343 Zenodo: 10.5281/zenodo.4616343 (2021)].
</ref>

<ref name="muhli_2021b">
'''H. Muhli, X. Chen, A.P. Bartók, P. Hernández-León, G. Csányi, T. Ala-Nissila, and M.A. Caro'''. ''Machine learning force fields based on local parametrization of dispersion interactions: Application to the phase diagram of C<sub>60</sub>''. [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.054106 Phys. Rev. B '''104''', 054106 (2021)].
</ref>

<ref name="bartok_2013">
'''AP Bartók, R Kondor, G Csányi'''. ''On representing chemical environments''. [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.184115 Phys. Rev. B '''87''', 184115 (2013)].
</ref>

<ref name="bartok_2015">
'''A.P. Bartók and G. Csányi'''. ''Gaussian Approximation Potentials: a brief tutorial introduction''. [https://doi.org/10.1002/qua.24927 Int. J. Quantum Chem. '''115''', 1051 (2015)].
</ref>

<ref name="caro_2019">
'''M.A. Caro'''. ''Optimizing many-body atomic descriptors for enhanced computational performance of machine learning based interatomic potentials''. [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.100.024112 Phys. Rev. B '''100''', 024112 (2019)].
</ref>

<ref name="tkatchenko_2009">
'''A. Tkatchenko and M. Scheffler'''. ''Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data''.
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.102.073005 Phys. Rev. Lett. '''102''', 073005 (2009)].
</ref>

<ref name="tkatchenko_2012">
'''A. Tkatchenko, R.A. DiStasio Jr, R. Car, and M. Scheffler'''. ''Accurate and efficient method for many-body van der Waals interactions''.
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.236402 Phys. Rev. Lett. '''108''', 236402 (2012)].
</ref>

<ref name="bartok_2018">
'''A.P. Bartók, J. Kermode, N. Bernstein, and G. Csányi'''. ''Machine learning a general-purpose interatomic potential for silicon''. [https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.041048 Phys. Rev. X '''8''', 041048 (2018)].
</ref>

<ref name="bartok_2017">
'''A.P. Bartók, S. De, C. Poelking, N. Bernstein, J.R. Kermode, G. Csányi, and M. Ceriotti'''. ''Machine learning unifies the modeling of materials and molecules''. [https://www.science.org/doi/10.1126/sciadv.1701816 Sci. Adv. '''3''', e1701816 (2017)].
</ref>

<ref name="de_2016">
'''S. De, A.P. Bartók, G. Csányi, and M. Ceriotti'''. ''Comparing molecules and solids across structural and alchemical space''. Phys. Chem. Chem. Phys. '''18''', 13754 (2016).
</ref>

</references>

Template:Reference list

2023-08-14T15:52:38Z

Miguel Caro: /* Reference list */

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-14T15:26:11Z

Miguel Caro: /* Construct the convex hull of NP stability */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

While we upload the PtAu GAP to a public repository (needed to run the workflow above), you can download a sample <code>db0.xyz</code> file [https://github.com/mcaroba/turbogap/blob/master/tutorials/PtAu_NPs_MDS/db0.xyz here].

Now, let's plot the convex hulls for the different NP sizes. First, create a directory called <code>hulls</code>, to keep the top directory a bit cleaner, and run the following code.

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

After extracting and collecting the NP energies in a suitable format, you can easily plot the results. This is a sample gnuplot code that can do the job.

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

In the figure below you can see how most of the alloy data (purple dots at <math>x \neq 0,1</math>) are higher in energy than the linearly interpolated values for the pure NP of the same size. The only exception are the NPs for <math>N = 30</math>. If the simulations had been carried out carefully, this would mean that alloying is energetically unfavorable since, for a given composition, a mixture of pure NPs is lower in energy than the alloyed NPs with the same average composition. In reality, 1) our annealing time was way too short to properly sample low minima of the alloy potential energy surface and 2) we are omitting the configurational entropy of the alloy which lowers its ''free'' energy of formation at finite temperature (we are only looking at the ''potential'' energy).

[[File:Hulls.png|800px]]

== Analyze the structure ==

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

# Read in the database
#db = []
#for i in range(1,660+1):
# atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
# db.append(atoms)

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

[[File:All_np.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-14T14:54:42Z

Miguel Caro: /* Construct the convex hull of NP stability */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with <code>.xyz</code> extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

[[File:Hulls.png]]

== Analyze the structure ==

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

# Read in the database
#db = []
#for i in range(1,660+1):
# atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
# db.append(atoms)

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

[[File:All_np.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-14T14:54:09Z

Miguel Caro: /* Construct the convex hull of NP stability */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

We are now going to collect all the 660 relaxed structures and put them into an ASE "database" XYZ file: this is simply a file with `.xyz` extension with all the individual XYZ files (containing only the final snapshot) concatenated. The following Python script will collect the data:

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

[[File:Hulls.png]]

== Analyze the structure ==

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

# Read in the database
#db = []
#for i in range(1,660+1):
# atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
# db.append(atoms)

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

[[File:All_np.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-14T06:19:06Z

Miguel Caro: /* Generating the nanoparticles */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png|800px]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

[[File:Hulls.png]]

== Analyze the structure ==

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

# Read in the database
#db = []
#for i in range(1,660+1):
# atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
# db.append(atoms)

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

[[File:All_np.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-11T14:00:08Z

Miguel Caro: /* Generating the nanoparticles */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here]), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This value scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

[[File:Hulls.png]]

== Analyze the structure ==

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

# Read in the database
#db = []
#for i in range(1,660+1):
# atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
# db.append(atoms)

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

[[File:All_np.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-11T13:59:26Z

Miguel Caro: /* Generating the nanoparticles */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
# print(num)
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here), and the PtAu and Au temperatures are estimated from that one scaling by the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). All temperatures are further scaled by a global parameter (<code>f = 1.1</code> below; you can try changing this number to check if it improves the structures). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This values scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

[[File:Hulls.png]]

== Analyze the structure ==

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

# Read in the database
#db = []
#for i in range(1,660+1):
# atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
# db.append(atoms)

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

[[File:All_np.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-11T13:56:41Z

Miguel Caro: /* Generating the nanoparticles */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the <code>init_xyz</code> directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
# print(num)
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here), and the PtAu and Au temperatures are estimated from that one scaling by a global parameter (<code>f = 1.1</code> below) and the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This values scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the <code>trajs/traj_1.1</code> directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

[[File:Hulls.png]]

== Analyze the structure ==

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

# Read in the database
#db = []
#for i in range(1,660+1):
# atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
# db.append(atoms)

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

[[File:All_np.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-11T13:55:17Z

Miguel Caro: /* Generating the nanoparticles */

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

'''WARNING1''': the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are ''most definitely'' too short to generate stable "publication-quality" NPs.

'''WARNING2''': because the PtAu GAP is still not published, you will not be able to run this step (yet). You can still try to follow the scripts and simulation logic, and download the database of relaxed NPs from the next section.

We are going to generate a database of PtAu NPs with variable size and composition. In particular, we will generate 10 NPs per size and composition, where the Pt and Au contents will be varied at 10% increments and the size will be varied at 5 atoms increments from 25 to 50. This will lead to 660 unique PtAu NPs (including pure Pt and Au NPs). The first step is to generate a set of random initial structures with the desired properties. The following Python code will take care of generating random distributions of atoms with roughly spherical shapes where the interatomic distances are not too small to avoid highly energetic starting configurations that may be prone to "blowing up". Make sure to create the `init_xyz` directory before proceeding.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
# print(num)
</syntaxhighlight>

The resulting structures will look something like this:

[[File:Init_all.png]]

Next, we will run some '''TurboGAP''' workflows consisting of high-temperature annealing, where the annealing temperature (1150 K) is the optimal temperature for Pt (as found [https://pubs.aip.org/aip/jcp/article/158/13/134704/2883270 here), and the PtAu and Au temperatures are estimated from that one scaling by a global parameter (`f = 1.1` below) and the ratio of experimental melting temperatures of pure Au and Pt (and interpolating linearly). High-<math>T</math> annealing is followed by a rapid quench down to 100 K, and this is again followed by [[gradient descent]] static relaxation. The whole workflow can be reproduced with the following Bash script.

<syntaxhighlight lang="bash" line="line">
# This values scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 1 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

The trajectory files with exactly three snapshots (system after annealing, quench and relaxation) will be save into the `trajs/traj_1.1` directory. As mentioned at the top of this section, the chosen parameters are way too lax to make properly relaxed NPs.

== Construct the convex hull of NP stability ==

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

[[File:Hulls.png]]

== Analyze the structure ==

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

# Read in the database
#db = []
#for i in range(1,660+1):
# atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
# db.append(atoms)

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

[[File:All_np.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

[[File:Mds_np.png]]

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

[[File:All_sites.png]]

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

[[File:Mds_sites.png]]

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-11T09:40:20Z

Miguel Caro:

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-11T09:38:40Z

Miguel Caro:

'''WARNING: Tutorial under construction!!!!'''

In this tutorial we are going to go beyond the core functionality of '''TurboGAP''' and focus instead of the analysis and interpretation of the results that can be obtained with '''TurboGAP'''. We will look at the energetics and structural characteristics of small PtAu alloy nanoparticles (NPs). Note: at the time of writing this tutorial the PtAu GAP used to generate the NPs is not publicly available. For this reason you will need to skip the step on NP generation, but can still download the pregenerated database so that you can run the rest of the tutorial.

'''Tutorial difficulty: <span style="color:white;background:red">HARD</span>'''

== Prerequisites for this tutorial ==

* A '''TurboGAP''' installation, only for the NP database generation step, which you can skip by downloading the database;
* An [https://wiki.fysik.dtu.dk/ase/ ASE] installation;
* Numpy;
* A plotting program. I will use gnuplot and provide the corresponding code, but it can be replaced by something else;
* An [https://github.com/mcaroba/ase_tools ase_tools] installation;
* A [https://github.com/libatoms/quip quippy] installation (Python interface for the QUIP code);
* A program for ball-and-stick visualization of atomistic structures. I will use the Python version of [https://www.ovito.org/python-downloads/ Ovito].

== Generating the nanoparticles ==

WARNING: the options below are just enough to generate example NPs for this tutorial. The annealing and quenching times are (quite probably) too short to generate stable "publication-quality" NPs.

<syntaxhighlight lang="python" line="line">
import numpy as np
from ase.io import write
from ase import Atoms

a = 3.9668
d_min = 2.

append = False
num = 1

nrand = 10
for n in range(25,50+1,5):
for f_Au in np.arange(0.,1.+1.e-10,0.1):
for i in range(0, nrand):
R = (a**3 / 4. * n * 3./4./np.pi)**(1./3.)

pos = []
while len(pos) < n:
this_pos = (2.*np.random.sample([3]) -1) * R
d = np.dot(this_pos, this_pos)**0.5
if d > R:
continue
if len(pos) == 0:
pos.append(this_pos)
else:
too_close = False
for other_pos in pos:
dv = this_pos - other_pos
d = np.dot(dv, dv)**0.5
if d < d_min:
too_close = True
break
if not too_close:
pos.append(this_pos)

n_Au = int(n*f_Au)
n_Pt = n - n_Au

atoms = Atoms("Pt%iAu%i" % (n_Pt, n_Au), positions = pos, pbc=True)
write("init_xyz/all.xyz", atoms, append = append)
atoms.center(vacuum = 10.)
write("init_xyz/%i.xyz" % num, atoms)
num += 1
append = True
# print(num)
</syntaxhighlight>

<syntaxhighlight lang="bash" line="line">
# This values scales the annealing temperature
f=1.1

mkdir -p trajs/
mkdir -p logs/
mkdir -p trajs/trajs_$f
mkdir -p logs/logs_$f

for i in $(seq 12 1 660); do

SECONDS=0

n_Au=$(awk '{if($1=="Au"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)
n_Pt=$(awk '{if($1=="Pt"){n+=1} else {n+=0}} END {print n}' init_xyz/$i.xyz)

T=$(echo $n_Au $n_Pt | awk -v f=$f '{print f*($1/($1+$2)*750.+$2/($1+$2)*1150.)}')

echo "Doing $i/660..."

##########################################
# Anneal at $T for 1 ps
cat>input<<eof
atoms_file = 'init_xyz/$i.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = $T
tau_t = 10.
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv thermo.log logs/logs_$f/anneal_$i.log
##########################################

##########################################
# Quench to 100K over 1 ps
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 250
md_step = 4.

optimize = "vv"
thermostat = "bussi"
t_beg = $T
t_end = 100.
tau_t = 100.

write_xyz = 250
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz > atoms.xyz
mv trajectory_out.xyz trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/quench_$i.log
##########################################

##########################################
# Relax
cat>input<<eof
atoms_file = 'atoms.xyz'
pot_file = 'gap_files/PtAuH.gap'

n_species = 3
species = Pt Au H

md_nsteps = 1000

optimize = "gd"
eof

mpirun -np 4 turbogap md &> /dev/null
n=$(head -1 trajectory_out.xyz | awk '{print $1+2}')
tail -$n trajectory_out.xyz >> trajs/trajs_$f/$i.xyz
mv thermo.log logs/logs_$f/relax_$i.log
rm trajectory_out.xyz
rm input
##########################################

echo " ... in $SECONDS s"

done
</syntaxhighlight>

== Construct the convex hull of NP stability ==

<syntaxhighlight lang="python" line="line">
from ase.io import read,write

# Read in the database
db = []
for i in range(1,660+1):
atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
db.append(atoms)

write("db0.xyz", db)
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
from ase.io import read
import numpy as np

db = read("../db0.xyz", index=":")

all = {}
Ns = []
xs = {}

for atoms in db:
N = len(atoms)
x = np.round(atoms.symbols.count("Pt")/N, decimals = 8)
if (N,x) not in all:
all[N,x] = [atoms.info["energy"]/N]
else:
all[N,x].append(atoms.info["energy"]/N)
if N not in Ns:
Ns.append(N)
if N not in xs:
xs[N] = [x]
elif x not in xs[N]:
xs[N].append(x)

for N in Ns:
x_vals = []
e_vals = []
for x in xs[N]:
for e in all[N,x]:
if x not in x_vals:
x_vals.append(x)
e_vals.append(e)
else:
for i in range(0, len(x_vals)):
if x == x_vals[i] and e < e_vals[i]:
e_vals[i] = e
if 0. in x_vals and 1. in x_vals:
i0 = np.where([x == 0. for x in x_vals])[0][0]
i1 = np.where([x == 1. for x in x_vals])[0][0]
e0 = e_vals[i0]
e1 = e_vals[i1]
idx = np.argsort(x_vals)
x_vals = np.array(x_vals)[idx]
e_vals = np.array(e_vals)[idx]
for i in range(0, len(x_vals)):
x = x_vals[i]
e_vals[i] -= e1*x + e0*(1.-x)
f = open("%i.dat" % N, "w")
for i in range(0, len(x_vals)):
if i == 0 or i == len(x_vals) or e_vals[i] <= 0.:
print(x_vals[i], e_vals[i], file=f)
print("", file=f)
print("", file=f)
for x in xs[N]:
for e in all[N,x]:
print(x, e-e1*x-e0*(1.-x), file=f)
f.close()
</syntaxhighlight>

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "hulls.png"

set multiplot layout 1,6

set xlabel "x in Pt_{x}Au_{1-x}"
set ylabel "Energy above hull (eV/atom)"

set yrange [-0.02:0.12]
set xtics 0.5
set mxtics 1

do for [i=25:50:5] {
set title "N = ".i
plot "".i.".dat" index 1 pt 7 not, "" index 0 pt 6 ps 1.2 lw 4 w lp not
}
</syntaxhighlight>

== Analyze the structure

<syntaxhighlight lang="python" line="line">
from ase import cluster
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids

# Read in the database
#db = []
#for i in range(1,660+1):
# atoms = read("trajs/trajs_1.1/%i.xyz" % i, index=-1)
# db.append(atoms)

db = read("db0.xyz", index=":")

###############################################################################################
# This builds the global kernel for the NP-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d1 = Descriptor(soap["Pt"])
d2 = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
q1 = d1.calc_descriptor(at)
q2 = d2.calc_descriptor(at)
if q1.shape == (1,0):
q = q2
elif q2.shape == (1,0):
q = q1
else:
q = np.concatenate((q1, q2))
qs.append(q)

kern = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(i,len(qs)):
k = 0.
for i2 in range(0, len(qs[i])):
for j2 in range(0, len(qs[j])):
k += np.dot(qs[i][i2], qs[j][j2])**zeta
kern[i,j] = k
kern[j,i] = k

kern_n = np.zeros([len(qs), len(qs)])
for i in range(0,len(qs)):
for j in range(0,len(qs)):
kern_n[i,j] = kern[i,j] / np.sqrt(kern[i,i] * kern[j,j])

dist = np.zeros([len(qs),len(qs)])
for i in range(0,len(qs)):
dist[i, i] = 0.
for j in range(i+1,len(qs)):
dist[i, j] = np.sqrt(1.-kern_n[i,j])
dist[j, i] = np.sqrt(1.-kern_n[i,j])

M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids

f = open("mds.dat", "w")
print("# X, Y, cluster, is_medoid, N, E, x in Pt_{x}Au_{1-x}", file=f)
for i in range(0, len(db)):
db[i].info["clmds_coordinates"] = XY_sparse[i]
db[i].info["clmds_cluster"] = C_sparse[i]
db[i].info["clmds_medoid"] = i in M_sparse
print(*XY_sparse[i], C_sparse[i], i in M_sparse, len(db[i]), db[i].info["energy"], db[i].symbols.count("Pt")/len(db[i]), file=f)

f.close()
write("db1.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db1.xyz", index=":")
cl = []
db = []

for atoms in db0:
if atoms.info["clmds_medoid"]:
db.append(atoms)
cl.append(atoms.info["clmds_cluster"])

dbt = np.array(db, dtype=object)
db = dbt[cl]

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

try:
del atoms.arrays["fix_atoms"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if atoms[i].symbol == "Pt":
colors[i] = np.array([0.8,0.8,0.8])
elif atoms[i].symbol == "Au":
colors[i] = np.array([1,1,0])
elif atoms[i].symbol == "H":
colors[i] = np.array([1,1,1])

pipeline.source.data.particles_.create_property("Color", data=colors)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
direction = (2, 3, -1)
vp.camera_dir = direction
vp.camera_pos = atoms.get_center_of_mass() - direction / np.dot(direction,direction)**0.5 * 30.
vp.render_image(filename = "np_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
sys.stdout.flush()
</syntaxhighlight>

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_np.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box

set title "PtAu NP according to cluster"
plot "../mds.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "PtAu NP according to composition"
set cblabel "x in Pt_{x}Au_{1-x}"
plot "../mds.dat" u 1:2:7 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set title "PtAu NP according to energy"
set cblabel "Total energy (eV/atom)"
plot "../mds.dat" u 1:2:($6/$5) palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

</syntaxhighlight>

== Analyzing the individual atomic sites ==

<syntaxhighlight lang="python" line="line">
import sys
from quippy.descriptors import Descriptor
from quippy.convert import ase_to_quip
import numpy as np
from ase.io import read,write
import cluster_mds as clmds
import kmedoids
from ase_tools import surface_list

# Read in the database
db = read("db1.xyz", index=":")

# Create mappings
which_structure = []
which_atom = []
map_back = {}
local_energy = []
k = 0
for i in range(0, len(db)):
le = db[i].get_array("local_energy")
for j in range(0, len(db[i])):
which_structure.append(i)
which_atom.append(j)
map_back[i,j] = k
local_energy.append(le[j])
k += 1

# Rolling sphere algorithm parameters
r_min = 4.
r_max = 4.5
n_tries = 1000000

# Build a list of surface atoms
surf_list_all = []
is_surface = np.full(len(which_atom), False, dtype=bool)
k = 0
for i in range(0, len(db)):
surf_list = surface_list(db[i], r_min, r_max, n_tries, cluster=True)
surf_list_all.append(surf_list)
for j in range(0,len(db[i])):
if j in surf_list:
is_surface[k] = True
k += 1
if True:
sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(db)) )
sys.stdout.flush()

###############################################################################################
# This builds the kernel for the site-wise cl-MDS map
n_cl = 22 # number of clusters
cutoff = 5.7; dc = 0.5; sigma = 0.5
zeta = 6
soap = {"Au": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=2 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma])),
"Pt": 'soap_turbo alpha_max={8 8} l_max=8 rcut_soft=%.4f rcut_hard=%.4f atom_sigma_r={%.4f %.4f} atom_sigma_t={%.4f %.4f} \
atom_sigma_r_scaling={0. 0.} atom_sigma_t_scaling={0. 0.} radial_enhancement=1 amplitude_scaling={1. 1.} \
basis="poly3gauss" scaling_mode="polynomial" species_Z={78 79} n_species=2 central_index=1 central_weight={1. 1.} \
compress_mode=trivial' % (cutoff-dc, cutoff, *(4*[sigma]))}

d = {}

d["Pt"] = Descriptor(soap["Pt"])
d["Au"] = Descriptor(soap["Au"])

qs = []
for atoms in db:
at = ase_to_quip(atoms)
N = {}
q = {}
for i in range(0,len(atoms)):
symb = atoms.symbols[i]
if symb not in N:
N[symb] = 0
q[symb] = d[symb].calc_descriptor(at)
this_q = q[symb][N[symb]]
qs.append(this_q)
N[symb] += 1

dist = np.zeros([len(qs),len(qs)])
n = 0
for i in range(0,len(qs)):
arg = 1. - np.dot(qs[:], qs[i])**zeta
arg2 = np.clip(arg, 0, 1)
dist[i,:] = np.sqrt(arg2)
# sys.stdout.write('\rBuilding list of surface atoms:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.write('\rBuilding distance matrix:%6.1f%%' % (float(i)*100./len(qs)) )
sys.stdout.flush()

print(" MBytes: ", dist.nbytes/1024/1024)
print("")

# Embedding
M, C = kmedoids.kMedoids(dist, n_cl, n_iso=n_cl//2, init_Ms="isolated", n_inits=10)
data = clmds.clMDS(dist_matrix = dist, sparsify="cur", n_sparse=500)
data.cluster_MDS([n_cl,1], weight_cluster_mds=2., weight_anchor_mds=10., init_medoids=M)
list_sparse = data.sparse_list
XY_sparse = data.sparse_coordinates
C_sparse = data.sparse_cluster_indices
M_sparse = data.sparse_medoids
M = []
for k in M_sparse:
M.append(list_sparse[k])

M = np.array( M, dtype=int )
indices = np.array( list(range(0,len(dist))), dtype=int )
data.compute_pts_estim_coordinates( indices=indices )
XY = data.estim_coordinates
C = data.estim_cluster_indices
# Transform to usual cluster array
C_usual = []
for i in range(0, n_cl):
C_usual.append([])

for i in range(0, len(C)):
C_usual[C[i]].append(i)

C_list = C
C = C_usual
#

k = 0
for atoms in db:
atoms.set_array("clmds_cluster", C_list[k:k+len(atoms)])
atoms.set_array("clmds_coordinates", XY[k:k+len(atoms)])
k += len(atoms)

k = 0
for atoms in db:
is_medoid = np.full(len(atoms), False)
is_surface2 = np.full(len(atoms), False)
for i in range(0, len(atoms)):
if is_surface[k]:
is_surface2[i] = True
if k in M:
is_medoid[i] = True
k += 1
atoms.set_array("clmds_medoid", is_medoid)
atoms.set_array("surface", is_surface2)

f = open("mds_sites.dat", "w")
print("# X, Y, cluster, is_medoid, is_surface, le, species, N of NP", file=f)
k = 0
for i in range(0, len(db)):
for j in range(0, len(db[i])):
print(*XY[k], C_list[k], k in M, is_surface[k], local_energy[k], db[i].symbols[j], len(db[i]), file=f)
k += 1

f.close()

write("db2.xyz", db)
###############################################################################################
</syntaxhighlight>

<syntaxhighlight lang="python" line="line">
import sys
from ase.io import read,write
from ase.visualize import view
import numpy as np
from ovito.io.ase import ase_to_ovito
from ovito.pipeline import StaticSource, Pipeline
from ovito.vis import Viewport, BondsVis, ParticlesVis, TachyonRenderer
from ovito.io import import_file
from ovito.modifiers import CreateBondsModifier, ColorCodingModifier
from scipy.special import erf

db0 = read("../db2.xyz", index=":")

db = []

for k in range(0, len(db0)):
atoms = db0[k]
medoid = atoms.get_array("clmds_medoid")
del atoms.arrays["clmds_medoid"]
for i in range(0, len(atoms)):
if medoid[i]:
this_medoid = np.full(len(atoms), False)
this_medoid[i] = True
atoms2 = atoms.copy()
atoms2.set_array("clmds_medoid", this_medoid)
db.append(atoms2)

for k in range(0, len(db)):

atoms = db[k].copy()

# atoms.center(vacuum=0.)
# atoms.pbc=False

medoid = atoms.get_array("clmds_medoid")
surface = atoms.get_array("surface")
slice = False
transparency = np.zeros(len(atoms))
cm = atoms.get_center_of_mass()
atoms.positions -= cm
for i in range(0, len(atoms)):
if medoid[i] and not surface[i]:
v = atoms[i].position
for j in range(0, len(atoms)):
u = atoms[j].position
if np.dot(v,u) > np.dot(v,v) and j != i:
transparency[j] = 0.9
direction = atoms.positions[i]
dist = 30.
elif medoid[i]:
direction = atoms.positions[i]
dist = 25.

try:
del atoms.arrays["fix_atoms"]
except:
pass
try:
del atoms.arrays["clmds_medoid"]
except:
pass
try:
del atoms.arrays["medoid"]
except:
pass
try:
del atoms.arrays["surface"]
except:
pass
try:
del atoms.arrays["clmds_cluster"]
except:
pass

atoms_data = ase_to_ovito(atoms)
pipeline = Pipeline(source = StaticSource(data = atoms_data))
pipeline.add_to_scene()

n_Pt = atoms.symbols.count("Pt")
n_Au = atoms.symbols.count("Au")
n_H = atoms.symbols.count("H")

types = pipeline.source.data.particles.particle_types_

radius = {"Au": 1.8, "Pt": 1.8, "H": 0.5}
for n in range(1,4):
try:
el = types.type_by_id_(n).name
types.type_by_id_(n).radius = radius[el]
except:
pass

colors = np.zeros([len(atoms),3])
for i in range(0, len(atoms)):
if medoid[i]:
k1 = 0.7; k2 = 0.3
else:
k1 = 1.; k2 = 0.
if atoms[i].symbol == "Pt":
colors[i] = k1*np.array([0.8,0.8,0.8]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "Au":
colors[i] = k1*np.array([1,1,0]) + k2*np.array([1,0,0])
elif atoms[i].symbol == "H":
colors[i] = k1*np.array([1,1,1]) + k2*np.array([1,0,0])

pipeline.source.data.particles_.create_property("Color", data=colors)
pipeline.source.data.particles_.create_property("Transparency", data=transparency)

tachyon = TachyonRenderer(shadows=False, direct_light_intensity=1.1)

pipeline.source.data.cell_.vis.render_cell = False

vp = Viewport()
vp.type = Viewport.Type.Perspective
# vp.camera_dir = (2, 3, -1)
# vp.zoom_all()
# direction = (2, 3, -1)
vp.camera_dir = -direction
vp.camera_pos = atoms.get_center_of_mass() + direction + direction / np.dot(direction,direction)**0.5 * dist
vp.render_image(filename = "sites_png/%i.png" % (k+1), size=(120,120), alpha=False, renderer=tachyon)
pipeline.remove_from_scene()

# Print progress
# sys.stdout.write('\rProgress:%6.1f%%' % ((k + 1)*100./len(db)) )
# sys.stdout.flush()
</syntaxhighlight>

<syntaxhighlight lang="bash" line="line">
set term pngcairo size 1200,480

set output "mds_sites.png"

set multiplot layout 1,3

set size ratio -1
set format x ""
set format y ""

unset tics

set key right box opaque

set title "Atomic sites according to cluster"
plot "../mds_sites.dat" u 1:2:3 lc var pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 7 ps 3 lc "green" t "Medoids", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2:(sprintf("%i", $3+1)) w labels not

set title "Sites according to composition and surface/bulk"
plot "../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Pt (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Pt" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Pt (bulk)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "True" ? $2 : 1/0) pt 7 t "Au (surface)", \
"../mds_sites.dat" u 1:(stringcolumn(7) eq "Au" && stringcolumn(5) eq "False" ? $2 : 1/0) pt 7 t "Au (bulk)", \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"

set tics

set title "Local site energy"
set cblabel "Local GAP energy (eV/atom)"
plot "../mds_sites.dat" u 1:2:6 palette pt 7 not, \
"" u (stringcolumn(4) eq "True" ? $1 : 1/0):2 pt 6 ps 1.4 lw 4 lc "green" t "Medoids"
</syntaxhighlight>

Energetic and structural analysis of a database of PtAu nanoclusters

2023-08-11T04:37:33Z

Miguel Caro: