Forces - Revision history

Miguel Caro at 11:58, 1 August 2021

2021-08-01T11:58:55Z

Miguel Caro at 14:11, 25 July 2021

2021-07-25T14:11:42Z

Miguel Caro at 14:08, 25 July 2021

2021-07-25T14:08:53Z

Miguel Caro at 13:42, 25 July 2021

2021-07-25T13:42:00Z

Miguel Caro at 13:41, 25 July 2021

2021-07-25T13:41:33Z

Miguel Caro: Created page with "The forces acting on the atoms (technically, the ''atomic nuclei'') are the negative gradients of the total energy with respect to the atomic positions, i.e., the usual de..."

2021-07-25T13:38:09Z

Created page with "The forces acting on the atoms (technically, the ''atomic nuclei'') are the negative gradients of the total energy with respect to the atomic positions, i.e., the usual de..."

New page

The [[forces]] acting on the atoms (technically, the ''atomic nuclei'') are the negative gradients of the total energy with respect to the atomic positions, i.e., the usual definition:

<math>
F_i^k = - \frac{\partial E}{\partial r_i^k},
</math>

where ''i'' is the atom index and ''k'' is each of the three Cartesian dimensions. Perhaps unintuitively, in [[GAP theory]] the force acting on atom ''i'' does not only depend on the atomic environment of ''i'' within its cutoff sphere or radius <math>r_\text{cut}</math>; it depends on the atomic environment within a sphere of radius <math>2 r_\text{cut}</math> (i.e., the volume of the environment for force calculation is 8 times larger than for local energy calculation). This is because the force will depend on the derivatives of the local energies <math>\epsilon_j</math> like so:

<math>
F_i^k = - \sum_j \frac{\partial \epsilon_j}{\partial r_i^k},
</math>

and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine thee atoms on a straight line, ''i'', ''j'' and ''l'', where ''j'' is in the middle and ''i'' and ''l'' are separated by <math>2 r_\text{cut}</math>. <math>\epsilon_i</math> does not depend on ''l'', since ''l'' is outside of ''i'''s cutoff sphere. However, the derivative <math>\frac{\partial \epsilon_j}{\partial r_i^k}</math> depends on ''l'' because ''l'' is within ''j'''s cutoff sphere.

← Older revision		Revision as of 11:58, 1 August 2021
Line 13:		Line 13:
	and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine three atoms on a straight line, ''i'', ''j'' and ''l'', where ''j'' is in the middle and ''i'' and ''l'' are separated by <math>2 r_\text{cut}</math>. <math>\epsilon_i</math> does not depend on ''l'', since ''l'' is outside of ''i''<nowiki>'</nowiki>s cutoff sphere. However, the derivative <math>\partial \epsilon_j / \partial r_i^k</math> depends on ''l'' because ''l'' is within ''j'''s cutoff sphere.		and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine three atoms on a straight line, ''i'', ''j'' and ''l'', where ''j'' is in the middle and ''i'' and ''l'' are separated by <math>2 r_\text{cut}</math>. <math>\epsilon_i</math> does not depend on ''l'', since ''l'' is outside of ''i''<nowiki>'</nowiki>s cutoff sphere. However, the derivative <math>\partial \epsilon_j / \partial r_i^k</math> depends on ''l'' because ''l'' is within ''j'''s cutoff sphere.

−	Currently, '''TurboGAP''' reports forces ~~in the second to fourth columns of the [[energies.dat]] file. In the future, forces will be reported~~ in the [[trajectory_out.xyz]] file, as an array property following ASE's [[extended XYZ]] format.	+	Currently, '''TurboGAP''' reports forces in the [[trajectory_out.xyz]] file, as an array property following ASE's [[extended XYZ]] format.

← Older revision		Revision as of 14:11, 25 July 2021
Line 12:		Line 12:

	and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine three atoms on a straight line, ''i'', ''j'' and ''l'', where ''j'' is in the middle and ''i'' and ''l'' are separated by <math>2 r_\text{cut}</math>. <math>\epsilon_i</math> does not depend on ''l'', since ''l'' is outside of ''i''<nowiki>'</nowiki>s cutoff sphere. However, the derivative <math>\partial \epsilon_j / \partial r_i^k</math> depends on ''l'' because ''l'' is within ''j'''s cutoff sphere.		and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine three atoms on a straight line, ''i'', ''j'' and ''l'', where ''j'' is in the middle and ''i'' and ''l'' are separated by <math>2 r_\text{cut}</math>. <math>\epsilon_i</math> does not depend on ''l'', since ''l'' is outside of ''i''<nowiki>'</nowiki>s cutoff sphere. However, the derivative <math>\partial \epsilon_j / \partial r_i^k</math> depends on ''l'' because ''l'' is within ''j'''s cutoff sphere.
		+
		+	Currently, '''TurboGAP''' reports forces in the second to fourth columns of the [[energies.dat]] file. In the future, forces will be reported in the [[trajectory_out.xyz]] file, as an array property following ASE's [[extended XYZ]] format.

← Older revision		Revision as of 14:08, 25 July 2021
Line 11:		Line 11:
	</math>		</math>

−	and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine ~~thee~~ atoms on a straight line, ''i'', ''j'' and ''l'', where ''j'' is in the middle and ''i'' and ''l'' are separated by <math>2 r_\text{cut}</math>. <math>\epsilon_i</math> does not depend on ''l'', since ''l'' is outside of ''i''<nowiki>'</nowiki>s cutoff sphere. However, the derivative <math>\partial \epsilon_j / \partial r_i^k</math> depends on ''l'' because ''l'' is within ''j'''s cutoff sphere.	+	and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine three atoms on a straight line, ''i'', ''j'' and ''l'', where ''j'' is in the middle and ''i'' and ''l'' are separated by <math>2 r_\text{cut}</math>. <math>\epsilon_i</math> does not depend on ''l'', since ''l'' is outside of ''i''<nowiki>'</nowiki>s cutoff sphere. However, the derivative <math>\partial \epsilon_j / \partial r_i^k</math> depends on ''l'' because ''l'' is within ''j'''s cutoff sphere.

← Older revision		Revision as of 13:42, 25 July 2021
Line 11:		Line 11:
	</math>		</math>

−	and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine thee atoms on a straight line, ''i'', ''j'' and ''l'', where ''j'' is in the middle and ''i'' and ''l'' are separated by <math>2 r_\text{cut}</math>. <math>\epsilon_i</math> does not depend on ''l'', since ''l'' is outside of ''i''<nowiki>'</nowiki>s cutoff sphere. However, the derivative <math>~~\frac{~~\partial \epsilon_j}{\partial r_i^k}</math> depends on ''l'' because ''l'' is within ''j'''s cutoff sphere.	+	and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine thee atoms on a straight line, ''i'', ''j'' and ''l'', where ''j'' is in the middle and ''i'' and ''l'' are separated by <math>2 r_\text{cut}</math>. <math>\epsilon_i</math> does not depend on ''l'', since ''l'' is outside of ''i''<nowiki>'</nowiki>s cutoff sphere. However, the derivative <math>\partial \epsilon_j / \partial r_i^k</math> depends on ''l'' because ''l'' is within ''j'''s cutoff sphere.

@@ Line 5: / Line 5: @@
 </math>
-where ''i'' is the atom index and ''k'' is each of the three Cartesian dimensions. Perhaps unintuitively, in [[GAP theory]] the force acting on atom ''i'' does not only depend on the atomic environment of ''i'' within its cutoff sphere or radius <math>r_\text{cut}</math>; it depends on the atomic environment within a sphere of radius <math>2 r_\text{cut}</math> (i.e., the volume of the environment for force calculation is 8 times larger than for local energy calculation). This is because the force will depend on the derivatives of the local energies <math>\epsilon_j</math> like so:
+where ''i'' is the atom index and ''k'' is each of the three Cartesian dimensions. Perhaps unintuitively, in [[GAP theory]] the force acting on atom ''i'' does not only depend on the atomic environment of ''i'' within its cutoff sphere of radius <math>r_\text{cut}</math>; it depends on the atomic environment within a sphere of radius <math>2 r_\text{cut}</math> (i.e., the volume of the environment for force calculation is 8 times larger than for local energy calculation). This is because the force will depend on the derivatives of the local energies <math>\epsilon_j</math> like so:
 <math>
@@ Line 11: / Line 11: @@
 </math>
-and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine thee atoms on a straight line, ''i'', ''j'' and ''l'', where ''j'' is in the middle and ''i'' and ''l'' are separated by <math>2 r_\text{cut}</math>. <math>\epsilon_i</math> does not depend on ''l'', since ''l'' is outside of ''i'''s cutoff sphere. However, the derivative <math>\frac{\partial \epsilon_j}{\partial r_i^k}</math> depends on ''l'' because ''l'' is within ''j'''s cutoff sphere.
+and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine thee atoms on a straight line, ''i'', ''j'' and ''l'', where ''j'' is in the middle and ''i'' and ''l'' are separated by <math>2 r_\text{cut}</math>. <math>\epsilon_i</math> does not depend on ''l'', since ''l'' is outside of ''i''<nowiki>'</nowiki>s cutoff sphere. However, the derivative <math>\frac{\partial \epsilon_j}{\partial r_i^k}</math> depends on ''l'' because ''l'' is within ''j'''s cutoff sphere.