Difference between revisions of "Forces"
Miguel Caro (talk | contribs) (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...") |
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| − | 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 | + | 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: |
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| − | and the local energies in GAP generally depend on the ''entire'' atomic environment. Imagine | + | 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. |
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| + | Currently, '''TurboGAP''' reports forces in the [[trajectory_out.xyz]] file, as an array property following ASE's [[extended XYZ]] format. | ||
Latest revision as of 11:58, 1 August 2021
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]\displaystyle{ 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 of radius [math]\displaystyle{ r_\text{cut} }[/math]; it depends on the atomic environment within a sphere of radius [math]\displaystyle{ 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]\displaystyle{ \epsilon_j }[/math] like so:
[math]\displaystyle{ 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 three atoms on a straight line, i, j and l, where j is in the middle and i and l are separated by [math]\displaystyle{ 2 r_\text{cut} }[/math]. [math]\displaystyle{ \epsilon_i }[/math] does not depend on l, since l is outside of i's cutoff sphere. However, the derivative [math]\displaystyle{ \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 trajectory_out.xyz file, as an array property following ASE's extended XYZ format.
