Forces

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 or 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 thee 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 is cutoff sphere. However, the derivative [math]\displaystyle{ \frac{\partial \epsilon_j}{\partial r_i^k} }[/math] depends on l because l is within js cutoff sphere.