# Forces

$\displaystyle{ F_i^k = - \frac{\partial E}{\partial r_i^k}, }$
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 $\displaystyle{ r_\text{cut} }$; it depends on the atomic environment within a sphere of radius $\displaystyle{ 2 r_\text{cut} }$ (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 $\displaystyle{ \epsilon_j }$ like so:
$\displaystyle{ F_i^k = - \sum_j \frac{\partial \epsilon_j}{\partial r_i^k}, }$
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 $\displaystyle{ 2 r_\text{cut} }$. $\displaystyle{ \epsilon_i }$ does not depend on l, since l is outside of i's cutoff sphere. However, the derivative $\displaystyle{ \partial \epsilon_j / \partial r_i^k }$ depends on l because l is within j's cutoff sphere.