Nuclear Gradient

The derivative of the total energy with respect to nuclear positions is needed for structure optimization and in molecular dynamics, that is

$${\partial E_{\rm total} \over \partial {\bf R}_{I,s}} = {\partial... ... {\bf R}_{I,s}} + {\partial E_{\rm ES} \over \partial {\bf R}_{I,s}} \enspace ,$$ (217)

as the kinetic energy $E_{\rm kin}$ and the exchange and correlation energy $E_{\rm xc}$ do not depend directly on the atomic positions, the relevant parts are

$${\partial E^{\rm PP}_{\rm local} \over \partial {\bf R}_{I,s}} = - \Omega \sum_{\bf G} i{\bf G}_{s} \Delta V_{\rm local}^{I}({\bf G}) S_{I}({\bf G}) n^\star({\bf G})$$ (218)

$${\partial E^{\rm PP}_{\rm nonlocal} \over \partial {\bf R}_{I,s}} = \sum_{i} f_{i} \sum_{\alpha, \beta \in {I}} \left\{ \left(F_{I,i}... ...ar \over \partial {\bf R}_{I,s}} h_{\alpha, \beta}^{I} F_{I,i}^{\beta} \right\}$$ (219)

$${\partial E_{\rm ES} \over \partial {\bf R}_{I,s}} = - \Omega \sum_{{\bf G} \neq 0} i {\bf G}_{s} {n^\star_{\rm ... ..._{I}({\bf G}) + {\partial E_{\rm ovrl} \over \partial {\bf R}_{I,s}} \enspace .$$ (220)

The contribution of the projectors of the non-local pseudopotentials is calculated from

$${\partial F_{I,i}^{\alpha} \over \partial {\bf R}_{I,s}} = - {1 \... ..., P_{\alpha}^{I} ({\bf G}) S_{I}({\bf G}) c_{i}^\star({\bf G}) \enspace .$$ (221)

Finally, the real space part contribution of the Ewald sum is

$${\partial E_{\rm ovrl} \over \partial {\bf R}_{I,s}} = \mathop{{\sum}'}_{J} \sum_{\bf L} \left\{ {Z_{I} Z_{J} \over \ver... ... \over \sqrt{{{\rm R}_{I}^{\rm c}}^2+{{\rm R}_{J}^{\rm c}}^2} } \right] \right.$$

$$\left. + {2 \over \sqrt{\pi}} {1 \over \sqrt{{{\rm R}_{I}^{\rm c}... ...\over \sqrt{{{\rm R}_{I}^{\rm c}}^2+{{\rm R}_{J}^{\rm c}}^2} } \right] \right\}$$

$$\times ({\bf R}_{I,s} - {\bf R}_{J,s} - {\bf L}_{s}) \enspace .$$ (222)

The self energy $E_{\rm self}$ is independent of the atomic positions and does not contribute to the forces.