Forces in BOMD

The forces needed in an implementation of BOMD are

$${d \over d{\bf R}_I} \left[ \min_{\{ \phi_i \} } E^{\rm KS} [\{ \phi_i \}; {\bf R}^N] \right] \enspace .$$ (32)

They can be calculated from the extended energy functional

$${\cal E}^{\rm KS} = E^{\rm KS} + \sum_{ij} \Lambda_{ij} \left( \langle \phi_i \mid \phi_j \rangle - \delta_{ij} \right)$$ (33)

to be

$${ d {\cal E}^{\rm KS} \over d {\bf R}_I } = {\partial E^{\rm KS} \over \partial {\bf R}_I } + \sum_{ij} \Lambda_{ij} { \partial \over \partial {\bf R}_I } \langle \phi_i \mid \phi_j \rangle$$

$$+ \sum_i \left[ {\partial E^{\rm KS} \over \partial \langle \phi_... ...gle \right] {\partial \langle \phi_i \mid \over \partial {\bf R}_I } \enspace .$$ (34)

The Kohn-Sham orbitals are assumed to be optimized, i.e. the term in brackets is (almost) zero and the forces simplify to

$$F^{\rm KS}({\bf R}_I) = - {\partial E^{\rm KS} \over \partial {\... ...rtial \over \partial {\bf R}_I } \langle \phi_i \mid \phi_j \rangle \enspace .$$ (35)

The accuracy of the forces used in BOMD depends linearly on the accuracy of the minimization (see Fig. 1) of the Kohn-Sham energy. This is an important point we will further investigate when we compare BOMD to the Car-Parrinello method.

Figure 1: Accuracy of nuclear forces for a system of 8 silicon atoms in a cubic unit cell at 10 Ry cutoff using norm-conserving pseudopotentials