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Forces in BOMD

The forces needed in an implementation of BOMD are

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

They can be calculated from the extended energy functional

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

to be
$\displaystyle { d {\cal E}^{\rm KS} \over d {\bf R}_I }$ $\displaystyle =$ $\displaystyle {\partial E^{\rm KS} \over \partial {\bf R}_I } +
\sum_{ij} \Lambda_{ij} { \partial \over \partial {\bf R}_I }
\langle \phi_i \mid \phi_j \rangle$  
    $\displaystyle + \sum_i \left[ {\partial E^{\rm KS} \over \partial \langle \phi_...
...gle \right]
{\partial \langle \phi_i \mid \over \partial {\bf R}_I } \enspace .$ (34)

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

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


next up previous contents index
Next: Car-Parrinello Molecular Dynamics Up: Born-Oppenheimer Molecular Dynamics Previous: Born-Oppenheimer Molecular Dynamics   Contents   Index
Costas Bekas 2008-07-04