Born-Oppenheimer Molecular Dynamics

The interaction energy $U({\bf R}^N)$ in the molecular dynamics method has the same physical meaning as the Kohn-Sham energy within the Born-Oppenheimer (BO) approximation. The Kohn-Sham energy depends only on the nuclear positions and defines the hypersurface for the movement of the nuclei. The Lagrangian for BO dynamics is therefore

$${\cal L}_{\rm BO}({\bf R}^N,{\bf\dot R}^N) = \sum_{I=1}^N {1 \ov... ...R}_I^2} - \min_{\{ \phi_i \} } E^{\rm KS} [\{ \phi_i \}; {\bf R}^N] \enspace ,$$ (30)

and the minimization is constraint to orthogonal sets of $\{ \phi_i \}$ . The equations of motions are

$$M_I {\bf\ddot R}_I = -\nabla_I \left[ \min_{\{ \phi_i \} } E^{\rm KS} [\{ \phi_i \}; {\bf R}^N] \right] \enspace .$$ (31)

The BOMD program in pseudocode is simply derived from the previous version.

       V(:) := V(:) + dt/(2*M(:))*F(:)
       R(:) := R(:) + dt*V(:)
       Optimize Kohn-Sham Orbitals (EKS)
       Calculate forces F(:) = dEKS/dR(:)
       V(:) := V(:) + dt/(2*M(:))*F(:)

Extensions to other ensembles along the ideas outlined in the last section are straight forward. In fact, a classical molecular dynamics program can easily be turned into a BOMD program by replacing the energy and force routines by the corresponding routines from a quantum chemistry program.