Forces in CPMD

The forces needed in a CPMD calculation are the partial derivative of the Kohn-Sham energy with respect to the independent variables, i.e. the nuclear positions and the Kohn-Sham orbitals. The orbital forces are calculated as the action of the Kohn-Sham Hamiltonian on the orbitals

$$F(\Phi_i) = - f_i H^{\rm KS} \mid \phi_i \rangle \enspace .$$ (46)

The forces with respect to the nuclear positions are

$$F({\bf R}_I) = - {\partial E^{\rm KS} \over \partial {\bf R}_I } \enspace .$$ (47)

These are the same forces as in BOMD, but there we derived the forces under the condition that the wavefunctions were optimized and therefore they are only correct up to the accuracy achieved in the wavefunction optimization. In CPMD these are the correct forces and calculated from analytic energy expressions are correct to machine precision.

Constraint forces are

$$F_c(\Phi_i) = \sum_j \Lambda_{ij} \mid \Phi_j \rangle$$ (48)

$$F_c({\bf R}_I) = \sum_{ij} \Lambda_{ij} {\partial \over \partial {\bf R}_I } \langle \Phi_i \mid \Phi_j \rangle \enspace ,$$ (49)

where the second force only appears for basis sets (or metrics) with a nuclear position dependent overlap of wavefunctions.