Total Energy

Molecular dynamics calculations with interaction potentials derived from density functional theory require the evaluation of the total energy and derivatives with respect to the parameters of the Lagrangian.

The total energy can be calculated as a sum of kinetic, external (local and non-local pseudopotential), exchange and correlation, and electrostatic energy

$$E_{\rm total} = E_{\rm kin} + E^{\rm PP}_{\rm local} + E^{\rm PP}_{\rm nonlocal} + E_{\rm xc} + E_{\rm ES} \enspace .$$ (208)

The individual terms are defined by

$$E_{\rm kin} = \sum_{i} \sum_{\bf G} {1 \over 2} f_{i} \vert{\bf G}\vert^2 \vert c_{i}({\bf G})\vert^2$$ (209)

$$E^{\rm PP}_{\rm local} = \sum_{I} \sum_{\bf G} \Delta {V_{\rm local}^{I}({\bf G})} S_{I}({\bf G}) n^\star({\bf G})$$ (210)

$$E^{\rm PP}_{\rm nonlocal} = \sum_{i} f_{i} \sum_{I} \sum_{\alpha, \beta \in {I}} \left(F_{I,i}^{\alpha} \right)^\star h_{\alpha \beta}^{I} F_{I,i}^{\beta}$$ (211)

$$E_{\rm xc} = \Omega \sum_{\bf G} {\epsilon_{\rm xc}({\bf G})} n^\star({\bf G})$$ (212)

$$E_{\rm ES} = 2 \pi \Omega \sum_{{\bf G} \neq 0} {\vert n_{\rm tot}({\bf G})\vert^2 \over G^2} + E_{\rm ovrl} - E_{\rm self} .$$ (213)

The overlap between the projectors of the non-local pseudopotential and the Kohn-Sham orbitals has been introduced in the equation above

$$F_{I,i}^{\alpha} = {1 \over \sqrt{\Omega}} \sum_{\bf G} P_{\alpha}^{I} ({\bf G}) S_{I}({\bf G}) c_{i}^\star({\bf G}) \enspace .$$ (214)