Accurate computation of the electron-phonon interaction contribution to the total energy

The standard Hamiltonian of a coupled electron-phonon system is based on second-order perturbation theory. The EPI contribution in the standard Hamiltonian consists of two terms, the EPI contribution to the band-structure energy and the partial-Fan-Migdal (FM)-occupied contribution. Within the non-adiabatic approximation, we derive a new expression for the partial-FM-occ contribution and show that it has the structure of a higher-order term, and not a second-order term. Along similar lines, we derive new expressions for the computation of the partial-FM-occ term. The new expressions for the partial-FM-occ term must be preferred over the standard expressions, in theoretical and computational studies, because they incorporate the complete physics underlying this term. Unlike the EPI contribution to individual eigenstates, the EPI contribution to the total energy must be computed in the non-adiabatic approximation for all materials, Infra-red (IR) active and IR-inactive. We report the computation of the standard Hamiltonian, for the first time, for Carbon polymorphs (diamond and hexagonal lonsdaleite) by including the EPI contribution to the total energy. This is the most accurate ab initio total energy reported till date. The present work also opens the way to compute the ab initio free-energy more accurately at finite temperatures by including the EPI contribution.