In search of the electron-phonon contribution to total energy

The total energy is a fundamental characteristic of solids, molecules, and nanostructures. In most first-principles calculations of the total energy, the nuclear kinetic operator is decoupled from the many-body electronic Hamiltonian and the dynamics of the nuclei is reintroduced afterwards. This two-step procedure introduced by Born and Oppenheimer (BO) is approximate. Energies beyond the electronic and vibrational (or phononic) main contributions might be relevant when small energy differences are important, such as when predicting stable polymorphs or describing magnetic energy landscape. We clarify the different flavors of BO decoupling and give an exact formulation for the total energy in the basis of BO electronic wavefunctions. Then, we list contributions, beyond the main ones, that appear in a perturbative expansion in powers of M₀^-1/4, where M₀ is a typical nuclear mass, up to sixth order. Some of these might be grouped and denoted the electron-phonon contribution to total energy,  E^elph, that first appears at fourth order. The electronic inertial mass contributes at sixth order. We clarify that the sum of the Allen-Heine-Cardona zero-point renormalization of eigenvalues over occupied states is not the electron-phonon contribution to the total energy but a part of the phononic contribution. The computation of the lowest-order E^elph is implemented and shown to be small but non-negligible (3.8 meV per atom) in the case of diamond and its hexagonal polymorph. We also estimate the electronic inertial mass contribution and confirm the size-consistency of all computed terms.