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 nuclear potential, and the dynamics of the nuclei is reintroduced afterward. 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 the 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 as the electron-phonon contribution to total energy, $E^{\mathrm{elph}}$, which 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 rather part of the phononic contribution. The computation of the lowest-order Eelph is implemented and shown to be small but non-negligible in the case of diamond (3.8~meV per atom) and GaN (9.7~meV per atom). We also estimate the electronic inertial mass contribution and the quasi-harmonic one for diamond. We confirm the size consistency of all computed terms.