In search of the electron-phonon contribution to total energy

TL;DR

This paper provides an exact formulation for the total energy in solids, clarifies the electron-phonon contribution, and demonstrates its small but non-negligible impact in diamond structures.

Contribution

It introduces a precise formulation for total energy beyond Born-Oppenheimer approximation and identifies the electron-phonon contribution at fourth order.

Findings

Electron-phonon contribution to total energy is small (~3.8 meV per atom) but significant.

Clarifies that zero-point renormalization is not the same as electron-phonon energy contribution.

Provides an implementation and size consistency check for the lowest-order electron-phonon energy calculation.

Abstract

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 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 $M_0^{-1/4}$, where $M_0$ is a typical nuclear mass, up to sixth order. Some of these might be grouped and denoted the electron-phonon contribution to total energy, $E^{\textrm{elph}}$, 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 $E^{\textrm{elph}}$ 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 the quasi-harmonic one. We confirm the size consistency of all computed terms.