In search of the electron-phonon contribution to total energy

TL;DR

The authors address whether and how electron–phonon effects contribute to the total energy beyond the standard electronic and phononic terms within the Born–Oppenheimer framework. They formulate an exact total-energy expression in the BO basis and perform a systematic mass-scaling analysis using the small parameter $\lambda = M_0^{-1/4}$ to sixth order, identifying a genuine fourth-order electron–phonon term $E^{\textrm{elph}}$ and clarifying that Allen’s energy is part of the phonon contribution rather than an independent correction. Through implementation in first-principles DFPT/DFT workflows and validation against finite-difference calculations for diamond, they show the elph term is small but non-negligible (a few meV per atom) and that the approach is size-consistent. Application to diamond and lonsdaleite demonstrates modest differences in elph-related energies and a slight reordering of stability when all terms are included, underscoring the practical relevance for precise energy predictions in systems with subtle energy differences. The work provides a principled, implementable route to incorporate the lowest-order electron–phonon contribution into total-energy calculations, improving accuracy for defect formation, phase stability, and surface phenomena.

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 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 $M_0^{-1/4}$, 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 confirm the size-consistency of all computed terms.

Figures (5)

• Figure 1: Example of coordinate diagram for the BO energy hypersurfaces $E^{\rm BO}_j\{{\mathbf{R}}\}$ of the electronic ground state ($j=0$) and first- and second-excited states ($j=1$ and $j=2$). Eigenenergies of the total Hamiltonian are also shown, labelled with the $i$ index. The ground state of the total Hamiltonian has $i=0$ label. Usually, the corresponding wavefunctions will be made predominantly of the BO wavefunctions of one of the BO hypersurfaces, but there will be hybridized states. The chosen reference configuration is denoted by $\{ \mathbf{R}^{\rm ref}\}$. In this example, we present a specific case where the $i=6$ and $i=7$ energy levels are degenerate, being hybrids from the $j=0$ and $j=1$ BO wavefunctions, and where the $j=0$ BO hypersurface has three minima, one global one and two local ones, that might all be chosen to be the reference state $\{ \mathbf{R}_{j=0}^0\}$ in Sec. \ref{['sec:scaling']}. The $j=2$ one has one global minimum.
• Figure 2: Same coordinate diagram as Fig. \ref{['fig:hypersurface']} where we focus on the specific $j=0$ BO hypersurface (black curved line). This BO hypersurface presents here two minima denoted $\{\mathbf{R}_{j=0}^0\}$ and $\{\mathbf{R}_{j=0}^{0'}\}$. The exact total energy $\mathcal{E}_i$ (black horizontal line) of some level $i$ is obtained from a series expansion considering one or the other reference configuration as starting point, in the neighborhood of which the BO hypersurface is expanded to quadratic order (red-dashed parabola), defining an harmonic oscillator Hamiltonian. An eigenstate of this harmonic oscillator Hamiltonian is labeled with the index $\jmath$ (dotless-$j$). The corresponding eigenenergy is added to the reference BO energy giving the red dashed horizontal line. The series expansion (red dash) maps the corresponding black energy level if the series converges.
• Figure 3: Diamond fourth-order total energy contribution computed using DFPT, Eq. \ref{['eq:fourthorder2']}, and with a sum-over-state (SOS) expression, Eq. \ref{['eq:fourthorder3']}. The $\mathbf{k}$-point grid used is 8$\times$8$\times$8 with only a single $\mathbf{q}=\boldsymbol{\Gamma}$ point.
• Figure 4: Diamond and lonsdaleite convergence rates for the second- and electron-phonon fourth-order contributions to the total energy (per 2 atoms) with respect to $\mathbf{q}$-point grid integration. The $\mathbf{k}$-point grid used is 8$\times$8$\times$8 and 6$\times$6$\times$4 for lonsdaleite. In the case of lonsdaleite the reported grid size correspond to the first two numbers, e.g. 24$^3$ means a $\mathbf{q}$ grid of 24$\times$24$\times$16.
• Figure 5: Diamond energy dependence on the lattice parameter $a$ and lonsdaleite energy dependence on the internal parameter $u$. All the energies are presented with their minimal value adjusted to 0. Vertical lines indicates the position of the energy minimum.