Theoretical issues in the accurate computation of the electron-phonon interaction contribution to the total energy

TL;DR

This paper investigates the computation of electron-phonon interactions in solids, revealing fundamental issues with current perturbation theory approaches that question their validity for total energy calculations.

Contribution

It demonstrates that the standard Hamiltonian's electron-phonon energy contributions are size-dependent, challenging the validity of second-order perturbation theory for entire crystals.

Findings

Per-atom EPI energy depends on unit-cell size.

Energy differences between polytypes are meaningful, not absolute energies.

Supports Fan's 1951 suggestion about perturbation theory's limitations.

Abstract

We report the computation of the Standard Hamiltonian of a coupled electron-phonon system by accurately computing the electron-phonon interaction (EPI) contribution to the total energy. This gives the most accurate ab initio total energy till date. However, our results show that the per-atom EPI energy is unit-cell-size dependent due to the partial-Fan-Migdal term that arises from the antisymmetric nature of the crystal wavefunction. Due to this, only energy differences between polytypes, in supercells with identical number of atoms, are meaningful, rather than per-atom total energy. This violates our understanding of Quantum Mechanics applied to periodic solids and raises serious theoretical questions. In his original (1951) paper, Fan suggested, without specifying any reason, that second-order perturbation theory applied to the whole crystal is of questionable validity. Our results support Fan's suggestion for the reason that the partial-FM term makes the per-atom total energy unit-cell-size dependent. This leads to a new fundamental problem in condensed matter physics, viz. second-order perturbation theory is invalid for whole crystals, an entire class. It is essential to resolve this problem, especially because it causes the standard Hamiltonian, the starting point of EPI studies and the most accurate ab initio total energy, to be of questionable validity.