Coupled Kohn-Sham equations for electrons and phonons

TL;DR

This paper develops a theoretical framework for coupled electron-phonon Kohn-Sham equations, incorporating superconductivity, and introduces methods for diagonalization and potential approximation in density functional theory.

Contribution

It establishes the algebraic structure of coupled electron-phonon Kohn-Sham equations, including the superconducting order parameter, and provides practical solution methods and mean-field potential approximations.

Findings

Derived conditions for real bosonic eigenvalues.

Proposed a practical solution method for the coupled system.

Formulated approximate mean-field potentials based on density matrices.

Abstract

This work establishes the algebraic structure of the Kohn-Sham equations to be solved in a density formulation of electron and phonon dynamics, including the superconducting order parameter. A Bogoliubov transform is required to diagonalize both the fermionic and bosonic Kohn-Sham Hamiltonians since they both represent a non-interacting quantum field theory. The Bogoliubov transform for phonons is non-Hermitian in the general case, and the corresponding time-evolution is non-unitary. Several sufficient conditions for ensuring that the bosonic eigenvalues are real are provided and a practical method for solving the system is described. Finally, we produce a set of approximate mean-field potentials which are functionals of the electronic and phononic density matrices and depend on the electron-phonon vertex.