# Exact factorization-based density functional theory of electron-phonon systems

## TL;DR

This paper extends density functional theory to include electron-phonon interactions using an exact factorization approach, revealing the importance of Berry curvature and providing a functional approximation that captures nonadiabatic effects on band structures.

## Contribution

It introduces a novel exact factorization-based DFT framework for electron-phonon systems, incorporating Berry curvature effects and developing an orbital-dependent functional for nonadiabatic interactions.

## Key findings

- Berry curvature influences phonon modes beyond potential energy surfaces.
- The functional reproduces nonadiabatic band structure renormalization.
- The approach generalizes DFT to more accurately model electron-phonon coupling.

## Abstract

Density functional theory is generalized to incorporate electron-phonon coupling. A Kohn-Sham equation yielding the electronic density $n_U(\mathbf{r})$, a conditional probability density depending parametrically on the phonon normal mode amplitudes $U=\{U_{\mathbf{q}\lambda}\}$, is coupled to the nuclear Schr\"odinger equation of the exact factorization method. The phonon modes are defined from the harmonic expansion of the nuclear Schr\"odinger equation. A nonzero Berry curvature on nuclear configuration space affects the phonon modes, showing that the potential energy surface alone is generally not sufficient to define the phonons. An orbital-dependent functional approximation for the non\-adiabatic exchange-correlation energy reproduces the leading-order nonadiabatic electron-phonon-induced band structure renormalization in the Fr\"ohlich model.