Density functional theory is generalized to incorporate electron-phonon coupling. A Kohn-Sham equation yielding the electronic density $n_U\left(\mathbf{r}\right)$, a conditional probability density depending parametrically on the phonon normal mode amplitudes $U=\left\{U_{\mathbf{q}\lambda }\right\}$, is coupled to the nuclear Schrödinger equation of the exact factorization method. The phonon modes are defined from the harmonic expansion of the nuclear Schrödinger 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 nonadiabatic exchange-correlation energy reproduces the leading-order nonadiabatic electron-phonon-induced band structure renormalization in the Fröhlich model.

• Received 4 February 2019

DOI:https://doi.org/10.1103/PhysRevB.99.165136