First-principles theory of nonlinear long-range electron-phonon interaction

Describing electron-phonon interactions in a solid requires knowledge of the electron-phonon matrix elements in the Hamiltonian. State-of-the-art first-principles calculations for the electron-phonon interaction are limited to the one-electron-one-phonon matrix element, which is suitable for harmonic materials. However, there is no first-principles theory for one-electron-two-phonon interactions, which occur in anharmonic materials with significant electron-phonon interaction such as halide perovskites and quantum paraelectrics. Here we derive an analytical expression for the long-range part of the one-electron-two-phonon matrix element, written in terms of microscopic quantities that can be calculated from first principles. We show that the long-range one-electron-two-phonon interaction is described by the derivative of the phonon dynamical matrix with respect to an external electric field. We calculate the quasiparticle energy of a large polaron including one-electron-two-phonon interaction and show that it can be written in terms of a one-electron-two-phonon spectral function ${\mathcal{T}}_{\ensuremath{\alpha}\ensuremath{\beta}}(\ensuremath{\omega})$. We demonstrate how to calculate this spectral function and its temperature dependence for the benchmark materials LiF and ${\mathrm{KTaO}}_{3}$, where it turns out that the effect is very small. The first-principles framework developed in this article is general, paving the way for future calculations of one-electron-two-phonon interactions in materials where the effect may be larger.