First-principles phonon self-energy including one-electron-two-phonon coupling

We derive the phonon self-energy including both one-electron–one-phonon and one-electron–two-phonon interaction vertices, with all couplings obtained from first-principles calculations. The full self-energy matrix, including off-diagonal terms, is computed and evaluated on-shell. We find that the two-phonon contribution grows with temperature and significantly affects the phonon renormalization at high temperatures. Results for LiF and CsPbBr3 demonstrate the quantitative impact of the two-phonon processes on phonon energies and linewidths.