First-Principles Approach to Electron-Vibration Interaction in Molecules from an Atomic Orbital Basis: The Allen–Heine–Cardona Theory and Beyond

approximation to the electronic self-energy to incorporate the many-body electronic effects. The agreement between computed vibronic spectral functions and experimental photoemission spectroscopy supports the proposed methodology. This work applies and implements first-principles electron-vibration renormalization in molecules while offering insights into understanding the role of many-body effects in phonon-induced band renormalization for periodic systems.