Exact Factorization of the Electron-Nuclear Wavefunction: Fundamentals and Algorithms

This Chapter provides an overview on the exact factorization of the electron-nuclear wavefunction, from the presentation of the fundamental theory to its application in the domain of photochemistry. The exact factorization is presented in relation to the more standard Born-Oppenheimer picture, often employed to interpret and simulate excited-state processes that are at the heart of photochemistry. Numerical studies on a two-mode two-state model system are reported focusing on the photo-excitation process by an ultrashort laser pulse and on the subsequent relaxation dynamics through a conical intersection. The aim here is to analyze the time-dependent potentials of the theory and to introduce the concept of nuclear trajectories in the context of the exact factorization. Various applications in photochemistry are then presented, namely the cis-trans photo-isomerization of 2-cis-penta-2,4-dieniminium cation, the photo-dissociation of IBr with the inclusion of spin-orbit coupling, different examples of proton-coupled electron transfer. Those studies are performed by applying the nonadiabatic coupled-trajectory algorithms derived from the exact factorization.