Phase-space averaging and natural branching of nuclear paths for nonadiabatic electron wavepacket dynamics

Abstract

We propose a simple and tractable method to treat quantum electron wavepacket dynamics that nonadiabatically couples with "classical" nuclear motions in mixed quantum-classical representation. The electron wavepacket is propagated inducing electronic-state mixing along our proposed paths. It has been shown in our previous studies that classical force working on nuclei in a nonadiabatic region is represented in a matrix form (called the force matrix), and the solutions of the Hamilton canonical equations of motion for nuclei based on this force matrix give rise to a cascade of infinitely many branching paths when solved simultaneously with electronic-state mixing. As a tractable approximation to these rigorous solutions, we here devise a method to provide much simpler nonadiabatic paths: (i) extract one or a few number of representative paths by taking an average over the paths in phase space (not averaging over the forces) that should be otherwise undergo the fine branching. (ii) After the nonadiabatic coupling becomes sufficiently small, let these paths naturally branch by running them with their own individual eigenforces (the eigenvalues of the force matrix). Since the eigenforces coincide with the forces of adiabatic potential energy surfaces in the limit of zero nonadiabatic coupling, these branching paths eventually run on one of possible adiabatic potential energy surfaces, converging to a classical path (Born-Oppenheimer path). The paths thus created are theoretically satisfactory in that they realize the coherent mixing of electronic states in the manner of quantum entanglement and yet eventually become consistent with the Born-Oppenheimer classical trajectories. We test the present method numerically with the use of two- and three-state systems that are extracted from ab initio calculations for the excited states of LiH molecule.