Path-branching representation for nonadiabatic electron dynamics in conical intersection

Path-branching representation (or phase-space averaging and natural branching method (PSANB) as its approximation) of nonadiabatic electron wavepacket dynamics is now known to work well for avoided crossings in many dimensional nonadiabatic transitions [Yonehara, T.; Hanasaki, K.; Takatsuka, K. Chem. Rev. 2012, 112, 499]. In this paper we examine feasibility of the path-branching representation in the theoretical studies of conical intersection (CI). The most characteristic feature of CI is the Herzberg-Longuet-Higgins phase (or the Berry phase) arising from the electronic part of the total wave function, and accordingly quantum phases of both electronic and nuclear dynamics should be taken into account in a balanced manner. We first show the PSANB can well capture the essential feature of the phase dynamics of CI. However, the nuclear phases, the wavelength of which is far shorter than that of the electronic phases, make the computation of nonadiabatic transition extremely oscillatory, resulting in very slow convergence with respect to the number of sampling paths. A similar difficulty quite often takes place in theoretical chemical dynamics. To cope with this situation, we devise a simple and tractable approximation in the application of PSANB resting on the fact that a small number of PSANB paths already reproduce accurate nonadiabatic transition probability.