Path integrals for nonadiabatically coupled electrons and nuclei in molecules: Force analysis for branching nuclear paths and conservation laws

Real-time dynamics in electron-nucleus coupled systems in molecules is studied using the path-integral formalism, with a special emphasis on nonadiabatic interactions. We first establish a formal path-integral description of the entire system. Applying the stationary phase approximation, we then derive coupled equations for the mixed quantum-classical treatment of the system: the equations of motion for electron wave-packet dynamics and those for nuclear dynamics driven by what we call the force form. Thus the present theory also serves as a general theory for dynamics in mixed quantum and classical systems. On this theoretical foundation, we analyze two theories of nonadiabatic electron-nucleus coupled systems from the viewpoint of path branching: the semiclassical Ehrenfest theory and the recently developed method of phase-space averaging and natural branching [T. Yonehara, S. Takahashi, and K. Takatsuka, J. Chem. Phys. 130, 214113 (2009)]. We give a unified account of the essential feature of their physical implications and limitations. Path-integral formalism leads to further refinement of the idea of path branching caused by nonadiabatic coupling, thus giving deeper insight into the nonadiabatic dynamics. Further, we study the conservation laws for energy, linear momentum, and angular momentum in the general mixed quantum-classical representation. We also extend the present path-integral formulation so as to handle nonadiabatic dynamics in laser fields.