Low-Temperature Quantum Fokker–Planck and Smoluchowski Equations and Their Extension to Multistate Systems

Simulating
electron–nucleus coupled dynamics poses a nontrivial
challenge and an important problem in the investigation of ultrafast
processes involving coupled electronic and vibrational dynamics. Because
irreversibility of the system dynamics results from thermal activation
and dissipation caused by the environment, in dynamical studies, it
is necessary to include heat bath degrees of freedom in the total
system. When the system dynamics involves high-energy electronic transitions,
the environment is regarded to be in a low-temperature regime and
we must treat it quantum mechanically. In this Article, we present
rigorous and versatile approaches for investigating the dynamics of
open systems with coupled electronic and vibrational degrees of freedom
within a fully quantum mechanical framework. These approaches are
based on a quantum Fokker–Planck equation and a quantum Smoluchowski
equation employing a heat bath with an Ohmic spectral density, with
non-Markovian low-temperature correction terms, and extensions of
these equations to the case of multistate systems. The accuracy of
these equations was numerically examined for a single-state Brownian
system, while their applicability was examined for multistate double-well
systems by comparing their results with those of the fewest-switch
surface hopping and Ehrenfest methods with a classical Markovian Langevin
force. Comparison of the transient absorption spectra obtained using
these methods clearly reveals the importance of the quantum low-temperature
correction terms. These equations allow us to treat nonadiabatic dynamics
in an efficient way, while maintaining numerical accuracy. The C++
source codes that we developed, which allow for the treatment of the
phase and coordinate space dynamics with any single-state or multistate
potential forms, are provided as Supporting Information.