Simulation of nonlinear optical signals via approximate solutions of the quantum–classical Liouville equation: Application to the pump–probe spectroscopy of a condensed phase electron transfer reaction

The equation-of-motion phase-matching approach provides an efficient way for calculating a system’s time-dependent polarization in any phase-matching direction subject to weak laser fields of arbitrary shape. Within this approach, we introduce the Poisson Bracket Mapping Equation and Forward–Backward methods, two approximate solutions of the quantum–classical Liouville equation, for simulating the laser-induced response of a quantum subsystem in a classical environment. We illustrate this approach by calculating the transient absorption signal in a model photo-induced condensed phase electron transfer reaction. The results are found to be in reasonable agreement with the exact results, which is promising for applications to realistic systems.