Deriving the exact nonadiabatic quantum propagator in the mapping variable representation

TL;DR

This paper derives an exact quantum propagator in the mapping variable representation for nonadiabatic dynamics, providing a foundation for developing accurate classical-like methods to compute quantum observables in complex systems.

Contribution

It introduces an exact quantum propagator expressed as a Moyal series in the mapping variable framework, linking it to existing semiclassical and mixed quantum-classical methods.

Findings

Different truncations recover known approximate methods.

Provides an analytic expression for thermal quantum correlation functions.

Establishes a theoretical basis for accurate classical-like dynamics.

Abstract

We derive an exact quantum propagator for nonadiabatic dynamics in multi-state systems using the mapping variable representation, where classical-like Cartesian variables are used to represent both continuous nuclear degrees of freedom and discrete electronic states. The resulting expression is a Moyal series that, when suitably approximated, can allow for the use of classical dynamics to efficiently model large systems. We demonstrate that different truncations of the exact propagator lead to existing approximate semiclassical and mixed quantum-classical methods and we derive an associated error term for each method. Furthermore, by combining the imaginary-time path-integral representation of the Boltzmann operator with the exact propagator, we obtain an analytic expression for thermal quantum real-time correlation functions. These results provide a rigorous theoretical foundation for the development of accurate and efficient classical-like dynamics to compute observables such as electron transfer reaction rates in complex quantized systems.