Mapping approach to the semiclassical description of nonadiabatic quantum dynamics

A theoretical formulation is outlined that allows us to extend the semiclassical Van Vleck–Gutzwiller formulation to the description of nonadiabatic quantum dynamics on coupled potential-energy surfaces. In this formulation the problem of a classical treatment of discrete quantum degrees of freedom (DoF) such as electronic states is bypassed by transforming the discrete quantum variables to continuous variables. The mapping approach thus consists of two steps: an exact quantum-mechanical transformation of discrete onto continuous DoF (the “mapping”) and a standard semiclassical treatment of the resulting dynamical problem. Extending previous work [G. Stock and M. Thoss, Phys. Rev. Lett. 78, 578 (1997)], various possibilities for obtaining a mapping from discrete to continuous DoF are investigated, in particular the Holstein-Primakoff transformation, Schwinger’s theory of angular momentum [in Quantum Theory of Angular Momentum, edited by L. C. Biedenharn and H. V. Dam (Academic, New York, 1965)], and the spin-coherent-state representation. Although all these representations are exact on the quantum-mechanical level, the accuracy of their semiclassical evaluation is shown to differ considerably. In particular, it is shown that a generalization of Schwinger’s theory appears to be the only transformation that provides an exact description of a general N-level system within a standard semiclassical evaluation. Exploiting the connection between spin-coherent states and Schwinger’s representation for a two-level system, furthermore, a semiclassical initial-value representation of the corresponding spin-coherent-state propagator is derived. Although this propagator represents an approximation, its appealing numerical features make it a promising candidate for the semiclassical description of large molecular systems with many DoF. Adopting various spin-boson-type models (i.e., a two-level system coupled to a single or many DoF), computational studies are presented for Schwinger’s and the spin-coherent-state representation, respectively. The performance of the semiclassical approximation in the case of regular and chaotic classical dynamics as well as for multimode electronic relaxation dynamics is discussed in some detail.