Dynamics and relaxation in interacting systems: Semigroup methods

The dynamical study of chemical systems whose evolution is governed by quantum mechanics can now be computed fairly effectively for small systems in which the evolution is entirely Hamiltonian. When such Hamiltonian systems interact with their environment, however, relaxation and dephasing terms are introduced into the evolution. To include the effect of these terms, several methods are in current use. This paper is devoted to an exposition, analysis, and several simple applications of the semigroup technique for dealing with these non-Hamiltonian evolution terms. We discuss the nature of the semigroup terms, how they arise and how they are applied, and some of their advantages and disadvantages compared to other methods including dissipation. Specific applications to three simple two-site problems are given.