The Bloch wave operator: generalizations and applications: II. The time-dependent case

Part II of the review shows how the stationary Bloch wave operator of part I can be suitably modified to give a time-dependent wave operator. This operator makes it possible to use a relatively small active space in order to describe the dynamical processes which occur in quantum mechanical systems which have a time-dependent Hamiltonian. A close study is made of the links between the time-dependent and time-independent wave operators at the adiabatic limit; the analysis clarifies the way in which the wave operator formalism allows the time evolution of a system or a wave packet to be described in terms of a fast evolution inside the active space together. with weak transitions out of this space which can be treated by perturbation methods. Two alternative wave operator equations of motion are derived and analysed. The first one is a non-linear differential equation in the usual Hilbert space; the second one is a differential equation in an extended Hilbert space with an extra time variable added and becomes equivalent to the usual Bloch equation when the Floquet Hamiltonian is taken in place of the ordinary Hamiltonian. A study is made of the close relationships between the time-dependent wave operator formalism, the Floquet theory and the (t, t') theory. Some original methods of solution of the two forms of wave operator equation are proposed and lead to new techniques of integration for the time-dependent Schrodinger equation (e.g., the generalized Green equation procedure). Mixed procedures involving both the time-independent and time-dependent wave operators are shown to be applicable to the internal eigenstate problem for. large complex matrices. A detailed account is given of the description of inelastic and photoreactive processes by means of the time-dependent wave operator formalism, with particular attention to laser-molecule interactions. The emphasis is on projection operator techniques, with special attention being given to the method of selection of the active space and to the use of basis set expansions in terms of instantaneous or generalized Floquet eigenstates. The adiabatic transport principle and its representation in terms of the wave operator approach are described, as well as the way in which effective energy trajectories in the complex plan can act as indicators of the degree of adiabaticity or nonadiabaticity of a dynamical process.