16. Wave Operators and Active Subspaces: Tools for the Simplified Dynamical Description of Quantum Processes Involving Many-Dimensional State Spaces

Time-dependent and stationary wave operators are presented as tools to deene active spaces and simpliied dynamics for the integration of the time-dependent Schrr odinger equation in large quantum spaces. Within this framework a new light is thrown on the duality between time-dependent and time-independent approaches and a generalized version of the adiabatic theorem is given. For the Floquet treatment of photodissociation processes, the choice of the relevant subspaces and the construction of the eeective Hamiltonians are carried out using the Bloch wave operator techniques. Iterative solutions of the basic equations associated with these wave operators are given, based on Jacobi, Gauss-Seidel and variational schemes. 1. Molecular collisions: their computational representation and analysis .. 1.1. Statement of the problem. A basic numerical problem for chemistry and molecular physics is the study of collisions. A general collision process leads from some entrance (initial) continuum to some exit ((nal) continuum. In the case of a full collision, energy exchanges appear between molecular bound states and molecular continua, leading to products characterized by new energy distributions, or by new chemical structures in the case of reactions and dissociations. Half-collision processes are those in which one continuum (entrance or exit) is a photon continuum and the other one is an ionization or dissociation continuum. Typical half-collision processes are photoionization and photodissociation and their reverse processes, radiative re-combination and radiative association. In most cases these half-collision processes can be described by a time-dependent Schrr odinger equation 1]. (1) where r represents one or many dissociative coordinates, and q a group of bound coordinates. The Hamiltonian H is in part a diierential operator with respect to q and r. It depends explicitly on the time when some of the molecular coordinates or the electromagnetic eld are considered as classical variables 2, 3]. Many approaches involving the theory of wave packet dynamics have been proposed during the last ten years to treat equation (1) 4, 5, 6]. These treatments discretize the continua by working with a bounded range of radial coordinates in conjunction with nite basis sets of square integrable functions. They are well suited for numerical integration methods; the construction of both the partial propagator U(t + dt; t) and the global propagator U(t; t 0) involves the repeated formation of the matrix-vector product H as the principal operation in the description of the time evolution process. (2) Traditional methods of calculating the partial propagator include the …