Extended Gaussian wave packet dynamics

Wave packet dynamics has exposed interesting new phenomena in several fields. In femto-chemistry [1,2] we are now able to time-resolve chemical processes and also observe effects such as the breakup and revival of wave packets [3]. In atom optics wave packets are used to model matter waves [4] and electron wave packets are seen in the dynamics of Rydberg atoms [5]. The numerical modelling of wave packet dynamics has been achieved by a number of methods [6,2], but one of the earliest approaches was by Heller [7] who simply used the Ansatz of a time dependent Gaussian wave packet. This Gaussian approach is, usually, an approximation and can be quite wrong, for example at turning points. Several improvements were made: the method of generalised Gaussian wave packets [8] used complex classical trajectories for Gaussian wave packets, and the hybrid method [9] used an expansion in terms of a grid of Gaussian wave packets. The approach taken here is in the spirit of the Heller’s Gaussian wave packets, but we seek to derive corrections to the method in order to obtain the full quantum wave packet dynamics. The aim will be to move to a basis which matches the Gaussian wave packet, and which removes all quadratic operator dependence from the Hamiltonian. The result is a system Hamiltonian with factors depending only on anharmonic contributions from the potential. The method, which is in principle exact, is similar to a time-dependent perturbation theory in a time dependent basis. Indeed, it can be developed as a perturbation theory in the higher order derivatives of the system potential about the classical motion. The idea of a time dependent basis, resulting in ‘extended’ Gaussian wave packet dynamics, was put forward by Coalson and Karplus [10], who wrote down an expansion of the wave packet in a Gauss-Hermite basis. Parameters for the basis were treated in a variational method by Kay [11] and the phase space picture was explored by Møller and Henriksen [12]. The use of a Gauss-Hermite basis has recently been expanded by Billing [13–15] to examine non-adiabatic transitions and corrections to classical path equations. In these papers a complete basis set of states, based on the Heller Gaussian wave packet, is used for an expansion of the system wave function. The approach used here is similar in principle, but the focus is on the system Hamiltonian which is transformed by displacement and squeezing in a way that removes all operator dependence which is quadratic or less. The result is an evolution equation which depends on a time dependent ‘residual potential’ which is based on the original potential with harmonic terms removed. By using the displacement nd squeezing transformations we can find matrix elements of the residual potential that determine the dynamics of the system. In section II of this paper we set up the problem and perform a basis shift according to the classical dynamics. Section III examines Gaussian wave packet dynamics in this displaced basis, and in the original basis by two different approaches. The relationship between the two approaches is established. In section IV we establish the squeezing transformation necessary to map the time evolution of a Gaussian wave packet from its initial state. By using the same transformation for another change of basis we can find the equations for corrections to Gaussian wave packet dynamics. These equations are expressed in a Fock basis in section V, where we also compare our results to the GaussHermite basis. Some examples of useful matrix elements for potentials are given in section VI, and in section VII the results are applied to an example of an exponential potential.