Coherent state based solutions of the time-dependent Schrodinger equation: hierarchy of approximations to the variational principle

In this review, we give a comprehensive comparison of the most widely used coherent state (CS) based methods to solve the time-dependent Schrodinger equation (TDSE). Starting from the fully variational coherent states (VCS) method, after a first approximation, the coupled coherent states (CCS) method can be derived, whereas an additional approximation leads to the semiclassical Herman-Kluk (HK) method. We numerically compare the different methods with another one, based on a static rectangular grid of coherent states (SCS), by applying all of them to the revival dynamics in a 1D Morse oscillator, with a special focus on the number of basis states (for the CCS and HK methods the number of classical trajectories) needed for convergence and the related issue of tight frames, which in principle allow the usage of CSs as if they were orthogonal. Different discretisation strategies for the occurring phase space integrals for systems with more degrees of freedom are also discussed and the apoptosis procedure that allows to circumvent the linear dependency problem in the VCS method is reviewed. The Holstein molecular crystal model serves to further illustrate the latter point.

Automated summary

This review comprehensively compares CS methods to solve the TDSE, including VCS, CCS, and HK methods, by testing on a 1D Morse oscillator for revival dynamics, focusing on the number of basis states needed for convergence and the issue of tight frames. Different discretisation strategies for phase space integrals and the apoptosis procedure in the VCS method are also discussed, with the Holstein molecular crystal model serving as an illustration.