Trajectory-dependent cellularized frozen Gaussians, a new approach for semiclassical dynamics: Theory and application to He–naphtalene eigenvalues

A semiclassical cellular method is proposed. Signals generated by semiclassical techniques generally deteriorate over time as trajectories become chaotic. One approach to remedy this problem has been to have each trajectory weighted by an entire cell of nearby trajectories (Filinov transform). But even in this approach the exponential part of the propagator typically becomes large and positive over time. Here the cellularization (Filinov) parameter is subject to constraints which make it time dependent and trajectory dependent. It also depends on dimensionality, so it ends up as a matrix. Physically, the Filinov transform is done differently in different directions associated with the stability matrix for the phase—essentially a more confined integration in directions where the matrix diverges and a wider integration in other directions. This squelches the contribution from any part of a trajectory that becomes excessively chaotic. A trajectory-dependent cellurized frozen Gaussian is applied here within the Herman–Kluk semiclassical approach. It is tested by looking at a single-particle three-dimensional problem, He attached to a rigid immovable naphtalene, where it is shown to be more accurate than the original HK approach, without the divergence of the correlation function common in the usual cellular dynamics (HK) formulation, and is able to separate a low-lying excited state from the ground state.