Generalized quantum cumulant dynamics

A means of unifying some semiclassical models of computational chemistry is presented; these include quantized Hamiltonian dynamics, quantal cumulant dynamics, and semiclassical Moyal dynamics (SMD). A general method for creating the infinite hierarchy of operator dynamics in the Heisenberg picture is derived together with a general method for truncation (or closure) of that series, and in addition, we provide a simple link to the phase space methods of SMD. Operator equations of arbitrary order may be created readily, avoiding the tedious algebra identified previously. Truncation is based on a simple recurrence formula which is related to, but avoids the more complex contractions of, Wick's theorem. This generalized method is validated against a number of trial problems considered using the previous methods. We also touch on some of the limitations involved using such methods, noting, in particular, that any truncation will lead to a state which is in some sense unphysical. Finally, we briefly introduce our quantum algebra package QuantAL which provides an automated method for the generation of the required equation set, the initial conditions for all variables from any start, and all the higher order approximations necessary for truncation of the series, at essentially arbitrary order.