A comparison of local and global single Gaussian approximations to time dynamics: One‐dimensional systems

The detailed calculation of the dynamics of a chemical system is usually not considered due to the size and cost of the computation. It is thus useful to examine various approximation methods. Such methods first need to be tried out on simple systems, like one‐dimensional motion. Here two approaches to approximating the solutions of the Schrodinger and von Neumann equations by single time‐dependent Gaussians are explored and contrasted, explicitly for one‐dimensional barrier penetration. The first approach, in which no tunneling occurs, is local in nature and characterized by an expansion of the equations of motion to second order about the average position of the Gaussian wave packet or about the average position and momentum of the Gaussian Wigner function. This approach was first introduced by Heller [E. J. Heller, J. Chem. Phys. 62, 1544 (1975)]. Here both Heller’s approach and a more general truncation method are considered. Indeed tunneling can be incorporated if second‐order terms in the quantal vo...