Frozen Gaussian Wavepacket Study of the Ground State of the He Atom

The Rayleigh-Ritz functional is used in conjunction with an approximate time evolution to improve ab initio estimates of ground-state energies. The improvement is due in part to the introduction of a novel variational "normalization function" for the approximate propagator. An additional variational parameter was introduced in the form of a constant shift energy of the Hamiltonian. The approximate propagator used was the frozen Gaussian propagator; however, the trajectories evolved on the coherent-state averaged Hamiltonian (Q representation). For Coulombic forces, this removes the singularity, easing the computation. An additional variational parameter was the width parameter used for the coherent states appearing in the frozen Gaussian propagator. Using an initial combination of nine Gaussian functions for He, with an initial energy of -2.5115 au, the variational method, with a very short time interval of integration, led to an improved energy of -2.81 ± 0.04 au.