Numerical Solutions of the Many-Electron Ground State

We examine two wave function optimization methods for producing the ground state. The first is a novel method for optimizing the location of the nodes of the fermion ground-state using a combination of diffusion Monte Carlo (DMC) and projected gradient descent (PGD). A PGD iteration shifts the parameters of a node-fixing trial function in the opposite direction of the DMC energy gradient, while maintaining the cusp condition for atomic electrons if necessary. The energy gradient required for this is calculated from DMC walker distributions by one of three methods derived from an exact analytical expression. The energy gradient calculation methods are combined with different gradient descent algorithms and a projection operator that maintains the cusp condition of atomic systems. We apply this PGD method to trial functions with randomized variational parameters, for simple atomic systems and the homogeneous electron gas, and the nodes are dramatically improved. For atomic systems, PGD lowered the DMC energy to the same level as nodes optimized by variational Monte Carlo (VMC). This PGD method departs from the standard procedure of optimizing the nodes with a non-DMC scheme such as variational Monte Carlo, Density function theory, or configuration interaction based calculation, which do not directly minimize the DMC energy. The second ground state optimization method that we examine implements imaginary-time time-dependent Density functional theory (it-TDDFT) propagation to periodic systems by modifying the Quantum ESPRESSO (QE) package. This implementation of it-TDDFT propagation converges to the exact energy produced by the standard self consistent field (SCF) method in all but one case, where it converged to a slightly lower value than SCF. This suggests it-TDDFT is a useful alternative for systems where SCF has difficulty reaching the Kohn-Sham ground state.