Generalized Pairing Wave Functions and Nodal Properties for Electronic Structure Quantum Monte Carlo

Abstract

The quantum Monte Carlo (QMC) is one of the most promising many-body electronic structure approaches. It employs stochastic techniques for solving the stationary Schr" odinger equation and for evaluation of expectation values. The key advantage of QMC is its capability to use the explicitly correlated wave functions, which allow the study of many-body effects beyond the reach of mean-field methods. The most important limit on QMC accuracy is the fixed-node approximation, which comes from necessity to circumvent the fermion sign problem. The size of resulting fixed-node errors depends on the quality of the nodes (the subset of position space where the wave function vanishes) of a used wave function. In this dissertation, we analyze the nodal properties of the existing fermionic wave functions and offer new types of variational wave functions with improved nodal structure.

In the first part of this dissertation, we study the fermion nodes for spin-polarized states of a few-electron ions and molecules with $s$, $p$, $d$ and $f$ one-particle orbitals. We find exact nodes for some cases of two electron atomic and molecular states and also the first exact node for the three-electron atomic system in $ˆ4S(pˆ3)$ state using appropriate coordinate maps and wave function symmetries. We analyze the cases of nodes for larger number of electrons in the Hartree-Fock approximation and for some cases we find transformations for projecting the high-dimensional nodal manifolds into 3D space. The nodal topologies and other properties are studied using these projections. Finally, for two specific cases of spin-unpolarized states, we show how correlations reduce the nodal structure to only two maximal nodal cells.

In the second part, we investigate several types of trial wave functions with pairing orbitals and their nodal properties in the fixed-node quantum Monte Carlo. Using a set of first row atoms and molecules we find that the wave functions in the form of single Pfaffian provide very consistent and systematic behavior in recovering the correlation energies on the level of 95%. In order to get beyond this limit we explore the possibilities of expanding the wave function in linear combinations of Pfaffians.

We observe that molecular systems require much larger expansions than atomic systems and that the linear combinations of a few Pfaffians lead to rather small gains in correlation energy. Further, we test the wave function based on fully-antisymmetrized product of independent pair orbitals. Despite its seemingly large variational potential, we do not observe significant gains in correlation energy. Finally, we combine these developments with the recently proposed inhomogeneous backflow transformations.