Pfaffian Pairing Wave Functions in Electronic-Structure Quantum Monte Carlo Simulations

We investigate the accuracy of trial wave functions for quantum Monte Carlo based on Pfaffian functional form with singlet and triplet pairing. Using a set of first row atoms and molecules we find that these wave functions 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 multi-Pfaffian pairing wave functions. We show that a small number of Pfaffians recovers another large fraction of the missing correlation energy comparable to the larger-scale configuration interaction wave functions. We also find that Pfaffians lead to substantial improvements in fermion nodes when compared to Hartree-Fock wave functions.

Figure 1

Correlation energies obtained by QMC methods with the different trial wave functions: VMC and fixed-node DMC with HF nodes and STU Pfaffian nodes (PF). The lower plot shows the fixed-node DMC correlation energy gains over HF nodes for BCS and STU Pfaffian wave functions. The statistical error bars are of the symbol sizes or smaller. Except for the Be atom all the calculations used the same pseudopotentials [15].

Figure 2

A 3D cut through the fermion node hypersurface of an oxygen atom obtained by scanning the wave function with a pair of spin-up and spin-down of electrons, both sitting at the scanning point. The remaining electrons are fixed at a given VMC snaphsot position (small green spheres). The nucleus is depicted in the center of the cube by the black sphere. The three colors show nodes of: Hartree-Fock (red/dark gray), STU Pfaffian nodes (orange/medium gray), and the nodes of the CI wave function (yellow/light gray). The CI node is very close to the exact one (see text). The HF node clearly divides the space into four nodal cells while Pfaffian and CI wave functions partitioning leads to the minimal number of two nodal cells.