Approximate and exact nodes of fermionic wavefunctions: coordinate transformations and topologies

TL;DR

This paper investigates fermion nodes in spin-polarized states of few-electron systems, deriving exact and approximate nodes, analyzing their topologies, and proposing coordinate transformations to improve quantum Monte Carlo methods.

Contribution

It introduces new coordinate maps and symmetries to find exact fermion nodes and extends Feynman-Cohen backflow coordinates for better wavefunction descriptions.

Findings

Exact nodes for two-electron atomic and molecular states identified.

First exact node for three-electron atomic system in $^4S(p^3)$ state derived.

Proposed coordinate transformations simplify nodal structures and enhance variational wavefunctions.

Abstract

A study of fermion nodes for spin-polarized states of a few-electron ions and molecules with $s,p,d$ one-particle orbitals is presented. 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 $^4S(p^3)$ state using appropriate coordinate maps and wavefunction 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 node manifolds into 3D space. The node topologies and other properties are studied using these projections. We also propose a general coordinate transformation as an extension of Feynman-Cohen backflow coordinates to both simplify the nodal description and as a new variational freedom for quantum Monte Carlo trial wavefunctions.