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

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 $^{4}S({p}^{3})$ 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 node manifolds into three-dimensional 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 wave functions.