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 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.

QWalk: A quantum Monte Carlo program for electronic structure

“…We have developed a new program QWalk for general purpose QMC calculations written in C++ with modern programming techniques and incorporating state of the art algorithms in a fast and flexible code. QWalk has already been used in several publications [3,4,5,6,7,8], and we would like to present a summary of its current capabilities.…”

Nodal surfaces and interdimensional degeneracies

“…For the ground state, the number of nodal cells is minimal and Mitas 9 has shown that these systems have the minimal number of two nodal pockets. As explained by Bajdich et al 5 , any distortion of the node from the great circle leads to additional cells which can only increase energy by imposing higher curvature (kinetic energy) on the wave function. This argument has been used by Feynman to demonstrate that the energy of fermionic ground state is always higher than the energy of the bosonic state, and by Ceperley 1 to demonstrate the tiling property of the nodal surface.…”

“…For atoms, it includes two triplet -3 S e (1s2s) and 3 P e (2p 2 ) -and two singlet -1 S e (2s 2 ) and 1 P e (2p 2 ) -states of the helium atom and the three-electron atomic state 4 S(2p 3 ). 4,5,12 The quality of fermion nodes is of prime importance in quantum Monte Carlo (QMC) calculations due to the fermion sign problem, which continues to preclude the application of in principle exact QMC methods to large systems. The dependence of the diffusion Monte Carlo a) Electronic mail: Corresponding author: pf.loos@anu.edu.au b) Electronic mail: dario.bressanini@uninsubria.it (DMC) energy on the quality of the trial wave function is often significant in practice, and is due to the fixed-node approximation which segregates the walkers in regions defined by the trial or guiding wave function.…”

“…Except in some particular cases, electronic or more generally fermionic nodes are poorly understood due to their high dimensionality and complex topology. 1,5 The number of systems for which the exact nodes are known analytically is very limited. For atoms, it includes two triplet -3 S e (1s2s) and 3 P e (2p 2 ) -and two singlet -1 S e (2s 2 ) and 1 P e (2p 2 ) -states of the helium atom and the three-electron atomic state 4 S(2p 3 ).…”

“…In recent years, strong evidence has been gathered showing that, for the lowest state of any given symmetry, there is a single nodal hypersurface that divides configuration space into only two nodal domains (one positive and one negative). [1][2][3][4][5][6][7][8][9][10][11] In other words, to have any chance to have exact nodes, a wave function must have only two nodal cells. For simplicity, in the remainder of this paper, we will say that a wave function has exact nodes if it has the same nodes as the exact wave function.…”