Weighted nodal domain averages of eigenstates for quantum Monte Carlo and beyond

TL;DR

This paper introduces weighted nodal domain averages as a novel tool to analyze and improve the nodal surfaces of many-body eigenstates, enhancing the accuracy of quantum Monte Carlo methods.

Contribution

It proposes a new approach using weighted nodal domain averages to better understand and potentially improve the nodal surfaces in quantum Monte Carlo calculations.

Findings

Weighted nodal domain averages relate eigenvalue differences to nodal hypersurface integrals.

Application to Be atom demonstrates the method's effectiveness.

Variational formulations suggest pathways to improve nodal accuracy.

Abstract

We study the nodal properties of many-body eigenstates of stationary Schr\"odinger equation that affect the accuracy of real-space quantum Monte Carlo calculations. In particular, we introduce weighted nodal domain averages that provide a new probe of nodal surfaces beyond the usual expectations. Particular choices for the weight function reveal, for example, that the difference between two arbitrary fermionic eigenvalues is given by the nodal hypersurface integrals normalized by overlaps with the bosonic ground state of the given Hamiltonian. Noninteracting and fully interacting Be atom with corresponding almost exact and approximate wave functions are used to illustrate several aspects of these concepts. Variational formulations that employ different weights are proposed for prospective improvement of nodes in variational and fixed-node diffusion Monte Carlo calculations.