Symmetry Constraints and Variational Principles in Diffusion Quantum Monte Carlo Calculations of Excited-State Energies

TL;DR

This paper investigates the use of symmetry constraints in diffusion Monte Carlo for excited states, revealing that the energy bounds depend on the symmetry representation of the trial functions, especially in degenerate cases.

Contribution

It demonstrates how symmetry considerations affect variational bounds in fixed-node DMC for excited states, clarifying when energies can fall below exact eigenvalues.

Findings

DMC energy bounds depend on the symmetry of trial functions.

Degenerate states can lead to DMC energies below the true eigenvalues.

Using subgroup symmetries can provide weaker variational bounds.

Abstract

Fixed-node diffusion Monte Carlo (DMC) is a stochastic algorithm for finding the lowest energy many-fermion wave function with the same nodal surface as a chosen trial function. It has proved itself among the most accurate methods available for calculating many-electron ground states, and is one of the few approaches that can be applied to systems large enough to act as realistic models of solids. In attempts to use fixed-node DMC for excited-state calculations, it has often been assumed that the DMC energy must be greater than or equal to the energy of the lowest exact eigenfunction with the same symmetry as the trial function. We show that this assumption is not justified unless the trial function transforms according to a one-dimensional irreducible representation of the symmetry group of the Hamiltonian. If the trial function transforms according to a multi-dimensional irreducible representation, corresponding to a degenerate energy level, the DMC energy may lie below the energy of the lowest eigenstate of that symmetry. Weaker variational bounds may then be obtained by choosing trial functions transforming according to one-dimensional irreducible representations of subgroups of the full symmetry group.