QUANTUM MONTE CARLO SIMULATIONS OF FERMIONS: A MATHEMATICAL ANALYSIS OF THE FIXED-NODE APPROXIMATION

Abstract

The Diffusion Monte Carlo (DMC) method is a powerful strategy to estimate the ground state energy E0of an N-body Schrödinger Hamiltonian H = -½Δ + V with high accuracy. It consists of writing E0as the long-time limit of an expectation value of a drift-diffusion process with a source term, and numerically simulating this process by means of a collection of random walkers. As for a number of stochastic methods, a DMC calculation makes use of an importance sampling function ψIwhich hopefully approximates some ground state ψ0of H. In the fermionic case, it has been observed that the DMC method is biased, except in the special case when the nodal surfaces of ψIcoincide with those of a ground state of H. The approximation due to the fact that, in practice, the nodal surfaces of ψIdiffer from those of the ground states of H, is referred to as the Fixed Node Approximation (FNA). Our purpose in this paper is to provide a mathematical analysis of the FNA. We prove that, under convenient hypotheses, a DMC calculation performed with the importance sampling function ψI, provides an estimation of the infimum of the energy 〈ψ, Hψ〉 on the set of the fermionic test functions ψ that exactly vanish on the nodal surfaces of ψI.