Quantum Monte Carlo Simulations of Fermions: A Mathematical Analysis of the Fixed-Node Approximation

The Diffusion Monte Carlo (DMC) method is a powerful strategy to estimate the ground state energy E 0 of an N-body Schrödinger Hamiltonian H = -½Δ + V with high accuracy. It consists of writing E 0 as 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 ψ I which hopefully approximates some ground state ψ 0 of 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 ψ I coincide with those of a ground state of H. The approximation due to the fact that, in practice, the nodal surfaces of ψ I differ 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 ψ I , 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 ψ I .