On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes

TL;DR

This paper introduces a probabilistic approach to shape derivatives of Dirichlet groundstates, applying it to fermion systems and proposing a Monte Carlo method to compute these derivatives for fixed node approximations.

Contribution

It provides a novel probabilistic interpretation of shape derivatives and develops a Monte Carlo algorithm for their approximation in fermionic groundstate problems.

Findings

Shape derivative vanishes under symmetric boundary distribution or exact eigenfunction zeros.

Proposes Nodal Monte-Carlo (NMC) for practical computation of shape derivatives.

Characterizes conditions for the vanishing of the shape derivative in fermionic systems.

Abstract

This paper considers Schr\"odinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the case of the so-called fixed node approximation of fermion groundstates, defined by the bottom eigenelements of the Schr\"odinger operator of a fermionic system with Dirichlet conditions on the nodes (the set of zeros) of an initially guessed skew-symmetric function. We show that the shape derivative of the fixed node energy vanishes if and only if either (i) the distribution on the nodes of the stopped random process is symmetric; or (ii) the nodes are exactly the zeros of a skew-symmetric eigenfunction of the operator. We propose an approximation of the shape derivative of the fixed node energy that can be computed with a Monte-Carlo algorithm, which can be referred to as Nodal Monte-Carlo (NMC). The latter approximation of the shape derivative also vanishes if and only if either (i) or (ii) holds.