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This paper considers Schrödinger operators, and presents a prob- abilistic 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 com- putations 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 Schrödinger operator of a fermionic system with Dirichlet conditions

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