Shape and topology optimization for maximum probability domains in quantum chemistry

This article is devoted to the mathematical and numerical treatments of a shape optimization problem emanating from the desire to reconcile quantum theories of chemistry and classical heuristic models: we aim to identify Maximum Probability Domains (MPDs), that is, domains Ω of the 3d space where the probability Pν (Ω) to find exactly ν among the n constituent electrons of a given molecule is maximum. In the Hartree-Fock framework, the shape functional Pν (Ω) arises as the integral over ν copies of Ω and (n − ν) copies of the complement R 3 \ Ω of an analytic function defined over the space R 3n of all the spatial configurations of the n electron system. Our first task is to explore the mathematical well-posedness of the shape optimization problem: under mild hypotheses, we prove that global maximizers of the probability functions Pν (Ω) do exist as open subsets of R 3 ; meanwhile, we identify the associated necessary first-order optimality condition. We then turn to the numerical calculation of MPDs, for which we resort to a level set based mesh evolution strategy: the latter allows for the robust tracking of complex evolutions of shapes, and it leaves the room for accurate chemical computations, carried out on high-resolution meshes of the optimized shapes. The efficiency of this procedure is enhanced thanks to the addition of a fixed-point strategy inspired from the first-order optimality conditions resulting from our theoretical considerations. Several three-dimensional examples are presented and discussed to appraise the efficiency of our algorithms.