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 \(\Omega \) of the 3d space where the probability \({\mathbb {P}}_\nu (\Omega )\) to find exactly \(\nu \) among the n constituent electrons of a given molecule is maximum. In the Hartree-Fock framework, the shape functional \({\mathbb {P}}_\nu (\Omega )\) arises as the integral over \(\nu \) copies of \(\Omega \) and \((n-\nu )\) copies of the complement \({\mathbb {R}}^3 \setminus \Omega \) of an analytic function defined over the space \({\mathbb {R}}^{3n}\) 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 \({\mathbb {P}}_\nu (\Omega )\) do exist as open subsets of \({\mathbb {R}}^3\); 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, while leaving 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.
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