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.