Shape and Topology Optimization for Maximum Probability Domains in Quantum Chemistry
SHAPE AND TOPOLOGY OPTIMIZATION FOR MAXIMUM PROBABILITY DOMAINS IN QUANTUM CHEMISTRY B. BRAIDA1, J. DALPHIN2, C. DAPOGNY3, P. FREY2, Y. PRIVAT4 1 Laboratoire de Chimie Théorique, Sorbonne Université, UMR 7616 CNRS, 75005, Paris France. 2 Sorbonne Université, Institut des Sciences du Calcul et des Données, ISCD, F-75005 Paris, France, 3 Univ. Grenoble Alpes, CNRS, Grenoble INP1, LJK, 38000 Grenoble, France 4 Université de Strasbourg, CNRS UMR 7501, INRIA, Institut de Recherche Mathématique Avancée (IRMA), 7 rue René Descartes, 67084 Strasbourg, France, Abstract. 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 3 3n (n − ν) copies of the complement R n Ω of an analytic function defined over the space R 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 3 functions Pν (Ω) do exist as open subsets of R ; 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.
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