# Shape and topology optimization for maximum probability domains in quantum chemistry

3 EDP - Equations aux Dérivées Partielles
LJK - Laboratoire Jean Kuntzmann
5 TONUS - TOkamaks and NUmerical Simulations
IRMA - Institut de Recherche Mathématique Avancée, Inria Nancy - Grand Est
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 $\Omega$ 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 $\R^3$; 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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Journal articles
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https://hal.archives-ouvertes.fr/hal-02977023
Contributor : Yannick Privat Connect in order to contact the contributor
Submitted on : Wednesday, June 15, 2022 - 11:12:10 AM
Last modification on : Friday, July 22, 2022 - 12:27:09 PM

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### Citation

Benoît Braida, Jérémy Dalphin, Charles Dapogny, Pascal Frey, Yannick Privat. Shape and topology optimization for maximum probability domains in quantum chemistry. Numerische Mathematik, Springer Verlag, 2022, 151, pp.1017--1064. ⟨10.1007/s00211-022-01305-z⟩. ⟨hal-02977023v2⟩

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