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Shape and topology optimization for maximum probability domains in quantum chemistry

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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 \({\mathbb {R}}^3\); 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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Acknowledgements

The work of C.D. and Y.P. is partly supported by the project ANR-18-CE40-0013 SHAPO, financed by the French Agence Nationale de la Recherche (ANR).

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Authors and Affiliations

  1. Laboratoire de Chimie Théorique, Sorbonne Université, UMR 7616 CNRS, 75005, Paris, France

    B. Braida

  2. Institut des Sciences du Calcul et des Données, ISCD, Sorbonne Université, F-75005, Paris, France

    J. Dalphin & P. Frey

  3. Institute of Engineering Univ. Grenoble Alpes, LJK, Univ. Grenoble Alpes, CNRS, Grenoble INP, 38000, Grenoble, France

    C. Dapogny

  4. Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg, CNRS UMR 7501, INRIA, 7 rue René Descartes, 67084, Strasbourg, France

    Y. Privat

  5. Institut Universitaire de France (IUF), Paris, France

    Y. Privat

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Braida, B., Dalphin, J., Dapogny, C. et al. Shape and topology optimization for maximum probability domains in quantum chemistry. Numer. Math. 151, 1017–1064 (2022). https://doi.org/10.1007/s00211-022-01305-z

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