# Shape derivatives of the probability to find a fixed number of electrons chemically characterized by a wave function

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1 ICS - Institut du Calcul et de la Simulation 2 Departamento de Ingenieria Matematica, Centro Modelamiento Matematico

Abstract : In Quantum Chemistry, researchers are interested in finding new ways to describe well the electronic structures of molecules and their interactions. The model of Maximal Probability Domains MPDs is a developing method based on probabilities that allows such a geometrical and spatial characterization of the electronic structures of chemical systems.\smallskipIn this article, we consider quantum systems of $n$ electrons chemically characterized by general wave functions. For any integer $k \geqslant 1$, we derive a formula for the $k$-th-order shape derivative of the functional $p { u} : \Omega \mapsto p { u}\Omega$, with $p { u}\Omega$ the probability to find exactly a fixed number $ u$ of electrons in a given spatial region $\Omega \subseteq \mathbb{R}^{3}$, where \textit{exactly} means that the $n - u$ remaining ones are located in the complement $\mathbb{R}^{3} \backslash \Omega$.\smallskipThis explicit formula is computable by Quantum Monte-Carlo methods and it holds true with respect to the $W^{1,\infty}$-perturbations of a measurable domain for $H^{k}$-regular wave functions. Then, by restricting our analysis to the first- and second-order shape derivatives, we can make our statement more precise with respect to the regularity of the domain, and recover the usual structure expected from shape derivatives. \smallskipThe main ingredient of the proof consists in generalizing at any higher order the well-known expressions for the first- and second-order shape derivatives of a volume integral. Although we only need to assume that the domain is measurable to get the shape differentiability of a volume integral at any order, we also prove that the $C^{1,1}$-regularity is enough to provide a notion of partial derivative with respect to the domain at any order shape gradient, Hessian,

Keywords : shape optimization shape derivatives volume integral maximal probability domains geometry of wave functions quantum chemistry

Author: ** Jérémy Dalphin - **

Source: https://hal.archives-ouvertes.fr/