| Autor: |
Braida, B., Dalphin, J., Dapogny, C., Frey, P., Privat, Y. |
| Předmět: |
|
| Zdroj: |
Numerische Mathematik; Aug2022, Vol. 151 Issue 4, p1017-1064, 48p |
| Abstrakt: |
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 P ν (Ω) to find exactly ν among the n constituent electrons of a given molecule is maximum. In the Hartree-Fock framework, the shape functional P ν (Ω) arises as the integral over ν copies of Ω and (n - ν) copies of the complement R 3 \ Ω of an analytic function defined over the space R 3 n 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 P ν (Ω) 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. [ABSTRACT FROM AUTHOR] |
| Databáze: |
Complementary Index |
| Externí odkaz: |
|
Full text from SpringerLink