Solutions of the multiconfiguration time-dependent Hartree–Fock equations with Coulomb interactions

Saber Trabelsi

doi:10.1016/j.crma.2007.06.005

Partial Differential Equations

Solutions of the multiconfiguration time-dependent Hartree–Fock equations with Coulomb interactions
[Solutions des équations de multi-configurations dépendant du temps en chimie quantique]

Présenté par : Pierre-Louis Lions

Saber Trabelsi ^1,²

¹ WPI, Fak. f. Mathematik, Univ. Wien – UZA 4, Nordbergstrasse 15, A-1090 Wien, Austria
² Laboratoire J.-L. Lions, universié Pierre et Marie Curie, 175, rue du Chevaleret, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 3, pp. 145-150.

Résumé
Abstract

Les méthodes de multi-configurations améliorent des modèles simples d'approximation bien connus de l'équation de Schrödinger linéaire à N corps pour les systèmes moléculaires sous interactions Coulombiennes, tels que les modèles de Hartree et de Hartree–Fock. Dans cette Note, nous étudions le cas de la méthode dite de Multiconfigurations Hartree–Fock dépendante du temps, qui consiste à approcher les fonctions d'onde antisymétriques d'un espace de Hilbert de dimension infinie par une combinaison linéaire dépendante du temps de déterminants de Slater. Nous écrivons le système d'équations d'évolution et nous établissons que ce système est bien posé dans un cadre fonctionnel adéquat, et ceci tant que la matrice densité associée ne change pas de rang. Notre preuve recouvre et simplifie les résultats d'existence et unicité de solutions des problèmes de Cauchy associés aux approximations de Hartree et de Hartree–Fock obtenus ailleurs.

Multiconfiguration methods are a natural generalization of well-known simple models for approximating the linear N body Schrödinger equation for atomic and molecular systems with binary (Coulomb) interactions, like the Hartree and the Hartree–Fock theories. This Note discusses the case of the multiconfiguration time-dependent Hartree–Fock (MCTDHF in short) method which consists in approximating the high-dimensional wavefunction by a time-dependent linear combination of Slater determinants. We formulate the system of equations of motion and we establish the well-posedness of this system in a convenient Hilbert space framework, at least as long as the associated one-particle density matrix keeps the same rank. Our proof covers and simplifies previous well-posedness results of the Cauchy problems associated to the time-dependent Hartree and the time-dependent Hartree–Fock approximations obtained elsewhere.

Reçu le : 2006-02-18
Accepté le : 2007-06-05
Publié le : 2007-07-17

DOI : 10.1016/j.crma.2007.06.005

Affiliations des auteurs :

Saber Trabelsi ^{1, 2}

¹ WPI, Fak. f. Mathematik, Univ. Wien – UZA 4, Nordbergstrasse 15, A-1090 Wien, Austria
² Laboratoire J.-L. Lions, universié Pierre et Marie Curie, 175, rue du Chevaleret, 75013 Paris, France

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     author = {Saber Trabelsi},
     title = {Solutions of the multiconfiguration time-dependent {Hartree{\textendash}Fock} equations with {Coulomb} interactions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {145--150},
     publisher = {Elsevier},
     volume = {345},
     number = {3},
     year = {2007},
     doi = {10.1016/j.crma.2007.06.005},
     language = {en},
}

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Saber Trabelsi. Solutions of the multiconfiguration time-dependent Hartree–Fock equations with Coulomb interactions. Comptes Rendus. Mathématique, Volume 345 (2007) no. 3, pp. 145-150. doi : 10.1016/j.crma.2007.06.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.06.005/

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