Numerical analysis of the adiabatic variable method for the approximation of the nuclear hamiltonian
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 779-798.

De nombreux problèmes en chimie quantique portent sur le calcul d’états fondamentaux ou excités de molécules et conduisent à la résolution de problèmes aux valeurs propres. Une des difficultés majeures dans ces calculs est la très grande dimension des systèmes qui sont en présence lors des simulations numériques. En effet les modes propres recherchés sont fonctions de 3n variables où n est le nombre de particules (électrons ou noyaux) de la molécule. Afin de réduire la dimension des systèmes à résoudre les chimistes multiplient les idées intéressantes qui permettent d’approcher le système complet. La méthode des variables adiabatiques entre dans ce cadre et nous présentons ici une étude mathématique rigoureuse de cette approximation. En particulier nous proposons un estimateur a posteriori qui pourrait permettre de vérifier l’hypothèse d’adiabaticité faite sur certaines variables ; des simulations numériques qui implémentent cet estimateur sont aussi présentées.

Many problems in quantum chemistry deal with the computation of fundamental or excited states of molecules and lead to the resolution of eigenvalue problems. One of the major difficulties in these computations lies in the very large dimension of the systems to be solved. Indeed these eigenfunctions depend on 3n variables where n stands for the number of particles (electrons and/or nucleari) in the molecule. In order to diminish the size of the systems to be solved, the chemists have proposed many interesting ideas. Among those stands the adiabatic variable method; we present in this paper a mathematical analysis of this approximation and propose, in particular, an a posteriori estimate that might allow for verifying the adiabaticity hypothesis that is done on some variables; numerical simulations that support the a posteriori estimators obtained theoretically are also presented.

Classification : 65N25, 35P15, 81V55
Mots clés : a posteriori estimator, adiabatic variable method, computational quantum chemistry, nuclear hamiltonian
Maday, Yvon  ; Turinici, Gabriel 1

1 INRIA Rocquencourt, Rocquencourt B.P. 105, 78153 Le Chesnay Cedex, France.
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Maday, Yvon; Turinici, Gabriel. Numerical analysis of the adiabatic variable method for the approximation of the nuclear hamiltonian. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 779-798. http://archive.numdam.org/item/M2AN_2001__35_4_779_0/

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