Numerical analysis of the adiabatic variable method for the approximation of the nuclear hamiltonian
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 4, p. 779-798

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.

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.

Classification:  65N25,  35P15,  81V55
Keywords: a posteriori estimator, adiabatic variable method, computational quantum chemistry, nuclear hamiltonian
@article{M2AN_2001__35_4_779_0,
     author = {Maday, Yvon and Turinici, Gabriel},
     title = {Numerical analysis of the adiabatic variable method for the approximation of the nuclear hamiltonian},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {4},
     year = {2001},
     pages = {779-798},
     zbl = {0995.65112},
     mrnumber = {1863280},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_4_779_0}
}
Maday, Yvon; Turinici, Gabriel. Numerical analysis of the adiabatic variable method for the approximation of the nuclear hamiltonian. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 4, pp. 779-798. http://www.numdam.org/item/M2AN_2001__35_4_779_0/

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