Numerical Analysis of the Adiabatic Variable Method for the Approximation of the Nuclear Hamiltonian

Abstract

Abstract. 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. Résumé.De nombreux problèmes en chimie quantique portent sur le calcul d'états fondamentaux ou excités de molécules et conduisentà 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 systèmesà 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 uné etude 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.Mathematics Subject Classification. 65N25, 35P15, 81V55.

A posteriori error estimator for the eigenvalue problem associated to the Schr�dinger operator with magnetic field

“…These numerical results developed in [8,10,11] are useful to determine where the superconductivity appears. The a posteriori error estimate we obtain is a prolongation of results by Larson [20], Bernardi-Métivet [5], Bernardi-Méti-vet-Verfürth [6], Maday-Turinici [22] or Verfürth [27] for another framework of operator with the Schrödinger operator with magnetic field and a Neumann magnetic boundary condition. We propose for this operator a better estimate than Verfürth [27] for −∇ · (A∇) + d and the a posteriori error estimate in Theorem 2.1 has the same order of convergence than for the a priori error estimate proposed by Babuška-Osborn [3] and recalled in Theorem 3.1.…”

“…For the Schrödinger operator with magnetic field, [11] proves that the first eigenvectors are localized in the boundary ; therefore adaptative mesh refinement techniques seem to be appropriate to gain computation time and for this, we need local error estimates. In this spirit, we can quote works of Babuska [2,4], Bernardi-Métivet [5], Bernardi-Métivet-Verfürth [6], Larson [20] who proposes a posteriori error estimates for the Laplacian operator with Dirichlet boundary conditions which can be extended to operators such d i,j =1 ∂ x j a ij (x)∂ x i +b(x) with Robin boundary conditions, Maday-Turinici [22] who work more specifically about the nuclear hamiltonian. Some of these articles are based on the work of Verfürth [27] who tries to make a more systematic analysis of any problem.…”