Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T18:00:24.636Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 April 2002

Yvon Maday  and
Gabriel Turinici
Yvon Maday
Affiliation:
ASCI, UPR 9029, bâtiment 506, Université Paris-Sud, 91405 Orsay. ([email protected]) Analyse numérique, Université Paris VI, place Jussieu, 75252 Paris Cedex 05, France. ([email protected])
Gabriel Turinici
Affiliation:
ASCI, UPR 9029, bâtiment 506, Université Paris-Sud, 91405 Orsay. ([email protected])
Get access

Abstract

Many problems in quantumchemistry 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 largedimension of the systems to be solved. Indeed these eigenfunctions dependon 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 manyinteresting ideas. Among those stands the adiabatic variable method; we present in this paper a mathematical analysis of thisapproximation and propose, in particular, an a posteriori estimate thatmight allow for verifying the adiabaticity hypothesis that is done on some variables; numerical simulations that support thea posteriori estimators obtained theoretically are also presented.

Keywords

A posteriori estimatoradiabatic variable methodcomputational quantum chemistrynuclear Hamiltonian.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 4 , July 2001 , pp. 779 - 798
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antihainen, J., Friesner, R. and Leforestier, C., Adiabatic pseudospectral calculation of the vibrational states of the four atom molecules: Application to hydrogen peroxide. J. Chem. Phys. 102 (1995) 1270.
M. Azaiez, M. Dauge and Y. Maday, Méthodes spectrales et les éléments spectraux. Institut de Recherche Mathématique de Rennes, Prépublications 1994-17 (1994).
Babuska, I. and Schwab, C., A posteriori error estimation for hierarchic models of elliptic boundary value problems on thin domains. SIAM J. Numer. Anal. 33 (1996) 241-246. CrossRef
C. Bernardi and Y. Maday, Spectral methods, in Handbook of numerical analysis, Vol. V, Part 2, Ph. G. Ciarlet and J.L. Lions Eds., North-Holland, Amsterdam (1997).
C. Bernardi and Y. Maday, Approximations spectrales de problèmes aux limites elliptiques. Springer, Paris, Berlin, New York (1992).
G. Caloz and J. Rappaz, Numerical analysis for nonlinear and bifurcation problems, in Handbook of numerical analysis, Vol. V, Part 2, Ph.G. Ciarlet and J.L. Lions Eds., North-Holland, Amsterdam (1997).
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods in fluid dynamics. Springer, Berlin (1987).
R. Dutray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Tome 5. Masson, CEA, Paris (1984).
Friesner, R., Bentley, J., Menou, M. and Leforestier, C., Adiabatic pseudospectral methods for multidimensional vibrational potentials. J. Chem. Phys. 99 (1993) 324. CrossRef
Kosloff, R., Time-dependent quantum-mecanical methods for molecular dynamics. J. Chem. Phys. 92 (1988) 2087. CrossRef
Kosloff, D. and Kosloff, R., Fourier method for the time dependent Schrödinger equation as a tool in molecular dynamics. J. Comp. Phys. 52 (1983) 35. CrossRef
Leforestier, C., Grid representation of rotating triatomics. J. Chem. Phys. 94 (1991) 6388. CrossRef
J.L. Lions and E. Magenes, Problèmes aux limites non-homogènes et applications. Dunod, Paris (1968).
R. Verfürth, A posteriori error estimates for non-linear problems. Finite element discretisations of elliptic equations. Math. Comp. 62 (1994) 445-475
R. Verfürth, A review of a posteriori error estimates and adaptative mesh-refinement techniques. Wiley-Teubner, Stuttgart (1997).
Yamashita, K., Mokoruma, K. and Leforestier, C., Theoretical study of the highly vibrationally excited states of FHF -: Ab initio potential energy surface and hyperspherical formulation. J. Chem. Phys. 99 (1993) 8848. CrossRef