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

Yvon Maday; Gabriel Turinici

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 4, page 779-798
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Maday, Yvon, and Turinici, Gabriel. "Numerical Analysis of the Adiabatic Variable Method for the Approximation of the Nuclear Hamiltonian." ESAIM: Mathematical Modelling and Numerical Analysis 35.4 (2010): 779-798. <http://eudml.org/doc/197514>.

@article{Maday2010,
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. },
author = {Maday, Yvon, Turinici, Gabriel},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {A posteriori estimator; adiabatic variable method; computational quantum chemistry; nuclear Hamiltonian.; a posteriori error control; chemical kinetics; Jacobi variables method; eigenelements; nuclear Hamiltonian; multi-body problem; numerical results},
language = {eng},
month = {3},
number = {4},
pages = {779-798},
publisher = {EDP Sciences},
title = {Numerical Analysis of the Adiabatic Variable Method for the Approximation of the Nuclear Hamiltonian},
url = {http://eudml.org/doc/197514},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Maday, Yvon
AU - Turinici, Gabriel
TI - Numerical Analysis of the Adiabatic Variable Method for the Approximation of the Nuclear Hamiltonian
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 4
SP - 779
EP - 798
AB - 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.
LA - eng
KW - A posteriori estimator; adiabatic variable method; computational quantum chemistry; nuclear Hamiltonian.; a posteriori error control; chemical kinetics; Jacobi variables method; eigenelements; nuclear Hamiltonian; multi-body problem; numerical results
UR - http://eudml.org/doc/197514
ER -

References

top
  1. J. Antihainen, R. Friesner and C. Leforestier, Adiabatic pseudospectral calculation of the vibrational states of the four atom molecules: Application to hydrogen peroxide. J. Chem. Phys.102 (1995) 1270.  
  2. M. Azaiez, M. Dauge and Y. Maday, Méthodes spectrales et les éléments spectraux. Institut de Recherche Mathématique de Rennes, Prépublications 1994-17 (1994).  
  3. I. Babuska and C. Schwab, A posteriori error estimation for hierarchic models of elliptic boundary value problems on thin domains. SIAM J. Numer. Anal.33 (1996) 241-246.  Zbl0846.65056
  4. 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).  Zbl0991.65124
  5. C. Bernardi and Y. Maday, Approximations spectrales de problèmes aux limites elliptiques. Springer, Paris, Berlin, New York (1992).  
  6. 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).  
  7. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods in fluid dynamics. Springer, Berlin (1987).  Zbl0717.76004
  8. R. Dutray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Tome 5. Masson, CEA, Paris (1984).  
  9. R. Friesner, J. Bentley, M. Menou and C. Leforestier, Adiabatic pseudospectral methods for multidimensional vibrational potentials. J. Chem. Phys.99 (1993) 324.  
  10. R. Kosloff, Time-dependent quantum-mecanical methods for molecular dynamics. J. Chem. Phys.92 (1988) 2087.  
  11. D. Kosloff and R. Kosloff, Fourier method for the time dependent Schrödinger equation as a tool in molecular dynamics. J. Comp. Phys.52 (1983) 35.  Zbl0513.65079
  12. C. Leforestier, Grid representation of rotating triatomics. J. Chem. Phys.94 (1991) 6388.  
  13. J.L. Lions and E. Magenes, Problèmes aux limites non-homogènes et applications. Dunod, Paris (1968).  Zbl0165.10801
  14. R. Verfürth, A posteriori error estimates for non-linear problems. Finite element discretisations of elliptic equations. Math. Comp.62 (1994) 445-475  Zbl0799.65112
  15. R. Verfürth, A review of a posteriori error estimates and adaptative mesh-refinement techniques. Wiley-Teubner, Stuttgart (1997).  
  16. K. Yamashita, K. Mokoruma and C. Leforestier, Theoretical study of the highly vibrationally excited states of FHF-: Ab initio potential energy surface and hyperspherical formulation. J. Chem. Phys.99 (1993) 8848.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.