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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

## Access Full Article

top## Abstract

top## How to cite

topMaday, 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- 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.
- M. Azaiez, M. Dauge and Y. Maday, Méthodes spectrales et les éléments spectraux. Institut de Recherche Mathématique de Rennes, Prépublications 1994-17 (1994).
- 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
- 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
- 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). Zbl0717.76004
- R. Dutray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Tome 5. Masson, CEA, Paris (1984).
- R. Friesner, J. Bentley, M. Menou and C. Leforestier, Adiabatic pseudospectral methods for multidimensional vibrational potentials. J. Chem. Phys.99 (1993) 324.
- R. Kosloff, Time-dependent quantum-mecanical methods for molecular dynamics. J. Chem. Phys.92 (1988) 2087.
- 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
- C. Leforestier, Grid representation of rotating triatomics. J. Chem. Phys.94 (1991) 6388.
- J.L. Lions and E. Magenes, Problèmes aux limites non-homogènes et applications. Dunod, Paris (1968). Zbl0165.10801
- R. Verfürth, A posteriori error estimates for non-linear problems. Finite element discretisations of elliptic equations. Math. Comp.62 (1994) 445-475 Zbl0799.65112
- R. Verfürth, A review of a posteriori error estimates and adaptative mesh-refinement techniques. Wiley-Teubner, Stuttgart (1997).
- 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 ?

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