Page 154 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

Quantum Chemistry Computations in Momentum Space

(1)
(2)
M. DEFRANCESCHI (1) , J. DELHALLE , L. DE WINDT , P. FISCHER (1, 3) ,
J.G. FRIPIAT (2)
(1) Commissariat à l'Energie Atomique, CE-Saclay, DSM/DRECAM/SRSIM,
F-91191 Gif-sur-Yvette Cedex, France
(2) Facultés Universitaires Notre-Dame de la Paix, Laboratoire de Chimie Théorique
Appliquée, Rue de Bruxelles, 61, B-5000 Namur, Belgium
(3) Université de Paris-Dauphine, Ceremade, Place Maréchal de Lattre de Tassigny,
F-750I6 Paris, France

1. Introduction

In quantum mechanics, the state of a physical system is described by a vector of an
Hilbert space, represented by a linear superposition of eigenvectors of Hermitian
operators which result from a particular choice of a maximal set of commuting
observables [1,2]. The various representations obtained in this way are connected by a
generalized Fourier transformation. The so-called Schrödinger method, normally used for
systems of electrons and nuclei, starts in an Hilbert space by taking the components of
particle coordinates as a maximal set; consequently, the state function of the system is

written in the coordinate representation, and this leads to the familiar Schrödinger equation
for determining the possible energies of atoms and molecules as eigenvalues of the total
Hamiltonian operator in position space. The Schrödinger equation can be expressed in
other representations as well ; e.g. by referring to the various particles in terms of
momenta instead of position vectors The state function in momentum space
representation becomes the ordinary Fourier transform of the state function in position
space, with appropriate factor :

Taking the Fourier transform of the ordinary Schrodinger equation yields, in atomic units,
139
Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 139–158.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.