Page 155 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

140 M. DEFRANCESCHI ET AL

where E is the total energy and V(r) represents the electron-nucleus attraction potential and
the electron-electron repulsion potential.

Except for a few situations related to scattering problems where observables typically
involve momenta, physical quantities are defined in position space (r-representation) even
where the momentum space representation (p-representation) would be more natural. For
instance, experiments such as Compton profiles and (e,2e) measurements [3,4] are
compared with theoretical momentum space distribution obtained by Fourier
transformation of wavefunctions [5] computed in the position space. The lack of wave
functions directly evaluated in momentum space is no doubt due to the development of
techniques using the Schrödinger equation in the r-representation for a large variety of
situations. At least two other factors contribute to dissuade the physicists and chemists
from considering momentum space as an interesting direction for solving their problems.
First, interpretation and visualization can be more difficult in momentum space and,
second, the Schrödinger equation, and approximations to it, e.g. the Hartree-Fock (HF)
equation, are expressed as integral equations in the p-representation instead of differential
equations in the r-representation. In spite of these barriers, momentum space offers
advantages which should not be ignored. For instance, it provides an interesting
alternative way for solving electronic structure problems of atoms and molecules,
traditionally addressed in position space [6,7]. This aspect is central to this work.

As far as in the thirties the possibility of calculating wave functions in momentum space
has been recognized ; in 1932, Hylleraas [8] treated the problem of a one-electron atom,
the solutions of which for discrete and continuous spectra are well known [9]. In 1949,
McWeeny and Coulson [10,11] tried to generalize this approach to many-electron systems
involving electron repulsion terms. Starting with fixed trial functions, they applied the
iterative method developed by Svartholm [12] for the case of nuclear systems to solve
variationally the integral momentum space wave equation of helium atom and hydrogen
molecule and H 2 . Owing to convergence difficulties found in the simplest systems,

they concluded that direct calculations of electronic wave functions in momentum space
were hopeless ; and so the subject disappeared from Quantum Chemistry literature for
nearly 30 years. The situation changed in 1981, when two crystallographers, Navaza and