Page 156 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

QUANTUM CHEMISTRY COMPUTATIONS IN MOMENTUM SPACE 141

Tsoucaris, decided to treat by Fourier transformation, not the Schrödinger equation itself,
but one of its most popular approximate forms for electron systems, namely the Hartree-
Fock equations. The form of these equations was known before, in connection with
electron-scattering problems [13], but their advantage for Quantum Chemistry calculations
was not yet recognized.

The work by Navaza and Tsoucaris on the molecule [7] proved the feasibility of direct
numerical molecular orbitals computations, i.e. without atomic basis functions contrary to
what happens in r-space where it is difficult to obtain accurate Hartree-Fock solutions for
atoms, molecules and solids due to the need of representing the solutions in terms of a
finite basis of known functions, e.g. the linear combination of atomic orbitals (LCAO)
approximation. For chemists interested in polyatomic molecules, the momentum method
is quite attractive because it is not limited to systems whose geometry determines the
coordinates to be used for integrating the position space equations, as for example polar
coordinates for molecules [14] because they have approximate spherical symmetry
and/or spheroidal coordinates for diatomic molecules, see e.g. Ref. [15]. During the last
years, we have contributed to demonstrate that direct momentum space calculations are in
principle feasible for any molecule by studying hydrogen systems of increasing
complexity : the ground state at the SCF and MC-SCF level [16], an open-shell
system [17] and a chain of H atoms including an infinite number of electrons and nuclei
[18,19]. More complex systems have also been studied : atoms up to neon [20-28],
cations [22,23, 28-31], anions [22, 23, 27, 28], symmetric molecules [16, 17,32-36] as
well as asymmetric molecules such as or HF [38].

The advantages of the momentum approach are not only limited to the opportunity for
direct numerical calculations for chemical systems, but it also offers the prospect of
selecting better bases of atomic functions on which rely almost all first principle quantum
mechanical calculations.

2. MOMENTUM SPACE EQUATIONS FOR A CLOSED-SHELL SYSTEM

The Fourier transformation method enables us to immediately write the momentum space
equations as soon as the SCF theory used to describe the system under consideration
allows us to build one or several effective Fock Hamiltonians for the orbitals to be
determined. This includes a rather large variety of situations: