Page 157 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

142 M. DEFRANCESCHI ET AL.
Closed-shell systems as defined in the standard Hartree-Fock theory [39-40].
Unrestricted monodeterminantal treatments using different orbitals for different spins
for open-shell systems (free radicals, triplet states, etc.) [41,42].
Roothaan open-shell treatments involving a closed-shell subsystem and outer unpaired
electrons interacting through two-index integrals of Coulomb and exchange type only
[43].
MC-SCF treatments written in terms of coupled Fock equations [44]. The simplest
examples are the two-configuration SCF theory [45] used in atomic
mixing [46], or bonding-antibonding molecular problems [47], and more generally the
Clementi-Veillard electron-pair MC-SCF theory [48].
SCF treatments for infinite chains having translational symmetry [49,50],

In the recent past, we have investigated and published examples illustrating the different
cases. For instance in Ref. [17] a Roothaan open-shell system, has been detailed, in
Refs. [18, 19] a SCF treatment for infinite chains and finally in Ref [16] a MC-SCF
treatment were proposed.

In this contribution our purpose is to review the principles and the results of the
momentum space approach for quantum chemistry calculations of molecules and
polymers. To avoid unnecessary complications, but without loss of generality, we shall
consider in details the case of closed-shell systems.

2.1. RESTRICTED HARTREE-FOCK EQUATIONS

Since both position and momentum formulations contain exactly the same information, it
is convenient to start from the familiar position space expression and express it in
momentum space. In the case of a closed-shell system of electrons in the field of
M nuclear charges located at fixed positions (Born-Oppenheimer approximation),
the doubly occupied orbitals of the Hartree-Fock model in the position space are
obtained from the second-order differential equation of the form if we
assume -as usual - that the off-diagonal Lagrange multipliers ensuring the
orthogonality of the have been eliminated by an appropriate unitary transformation
inside the closed set. The F operator giving the orbitals iteratively is a one-electron
Hamiltonian including a kinetic term and an effective potential in which the electron-
nucleus attraction is balanced by the Coulomb-exchange potential approximating the real
electron-electron interaction. In atomic units, we have :