Page 288 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

AB INITIO CALCULATIONS OF POLARIZABILITIES IN MOLECULES 271

3.1. THE CPHF METHOD
The variational theorem which has been initially proved in 1907 (24), before the
birthday of the Quantum Mechanics, has given rise to a method widely employed in
Quantum calculations. The finite–field method, developed by Cohen and Roothan
(25), is connected to this method. The Stark Hamiltonian explicitly appears
in the Fock monoelectronic operator. The polarizability is derived from the second
derivative of the energy with respect to the electric field. The finite–field method has
been developed at the SCF and CI levels but the difficulty of such a method is the
well known loss in the numerical precision in the limit of small or strong fields. The
latter case poses several interconnected problems in the calculation of polarizability
at a given order, n :
• The strength of the field must not be so strong that higher order effects
come into play; and then, should we introduce the basis functions suited for
order m to get a correct response of the system up to order n, even if we are
concerned only with the nth–order ?
• The pointwise energies will be fitted by a Taylor espansion. What must be the
order of the expansion ? How much points must be considered ? It is necessary
to master the numerical techniques well.
By allowing the direct calculation of the successive derivatives (thus without resort-
ing to any effective value of the field), the perturbation methods offers an elegant