Page 282 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

AB INITIO CALCULATIONS OF POLARIZABILITIES IN MOLECULES 265
2.2. DIPOLAR FACTOR
The dipolar factor may be interpreted in terms of gauge invariance. The electric
observables usually are calculated in the gauge In the change to
the gauge the Hamiltonian is transformed and the wavefunction becomes
(11):

If the strength of the electric field is small enough, then :

As known (11), the gauge invariance is ensured i f :

By omitting time–dependent terms, as in the preceding paragraph, the function
may be read as the sum of the unperturbed wavefunction and a term which is the
product of this function by a linear combination of the electronic coordinates, i.e. the
Kirkwood’s function. Thus, the dipolar factor ensures gauge–invariance.
But the role of the dipolar factor in this mixed method is essential on the
following point : its contribution in the computation occurs in a complementary
(and sometimes preponderant) way to that calculated only from the excited states,
the number of which is unavoidably limited by the computation limits. But before
discussing their number, we have to comment the description of these states.

2.3. EXCITED STATES AND EXTRAPOLATION PROCEDURE
In a first approach, Rérat (10) described the excited states of Eq.15 through
Slater determinants, constructed by monoexcitation of the ground state
through the monoelectronic operator. By reason of orthogonality (deriving from
all those necessary to the description of were rejected. The lack of
such determinants does not allow to have a good description of the excited states
when they have a dominant configuration appearing also in If this approach led
to interesting static results with reduced basis sets, it could not reach the resonances
correctly.
It is the reason for which the Slater determinants have been replaced by the kets
accounting for the true spectral states (1). These states have been computed
independently by the CIPSI (4) program which treats the electronic correlation. Pre-
liminary calculations of energies have been made by the standard CIPSI algorithm
(4a) on small S subspaces of c.a. 400 determinants. Perturbation treatments involv-
ing larger subspaces (about 1000 for CO) have been achieved using the diagrammatic
version of CIPSI (4b).
The quality of the states has been tested through their energy and also their
transition moment. Moreover from the natural orbitals and Mulliken populations
analysis, we have determined the predominant electronic configuration of each
state and its Rydberg character. Such an analysis is particularly interesting since
it explains the contribution of each to the calculation of the static or dynamic
polarizability; it allows a better understanding in the case of the CO molecule : the
difficulty of the calculation and the wide range of published values for the parallel
component while the computation of the perpendicular component is easier. In effect
in the case of CO :