Page 280 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

AB INITIO CALCULATIONS OF POLARIZABILITIES IN MOLECULES 263

dipole moment The perturbed molecular wavefunction is expanded in terms of the
complete set of eigenfunctions of the unperturbed molecular Hamiltonian
and the components of the polarizability are given through an expansion over all
electronic excited states. For a static external electric field,

where u and v run over the cartesian electronic coordinates x, y and z.
For an oscillating electromagnetic field characterized by its pulsation

the dynamic polarizability is derived from the time–dependent perturbation theory :

In such an expression, must be read as the sum of two functions
like

where

The ket and its counterpart are calculated as a weighted sum over the
excited states; the weight of each state is well defined through its interaction
with the ground state by the operator. The function represents
the first–order perturbed wavefunction whose knowledge is essential in the variation–
perturbation treatment. Expression (5) has been proposed by Karplus and Kolker
(7).

2.1.2. The polynomial approach
Previously, Kirkwood(8) had suggested another choice: he deduced the first–order
perturbed wavefunction from the unperturbed one which was multiplied by a linear
combination of the electronic coordinates, i.e. :

with :

is the electric field component along v direction and some constants. In this
approach, the polarizability may be calculated very easily from the second–order
perturbed wavefunction which is simply given by :