The intensity of a UV-Visible absorption band is a function of the oscillator
strength and of the energy of the absorption band. Both quantities are
printed when a C.I. calculation is done, and MECI is used. In the output,
the oscillator strength, equation 1, in (electron-Angstroms) is printed under the heading "POLARIZATION".
This quantity is either zero or obligate positive, as it is calculated from the
square root of the square of the oscillator. If the State is
non-degenerate, then the "square root of the square" step could be omitted, but
if the State is degenerate, the oscillator is the square root of the sum of
squares of the oscillators for each component of the State, i.e., it is the
scalar of the vector of orthogonal oscillators.

Absorption intensity is proportional to the energy of the transition and to
the square of the oscillator strength, i.e., the absorption intensity is
proportional to: ν[(μ_{x})^{2}
+ (μ_{y})^{2 }+
(μ_{z})^{2}].

Theory

A system can go from the ground state to an excited state as the result of the
absorption of a photon. The probability of this happening, κ,
is
given (Wilson Decius and Cross, "Molecular Vibrations", p
163, McGraw-Hill (1955)) in terms of the oscillator integral:

1

by

For electronic photoexcitations, Φ,
are state functions:

In order to evaluate 1, a property of integrals of the type:

will be used several times. This property is:

From this, it follows that, if

then

To prove this relationship, consider the integral

Obviously, this integral has a value of zero, therefore

In this expression, the first and fourth terms are obviously zero, therefore

The starting point for evaluating 1 is to calculate the molecular
orbital oscillator:

In order to solve this integral, the following approximations will be made:

and

The operator
can be expanded into three terms:

where
,
and
are Cartesian position operators.
For each atomic orbital, the position would be that of the nucleus.

The zero or origin of the position operators is not immediately obvious.
In general it would not be the origin of the Cartesian coordinate
system used. The origin of the position operator would depend on the
individual M.O. and would be defined so that:

To determine the origin, the position operators are replaced by
,
where
is the distance from the origin of the Cartesian
coordinate system used to the origin of the position operator. Then:

from which it follows that:

The integrals over molecular orbitals are evaluated by first calculating