  We present an analytical and numerical solution of the calculation of the
transition moments for the exact semi-classical light-matter interaction for
wavefunctions expanded in a Gaussian basis. By a simple manipulation we show
that the exact semi-classical light-matter interaction of a plane wave can be
compared to a Fourier transformation of a Gaussian where analytical recursive
formulas are well known and hence making the difficulty in the implementation
of the exact semi-classical light-matter interaction comparable to the
transition dipole. Since the evaluation of the analytical expression involves a
new Gaussian we instead have chosen to evaluate the integrals using a standard
Gau{\ss}-Hermite quadrature since this is faster. A brief discussion of the
numerical advantages of the exact semi-classical light-matter interaction in
comparison to the multipole expansion along with the unphysical interpretation
of the multipole expansion is discussed. Numerical examples on
[CuCl$_4$]$^{2-}$ to show that the usual features of the multipole expansion is
immediately visible also for the exact semi-classical light-matter interaction
and that this can be used to distinguish between symmetries. Calculation on
[FeCl$_4$]$^{1-}$ is presented to demonstrate the better numerical stability
with respect to the choice of basis set in comparison to the multipole
expansion and finally Fe-O-Fe to show origin independence is a given for the
exact operator. The implementation is freely available in OpenMolcas.
