  The absorption spectroscopy of calcium-doped argon clusters is described in
terms of an effective diatomics molecule Ca-(Ar_n), in the framework of
semiclassical vertical transitions. We show how, upon choosing a suitable
reaction coordinate, the effective finite-temperature equilibrium properties
can be obtained for the ground- and excited-surfaces from the potential of mean
force (PMF). An extension of the recent multiple range random-walk method is
used to calculate the PMF over continuous intervals of distances. The
absorption spectra calculated using this single-coordinate description are
found to be in good agreement with the spectra obtained from high-statistics
Monte Carlo data, in various situations. For CaAr$_{13}$, we compare the
performances of two different choices of the reaction coordinate. For CaAr_37,
the method is seen to be accurate enough to distinguish between different
low-energy structures. Finally, the idea of casting the initial many-body
problem into a single degree of freedom problem is tested on the spectroscopy
of calcium in bulk solid argon.
