  The rigorous quantum mechanical description of the collective interaction of
many molecules with the radiation field is usually considered numerically
intractable, and approximation schemes must be employed. Standard spectroscopy
usually contains some levels of perturbation theory, but under strong coupling
conditions, other approximations are used. A common approximation is the
1-exciton model in which processes involving weak excitations are described
using a basis comprising of the ground state and singly excited states of the
molecule cavity-mode system. In another frequently used approximation in
numerical investigations, the electromagnetic field is described classically,
and the quantum molecular subsystem is treated in the mean-field Hartree
approximation with its wavefunction assumed to be a product of single
molecules' wavefunctions. The former disregards states that take long time to
populate, and is therefore essentially a short time approximation. The latter
is not limited in this way, but by its nature, disregards some intermolecular
and molecule-field correlations. In this work, we directly compare results
obtained from these approximations when applied to several prototype problems
involving the optical response of molecule-in-optical cavities systems. In
particular, we show that our recent model investigation (J. Chem. Phys. 157,
114108 (2022)) of the interplay between the electronic strong coupling and
molecular nuclear dynamics using the truncated 1-exciton approximation agrees
very well with the semiclassical mean-field calculation.
