We consider the problem of calculating the vibronic absorption spectrum of a diatomic molecule coupled to a condensed phase environment, where all nuclear degrees of freedom are taken in the quadratic approximation, and where the two electronic states couple differently to the solvent. This simple model is used to examine several commonly used semiclassical approximations. The method of Kubo-Toyozawa is adapted to enable exact calculation of the real-time dipole autocorrelation function for the quantum mechanical treatment. Alternatively, we derive an expression for this correlation function in terms of a path-integral influence functional, which is not limited to a finite number of bath modes and could be applied to treat anharmonic solutes in condensed matter. We then obtain an analytical solution for the classical treatment of nuclear dynamics, and develop a mixed quantum-classical approach, where the dynamics of the diatomic vibrational mode is treated quantum mechanically and the bath is treated classically. It is shown that the mixed quantum-classical treatment provides better agreement with the exact quantum treatment than the other approximations for a wide range of parameters. Exact analytical results similar to the pure dephasing theory of Skinner and Hsu are obtained for the asymptotic long time behavior of the dipole autocorrelation functions.