  A mixed semiclassical initial value representation expression for
spectroscopic calculations is derived. The formulation takes advantage of the
time-averaging filtering and the hierarchical properties of different
trajectory based propagation methods. A separable approximation is then
introduced that greatly reduces (about an order of magnitude) the computational
cost compared with a full Herman-Kluk time-averaging semiclassical calculation
for the same systems. The expression is exact for the harmonic case and it is
tested numerically for a Morse potential coupled to one or two additional
harmonic degrees of freedom. Results are compared to full Herman-Kluk
time-averaging calculations and exact quantum wavepacket propagations. We found
the peak positions of the mixed semiclassical approximations to be always in
very good agreement with full quantum calculations, while overtone peak
intensities are lower with respect to the exact ones. Given the reduced
computational effort required by this new mixed semiclassical approximation, we
believe the present method to make spectroscopic calculations available for
higher dimensional systems than accessible before.
