  The semiclassical Double Herman-Kluk Initial Value Representation is an
accurate approach to computing quantum real time correlation functions, but its
applications are limited by the need to evaluate an oscillatory integral. In
previous work, we have shown that this `sign problem' can be mitigated using
the modified Filinov filtration technique to control the extent to which
individual modes of the system contribute to the overall phase of the
integrand. Here we follow this idea to a logical conclusion: we analytically
derive a general expression for the mixed quantum-classical limit of the
semiclassical correlation function - AMQC-IVR, where the phase contributions
from the `classical' modes of the system are filtered while the `quantum' modes
are treated in the full semiclassical limit. We numerically demonstrate the
accuracy and efficiency of the AMQC-IVR formulation in calculations of quantum
correlation functions and reaction rates using three model systems with varied
coupling strengths between the classical and quantum subsystems. We also
introduce a separable prefactor approximation that further reduces the
computational cost, but is only accurate in the limit of weak coupling between
the quantum and classical subsystems.
