  We extend the Mixed Quantum-Classical Initial Value Representation (MQC-IVR),
a semiclassical method for computing real-time correlation functions, to
electronically nonadiabatic systems using the Meyer-Miller-Stock-Thoss (MMST)
Hamiltonian to treat electronic and nuclear degrees of freedom (dofs) within a
consistent dynamic framework. We introduce an efficient symplectic integration
scheme, the MInt algorithm, for numerical time-evolution of the nuclear and
electronic phase space variables as well as the Monodromy matrix, under the
non-separable MMST Hamiltonian. We then calculate the probability of
transmission through a curve-crossing in model two-level systems and show that
in the quantum limit MQC-IVR is in good agreement with the exact quantum
results, whereas in the classical limit the method yields results in keeping
with mean-field approaches like the Linearized Semiclassical IVR. Finally,
exploiting the ability of MQC-IVR to quantize different dofs to different
extents, we present a detailed study of the extents to which quantizing the
nuclear and electronic dofs improves numerical convergence properties without
significant loss of accuracy.
