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Although the quantum classical Liouville equation (QCLE) arises by cuttingoff the exact equation of motion for a coupled nuclear-electronic system atorder 1 (1 = $\hbar^0$ ), we show that the QCLE does include Berry's phaseeffects and Berry's forces (which are proportional to a higher order, $\hbar$ =$\hbar^1$ ). Thus, the fundamental equation underlying mixed quantum-classicaldynamics does not need a correction for Berry's phase effects and is valid forthe case of complex Hamiltonians. Furthermore, we also show that, even thoughTully's surface hopping model ignores Berry's phase, Berry's phase effects areincluded automatically within Ehrenfest dynamics. These findings should be ofgreat importance if we seek to model coupled nuclear-electronic dynamics forsystems with spin-orbit coupling, where the complex nature of the Hamiltonianis paramount.

A Demonstration of Consistency between the Quantum Classical Liouville Equation and Berry's Phase and Curvature