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