A Demonstration of Consistency between the Quantum Classical Liouville Equation and Berrys Phase and Curvature

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 Berrys phase effects and Berrys 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 Berrys phase effects and is valid for the case of complex Hamiltonians. Furthermore, we also show that, even though Tullys surface hopping model ignores Berrys phase, Berrys 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.