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

## TL;DR

This paper demonstrates that the quantum classical Liouville equation inherently includes Berry's phase effects and forces, validating its use for complex Hamiltonians with spin-orbit coupling without additional corrections.

## Contribution

It shows that the QCLE naturally incorporates Berry's phase and forces, and that Ehrenfest dynamics also include Berry's phase effects, clarifying their roles in mixed quantum-classical models.

## Key findings

- QCLE includes Berry's phase effects at order 
- Ehrenfest dynamics automatically incorporate Berry's phase
- Tully's surface hopping model ignores Berry's phase

## Abstract

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.