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

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 = ${\hslash }^0$ ), we show that the QCLE does include Berry's phase effects and Berry's forces (which are proportional to a higher order, $\hslash$ = ${\hslash }^1$ ). 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.

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Cited by in corpus (7)

• Electron Transfer and Spin-Orbit Coupling: How Strong are Berry Force Effects In and Out of Equilibrium in the Presence of Nuclear Friction?
• Chemical Reaction Rates for Systems with Spin-Orbit Coupling and an Odd Number of Electrons: Does Berry's Phase Lead to Meaningful Spin-Dependent Nuclear Dynamics for a Two State Crossing?
• A Phase-Space Semiclassical Approach for Modeling Nonadiabatic Nuclear Dynamics with Electronic Spin
• On the Proper Derivation of the Floquet-based Quantum Classical Liouville Equation and Surface Hopping Describing a Molecule or Material Subject to an External Field
• On the Meaning of Berry Force For Unrestricted Systems Treated With Mean-Field Electronic Structure
• The Effect of Duschinskii Rotations on Spin-Dependent Electron Transfer Dynamics
• A Quantum-Classical Liouville Formalism in a Preconditioned Basis and Its Connection with Phase-Space Surface Hopping