A demonstration of consistency between the quantum classical Liouville equation and Berry's phase and curvature for the case of complex Hamiltonians

Joseph Subotnik; Gaohan Miao; Nicole Bellonzi; Hung-Hsuan Teh; Wenjie Dou

doi:10.1063/1.5116210

A demonstration of consistency between the quantum classical Liouville equation and Berry's phase and curvature for the case of complex Hamiltonians

J Chem Phys. 2019 Aug 21;151(7):074113. doi: 10.1063/1.5116210.

Authors

Joseph Subotnik¹, Gaohan Miao¹, Nicole Bellonzi¹, Hung-Hsuan Teh¹, Wenjie Dou²

Affiliations

¹ Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323, USA.
² Department of Chemistry, University of California, Berkeley, California 94720, USA.

PMID: 31438699
DOI: 10.1063/1.5116210

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 = ℏ⁰), we show that the QCLE does include Berry's phase effects and Berry's forces (which are proportional to a higher order, ℏ = ℏ¹). 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 (i.e., not just real) Hamiltonians, where exotic features can arise in the course of electronic relaxation. 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 odd numbers of electrons and spin-orbit coupling, where the complex nature of the Hamiltonian is paramount.