Importance of eigenvector sign consistency in computations of expectation values via mixed quantum-classical surface-hopping dynamics

For simulations of systems undergoing nonadiabatic dynamics between multiple quantum states, one must often compute nonadiabatic coupling vectors, which involve the overlap of one adiabatic state with the gradient (with respect to the nuclear positions) of another. Because these states are usually computed numerically using a matrix diagonalization routine, the signs of the eigenvectors and the resulting nonadiabatic coupling vectors can change erratically over the course of a trajectory. To ensure a smooth evolution of the eigenvectors, one usually tracks the sign of the overlap of the current eigenvector with that of the previous time step and corrects the sign if needed. However, when computing time-dependent expectation values based on an ensemble of trajectories (which are typically initialized at different points in configuration space), one now has to ensure eigenvector sign continuity across the entire ensemble. In this work, we present a simple procedure for accomplishing this in mixed quantum-classical surface-hopping dynamics. The utility of this procedure is demonstrated by calculating both adiabatic state populations in a reduced model of a condensed phase proton-coupled electron transfer (PCET) reaction and subsystem state populations in an excitonic model of an alpha-helical chain undergoing vibrational energy transport (VET), using mixed quantum-classical Liouville surface-hopping dynamics. Our results show that when eigenvector sign continuity across the ensemble is neglected, as is commonly done, one can obtain large deviations from the correct values. However, after implementing this procedure, the results agree better with the exact results in the case of the PCET model and behave physically in the case of the VET model. These results illustrate the importance of a careful treatment of the signs in mixed quantum-classical dynamics calculations of expectation values in the adiabatic representation.