Quantum-Classical Liouville Approach to Molecular Dynamics: Surface Hopping Gaussian Phase-Space Packets
Illia Horenko, Christian Salzmann, Burkhard Schmidt, and Christof Schütte
In mixed quantum-classical molecular dynamics few but important degrees of freedom of a molecular system are modeled quantum-mechanically while the remaining degrees of freedom are treated within the classical approximation. Such models can be systematically derived as a first order approximation to the partial Wigner transform of the quantum Liouville-von Neumann equation. The resulting adiabatic quantum-classical Liouville equation (QCLE) can be decomposed into three individual propagators by means of a Trotter splitting:
Phase oscillations of the coherences resulting from the time evolution of the quantum-mechanical subsystem.
Exchange of densities and coherences reflecting non-adiabatic effects in quantum-classical dynamics.
Classical Liouvillian transport of densities and coherences along adiabatic potential energy surfaces or arithmetic means thereof.
A novel stochastic implementation of the QCLE is proposed in the present work. In order to substantially improve the traditional algorithm based on surface hopping trajectories [J. C. Tully, J. Chem. Phys. 93 (2), 1061 (1990)], we model the evolution of densities and coherences by a set of surface hopping Gaussian phase-space packets (GPPs) with variable width and with adjustable real or complex amplitudes, respectively. The dense sampling of phase-space offers two main advantages over other numerical schemes to solve the QCLE. First, it allows to perform a quantum-classical simulation employing a constant number of particles, i. e. the generation of new trajectories at each surface hop is avoided. Second, the effect of non-local operators in the exchange of densities and coherences can be treated without having to invoke the momentum jump approximation.
For the example of a single avoided crossing we demonstrate that convergence towards fully quantum-mechanical dynamics is much faster for surface hopping GPPs than for trajectory-based methods. For dual avoided crossings the Gaussian-based dynamics correctly reproduces the quantum-mechanical result even when trajectory-based methods not accounting for the transport of coherences fail qualitatively.
For animated graphics of prototypical quantum dynamical simulations, see the corresponding demonstration examples of our WavePacket software package: