Quantum-classical Liouville description of multidimensional nonadiabatic molecular dynamics

Abstract

The quantum-classical Liouville formulation gives a quantum-mechanical density-matrix description of the ''quantum'' particles of a problem ͑e.g., the electrons͒ and a classical phase-space-density description of the ''classical'' particles ͑e.g., the nuclei͒. In order to employ this formulation to describe multidimensional nonadiabatic processes in complex molecular systems, this work is concerned with an efficient Monte Carlo implementation of the quantum-classical Liouville equation. Although an exact stochastic realization of this equation is in principle available, in practice one has to cope with two major complications: ͑i͒ The representation of nonlocal phase-space operators in terms of local classical trajectories and ͑ii͒ the convergence of the Monte Carlo sampling which is cumbersome due to complex-valued trajectories with rapidly oscillating phases. Several strategies to cope with these problems are discussed, including various approximations to determine the momentum shift associated with a nonadiabatic transition, the on-the-fly generation of new trajectories at curve-crossings, and the localization of trajectories after irreversible electronic transitions. Employing several multidimensional model systems describing ultrafast photoinduced electron transfer and internal conversion, detailed numerical studies are performed which are compared to exact quantum calculations as well as to the ''fewest-switches'' surface-hopping method. In all cases under consideration, the Liouville calculations are in good agreement with the quantum reference. In particular, the approach is shown to provide a correct quantum-classical description of the electronic coherence.

A mixed quantum-classical Liouville study of the population dynamics in a model photo-induced condensed phase electron transfer reaction

“…The last several decades have witnessed the development of an array of methods for describing the quantum dynamics of these systems, including mean-field and surface-hopping schemes, 1-7 semi-classical path integral methods, [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and algorithms based on the quantum-classical Liouville equation (QCLE). [23][24][25][26][27][28][29][30][31][32][33][34][35] For systems of considerable size, those methods which treat a few crucial DOF (e.g., a) gabriel.hanna@ualberta.ca transferring protons and electrons, high-frequency vibrational modes, photoexcited electronic DOF of a chromophore, exciton states of a light harvesting system) fully quantum mechanically and their environment (e.g., remainder of chromophore, protein backbone, solvent) at a lower level of approximation, can provide computationally feasible and accurate algorithms for simulating nonadiabatic processes. For example, many surface-hopping schemes, mean-field methods, 17,36,37 and some semi-classical path integral formulations adopt a mixed quantum-classical approach in which a few DOF of primary interest are evolved quantum mechanically, while the rest are evolved according to Newton's equations of motion.…”

Mapping quantum-classical Liouville equation: Projectors and trajectories

“…(42) and (45) In this Appendix we establish the equality given in Eq. (52) that shows iL m commutes with the projection operator P. Inserting the definitions of B m (X ), B P m (X ), ρ m (X ) and ρ P m (X ) given in Eqs.…”

Simulation of environmental effects on coherent quantum dynamics in many-body systems

“…The evolution of these functions is then approximated by the dynamics of coupled classical trajectory ensembles. Similar approaches have been pursued by Kapral, Ciccotti, and co-workers, [20][21][22][23] Schofield and co-workers, 24, 25 Ando and co-workers, 26,27 Stock and co-workers, 28 and others.…”