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arXiv:1205.1283 · The Journal of Chemical Physics

Nonadiabatic dynamics in open quantum-classical systems: Forward-backward trajectory solution

Chang‐Yu Hsieh, Raymond Kapral

Published Jul 18, 2012• 112 citations
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Abstract

A new approximate solution to the quantum-classical Liouville equation is derived starting from the formal solution of this equation in forward-backward form. The time evolution of a mixed quantum-classical system described by this equation is obtained in a coherent state basis using the mapping representation, which expresses N quantum degrees of freedom in a 2N-dimensional phase space. The solution yields a simple dynamics in which a set of N coherent state coordinates evolves in forward and backward trajectories, while the bath coordinates evolve under the influence of the mean potential that depends on these forward and backward trajectories. It is shown that the solution satisfies the differential form of the quantum-classical Liouville equation exactly. Relations to other mixed quantum-classical and semi-classical schemes are discussed.

DOI
10.1063/1.4736841

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A new approximate solution to the quantum-classical Liouville equation is derived starting from the formal solution of this equation in forward-backward form. The time evolution of a mixed quantum-classical system described by this equation is obtained in a coherent state basis using the mapping representation, which expresses N quantum degrees of freedom in a 2N-dimensional phase space. The solution yields a simple dynamics in which a set of N coherent state coordinates evolves in forward and backward trajectories, while the bath coordinates evolve under the influence of the mean potential that depends on these forward and backward trajectories. It is shown that the solution satisfies the differential form of the quantum-classical Liouville equation exactly. Relations to other mixed quantum-classical and semi-classical schemes are discussed.
A new approximate solution to the quantum-classical Liouville equation is derived starting from the formal solution of this equation in forward-backward form. The time evolution of a mixed quantum-classical system described by this equation is obtained in a coherent state basis using the mapping representation, which expresses N quantum degrees of freedom in a 2N-dimensional phase space. The solution yields a simple non-Hamiltonian dynamics in which a set of N coherent state coordinates evolve in forward and backward trajectories while the bath coordinates evolve under the influence of the mean potential that depends on these forward and backward tra-jectories. It is shown that the solution satisfies the differential form of the quantum-classical Liouville equation exactly. Relations to other mixed quantum-classical and semi-classical schemes are discussed. 1 ar
The evolution of a mixed quantum-classical system is expressed in the mapping formalism where discrete quantum states are mapped onto oscillator states, resulting in a phase space description of the quantum degrees of freedom. By defining projection operators onto the mapping states corresponding to the physical quantum states, it is shown that the mapping quantum-classical Liouville operator commutes with the projection operator so that the dynamics is confined to the physical space. It is also shown that a trajectory-based solution of this equation can be constructed that requires the simulation of an ensemble of entangled trajectories. An approximation to this evolution equation which retains only the Poisson bracket contribution to the evolution operator does admit a solution in an ensemble of independent trajectories but it is shown that this operator does not commute with the projection operators and the dynamics may take the system outside the physical space. The dynamical instabilities, utility, and domain of validity of this approximate dynamics are discussed. The effects are illustrated by simulations on several quantum systems.
The quantum-classical Liouville equation describes the dynamics of a quantum subsystem coupled to a classical environment. It has been simulated using various methods, notably, surface-hopping schemes. A representation of this equation in the mapping Hamiltonian basis for the quantum subsystem is derived. The resulting equation of motion, in conjunction with expressions for quantum expectation values in the mapping basis, provides another route to the computation of the nonadiabatic dynamics of observables that does not involve surface-hopping dynamics. The quantum-classical Liouville equation is exact for the spin-boson system. This well-known model is simulated using an approximation to the evolution equation in the mapping basis, and close agreement with exact quantum results is found.
Solving quantum dynamics is an exponentially difficult problem. Thus, an exact numerical solution is inaccessible for any condensed matter system.\nA promising approach is to divide the system into a quantum subsystem containing degrees of freedom which are of greater interest or those which have more profound quantum character (e.g., have smaller mass) and a classical bath containing the rest of the system.\nImposing such a partition and treating the bath classically results in quantum-classical dynamics. The quantum-classical Liouville equation is a general equation in the Hilbert space of quantum degrees of freedom while it resides in the phase space of the classical degrees of freedom.\nAny numerical solution to this equation requires representation of the quantum subsystem in some basis. Solutions to this equation have been already proposed in the subsystem, adiabatic and force bases, each with its own cons and pros.\n\nIn this work, the quantum-classical equations of motion are cast in the subsystem basis and subsequently mapped to a number of fictitious harmonic oscillators.\nThe result is quantum-classical dynamics in the mapping basis which treats both quantum and classical degrees of freedom on the same footing, i.e., in phase space. Neglecting a portion of the back reaction of the quantum-subsystem to classical bath results in an expression for the time evolution of an operator (density matrix) equal to its Poisson bracket with the Hamiltonian.\nThis equation can be solved in terms of characteristics to provide a computationally tractable method for calculating quantum-classical dynamical properties. The expressions for expectation values and correlation functions in this formalism are derived.\nCalculations on spin-boson system, barrier crossing models---the so called Tully models---and the Fenna-Mathews-Olson pigments show very good agreement between the results of this method and numerical solutions to the Schrödinger equation.
The quantum-classical Liouville equation provides a description of the dynamics of a quantum subsystem coupled to a classical environment. Representing this equation in the mapping basis leads to a continuous description of discrete quantum states of the subsystem and may provide an alternate route to the construction of simulation schemes. In the mapping basis the quantum-classical Liouville equation consists of a Poisson bracket contribution and a more complex term. By transforming the evolution equation, term-by-term, back to the subsystem basis, the complex term (excess coupling term) is identified as being due to a fraction of the back reaction of the quantum subsystem on its environment. A simple approximation to quantum-classical Liouville dynamics in the mapping basis is obtained by retaining only the Poisson bracket contribution. This approximate mapping form of the quantum-classical Liouville equation can be simulated easily by Newtonian trajectories. We provide an analysis of the effects of neglecting the presence of the excess coupling term on the expectation values of various types of observables. Calculations are carried out on nonadiabatic population and quantum coherence dynamics for curve crossing models. For these observables, the effects of the excess coupling term enter indirectly in the computation and good estimates are obtained with the simplified propagation.
The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.