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Computer simulation of nonadiabatic dynamics by means of the quantum-classical Liouville equation.

Daniel A. Uken

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

Simulation of quantum dynamics for many-body systems is an open area of research. For interacting many-body quantum systems, the computer memory necessary to perform calculations has an astronomical value, so that approximated models are needed to reduce the required computational resources. A useful approximation that can often be made is that of quantum-classical dynamics, where the majority of the degrees are treated classically, while a few of them must be treated quantum mechanically. When energy is exchanged very quickly between the quantum subsystem and classical environment, the dynamics is nonadiabatic. Most theories for nonadiabatic dynamics are unsatisfactory, as they fail to properly describe the quantum backreaction of the subsystem on the environment. However, an approach based on the quantum-classical Liouville equation solves this problem. Even so, nonadiabatic dynamics is difficult to implement on a computer, and longer simulation times are often inaccessible due to statistical error. There is thus a need for improved algorithms for nonadiabatic dynamics. In this thesis, two algorithms that utilise the quantum-classical Liouville equation will be qualitatively and quantitatively compared. In addition, stochastic sampling schemes for nonadiabatic transitions will be studied, and a new sampling scheme is introduced [D. A. Uken et al., Phys. Rev. E. 88, 033301 (2013)] which proves to have a dramatic advantage over existing techniques, allowing far longer simulation times to be calculated reliably.

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Simulation of quantum dynamics for many-body systems is an open area of research. For interacting many-body quantum systems, the computer memory necessary to perform calculations has an astronomical value, so that approximated models are needed to reduce the required computational resources. A useful approximation that can often be made is that of quantum-classical dynamics, where the majority of the degrees are treated classically, while a few of them must be treated quantum mechanically. When energy is exchanged very quickly between the quantum subsystem and classical environment, the dynamics is nonadiabatic. Most theories for nonadiabatic dynamics are unsatisfactory, as they fail to properly describe the quantum backreaction of the subsystem on the environment. However, an approach based on the quantum-classical Liouville equation solves this problem. Even so, nonadiabatic dynamics is difficult to implement on a computer, and longer simulation times are often inaccessible due to statistical error. There is thus a need for improved algorithms for nonadiabatic dynamics. In this thesis, two algorithms that utilise the quantum-classical Liouville equation will be qualitatively and quantitatively compared. In addition, stochastic sampling schemes for nonadiabatic transitions will be studied, and a new sampling scheme is introduced [D. A. Uken et al., Phys. Rev. E. 88, 033301 (2013)] which proves to have a dramatic advantage over existing techniques, allowing far longer simulation times to be calculated reliably.
The simulation of the dynamics of quantum systems is very difficult, due to the fact that, in general, it cannot be calculated exactly for interacting many-body systems. Brute force simulations of quantum dynamics are simply not feasible, and approximations need to be made. In many instances a quantum system can be approximated as a quantum-classical system, where only a subsystem of interest is treated quantum mechanically, and the rest is considered as a classical bath. When energy is free to be exchanged between the subsystem and its environment, the dynamics that occur is said to be nonadiabatic. This type of dynamics is challenging to calculate on a computer, as it can lead to large statistical errors at long times. Hence, there is a need for improved algorithms for nonadiabatic dynamics. In this thesis, a recently introduced nonadiabatic sampling scheme [A. Sergi and F. Petruccione, Phys. Rev. E 81, 032101 (2010)] is used to calculate the long-time dynamics of a model system comprising a quantum spin coupled to a bath of harmonic oscillators. Also, various technical aspects of the algorithm are investigated.
Quantum rate processes in condensed phase systems are often computed by combining quantum and classical descriptions of the dynamics. An algorithm for simulating the quantum-classical Liouville equation, which describes the dynamics of a quantum subsystem coupled to a classical bath, is presented in this paper. The algorithm is based on a Trotter decomposition of the quantum-classical propagator, in conjunction with Monte Carlo sampling of quantum transitions, to yield a surface-hopping representation of the dynamics. An expression for the nonadiabatic propagator that is responsible for quantum transitions and associated bath momentum changes is derived in a form that is convenient for Monte Carlo sampling and exactly conserves the total energy of the system in individual trajectories. The expectation values of operators or quantum correlation functions can be evaluated by initial sampling of quantum states and use of quantum-classical Liouville dynamics for the time evolution. The algorithm is tested by calculations on the spin-boson model, for which exact quantum results are available, and is shown to reproduce the exact results for stronger nonadiabatic coupling and much longer times using fewer trajectories than other schemes for simulating quantum-classical Liouville dynamics.
We study a number of filtering schemes for the reduction of the statistical error in nonadiabatic calculations by means of the quantum-classical Liouville equation. In particular, we focus on a scheme based on setting a threshold value on the sampling weights, so that when the threshold is overcome the value of the weight is reset, and on another approach which prunes the ensemble of the allowed nonadiabatic transitions according to a generalized sampling probability. Both methods have advantages and drawbacks, however, their combination drastically improves the performance of an algorithm known as the sequential short-time step propagation [MacKernan et al., J. Phys: Condens. Matter 14, 9069 (2002)], which is derived from a simple first order expansion of the quantum-classical propagator. Such an algorithm together with the combined filtering procedures produce results that compare very well with those obtained by means of numerically accurate path integral quantum calculations for the spin-boson model, even for intermediate and strong coupling regimes.
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.
Methods for simulating the dynamics of composite systems, where part of the system is treated quantum mechanically and its environment is treated classically, are discussed. Such quantum-classical sys- tems arise in many physical contexts where certain degrees of freedom have an essential quantum character while the other degrees of freedom to which they are coupled may be treated classically to a good approxi- mation. The dynamics of these composite systems are governed by a quantum-classical Liouville equation for either the density matrix or the dynamical variables which are operators in the Hilbert space of the quantum subsystem and functions of the classical phase space variables of the classical environment. Solutions of the evolution equations may be formulated in terms of surface-hopping dynamics involving ensembles of tra- jectory segments interspersed with quantum transitions. The surface-hopping schemes incorporate quantum coherence and account for energy exchanges between the quantum and classical degrees of freedom. Various simulation algorithms are discussed and illustrated with calculations on simple spin-boson models but the methods described here are applicable to realistic many- body environments.
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.