Computer simulation of quantum dynamics in a classical spin environment

In this paper, a formalism for studying the dynamics of quantum systems coupled to classical spin environments is reviewed. The theory is based on generalized antisymmetric brackets and naturally predicts open-path off-diagonal geometric phases in the evolution of the density matrix. It is shown that such geometric phases must also be considered in the quantum–classical Liouville equation for a classical bath with canonical phase space coordinates; this occurs whenever the adiabatics basis is complex (as in the case of a magnetic field coupled to the quantum subsystem). When the quantum subsystem is weakly coupled to the spin environment, non-adiabatic transitions can be neglected and one can construct an effective non-Markovian computer simulation scheme for open quantum system dynamics in classical spin environments. In order to tackle this case, integration algorithms based on the symmetric Trotter factorization of the classical-like spin propagator are derived. Such algorithms are applied to a model comprising a quantum two-level system coupled to a single classical spin in an external magnetic field. Starting from an excited state, the population difference and the coherences of this two-state model are simulated in time while the dynamics of the classical spin is monitored in detail. It is the author’s opinion that the numerical evidence provided in this paper is a first step toward developing the simulation of quantum dynamics in classical spin environments into an effective tool. In turn, the ability to simulate such a dynamics can have a positive impact on various fields, among which, for example, nanoscience.     

Transport properties of quantum-classical systems

Correlation function expressions for calculating transport coefficients for quantum-classical systems are derived. The results are obtained by starting with quantum transport coefficient expressions and replacing the quantum time evolution with quantum-classical Liouville evolution, while retaining the full quantum equilibrium structure through the spectral density function. The method provides a variety of routes for simulating transport coefficients of mixed quantum-classical systems, composed of a quantum subsystem and a classical bath, by selecting different but equivalent time evolution schemes of any operator or the spectral density. The structure of the spectral density is examined for a single harmonic oscillator where exact analytical results can be obtained. The utility of the formulation is illustrated by considering the rate constant of an activated quantum transfer process that can be described by a many-body bath reaction coordinate.     

Generalization of classical mechanics for nuclear motions on nonadiabatically coupled potential energy surfaces in chemical reactions

Classical trajectory study of nuclear motion on the Born-Oppenheimer potential energy surfaces is now one of the standard methods of chemical dynamics. In particular, this approach is inevitable in the studies of large molecular systems. However, as soon as more than a single potential energy surface is involved due to nonadiabatic coupling, such a naive application of classical mechanics loses its theoretical foundation. This is a classic and fundamental issue in the foundation of chemistry. To cope with this problem, we propose a generalization of classical mechanics that provides a path even in cases where multiple potential energy surfaces are involved in a single event and the Born-Oppenheimer approximation breaks down. This generalization is made by diagonalization of the matrix representation of nuclear forces in nonadiabatic dynamics, which is derived from a mixed quantum-classical representation of the electron-nucleus entangled Hamiltonian [Takatsuka, K. J. Chem. Phys. 2006, 124, 064111]. A manifestation of quantum fluctuation on a classical subsystem that directly contacts with a quantum subsystem is discussed. We also show that the Hamiltonian thus represented gives a theoretical foundation to examine the validity of the so-called semiclassical Ehrenfest theory (or mean-field theory) for electron quantum wavepacket dynamics, and indeed, it is pointed out that the electronic Hamiltonian to be used in this theory should be slightly modified.     

Ehrenfest dynamics is purity non-preserving: A necessary ingredient for decoherence

We discuss the evolution of purity in mixed quantum/classical approaches to electronic nonadiabatic dynamics in the context of the Ehrenfest model. As it is impossible to exactly determine initial conditions for a realistic system, we choose to work in the statistical Ehrenfest formalism that we introduced in Alonso et al. [J. Phys. A: Math. Theor. 44, 396004 (2011)]. From it, we develop a new framework to determine exactly the change in the purity of the quantum subsystem along with the evolution of a statistical Ehrenfest system. In a simple case, we verify how and to which extent Ehrenfest statistical dynamics makes a system with more than one classical trajectory, and an initial quantum pure state become a quantum mixed one. We prove this numerically showing how the evolution of purity depends on time, on the dimension of the quantum state space D, and on the number of classical trajectories N of the initial distribution. The results in this work open new perspectives for studying decoherence with Ehrenfest dynamics.     

Simulating quantum dynamics in classical environments

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 systems 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 approximation. 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 trajectory 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.     

Measuring nonadiabaticity of molecular quantum dynamics with quantum fidelity and with its efficient semiclassical approximation

We propose to measure nonadiabaticity of molecular quantum dynamics rigorously with the quantum fidelity between the Born-Oppenheimer and fully nonadiabatic dynamics. It is shown that this measure of nonadiabaticity applies in situations where other criteria, such as the energy gap criterion or the extent of population transfer, fail. We further propose to estimate this quantum fidelity efficiently with a generalization of the dephasing representation to multiple surfaces. Two variants of the multiple-surface dephasing representation (MSDR) are introduced, in which the nuclei are propagated either with the fewest-switches surface hopping or with the locally mean field dynamics (LMFD). The LMFD can be interpreted as the Ehrenfest dynamics of an ensemble of nuclear trajectories, and has been used previously in the nonadiabatic semiclassical initial value representation. In addition to propagating an ensemble of classical trajectories, the MSDR requires evaluating nonadiabatic couplings and solving the Schro?dinger (or more generally, the quantum Liouville-von Neumann) equation for a single discrete degree of freedom. The MSDR can be also used in the diabatic basis to measure the importance of the diabatic couplings. The method is tested on three model problems introduced by Tully and on a two-surface model of dissociation of NaI.     

A local coherent-state approximation to system-bath quantum dynamics

A novel quantum method to deal with typical system-bath dynamical problems is introduced. Subsystem discrete variable representation and bath coherent-state sets are used to write down a multiconfigurational expansion of the wave function of the whole system. With the help of the Dirac-Frenkel variational principle, simple equations of motion--a kind of Schrodinger-Langevin equation for the subsystem coupled to (pseudo) classical equations for the bath--are derived. True dissipative dynamics at all times is obtained by coupling the bath to a secondary, classical Ohmic bath, which is modeled by adding a friction coefficient in the derived pseudoclassical bath equations. The resulting equations are then solved for a number of model problems, ranging from tunneling to vibrational relaxation dynamics. Comparison of the results with those of exact, multiconfiguration time-dependent Hartree calculations in systems with up to 80 bath oscillators shows that the proposed method can be very accurate and might be of help in studying realistic problems with very large baths. To this end, its linear scaling behavior with respect to the number of bath degrees of freedom is shown in practice with model calculations using tens of thousands of bath oscillators.     

A quantum-classical approach to the photoabsorption spectrum of pyrazine

We have used the time-dependent discrete variable representation (TDDVR) method to simulate the photoabsorption spectrum of pyrazine. The time-dependent molecular dynamics of pyrazine after excitation to the S2 electronic state is considered as a benchmark to investigate the S2 absorption spectrum. We have carried out the dynamics on a basic four-mode model of pyrazine with the inclusion of five major modes as well as the rest of the vibrational modes as bath modes. Investigations reveal the effect of bath modes such as energy and population transfer from the subsystem to the bath. Calculated results demonstrate excellent agreement with traditional quantum-mechanical findings during the entire propagation and converge to the exact quantum results when enough gridpoints are used. It appears that TDDVR, as a numerical quantum dynamics methodology, is a good compromise between accuracy and speed.     

Non-adiabatic molecular dynamics with quantum solvent effects

Three novel approaches extending quantum-classical non-adiabatic (NA) molecular dynamics (MD) to include quantum effects of solvent environments are described. In a standard NA-MD the solute subsystem is treated quantum mechanically, while the larger solvent part of a system is treated classically. The three novel approaches presented here are based on the Bohmian formulation of quantum mechanics, the stochastic Schrödinger equation for the evolution of open quantum systems and the quantized Hamilton dynamics generalization of classical mechanics. The approaches extend the standard NA-MD to incorporate the following quantum effects of the solvent. (1) Branching, i.e. the ability of solvent quantum wave packets to split and follow asymptotically diverging trajectories correlated with different quantum states of the solute. (2) Decoherence, i.e. loss of quantum interference within the solute subsystem induced by the diverging solvent trajectories. (3) Zero point energy that contributes to NA coupling and must be preserved during the energy exchange between solvent and solute degrees of freedom. The Bohmian quantum-classical mechanics, stochastic mean-field and quantized mean-field approximations incorporate the quantum solvent effects into the standard quantum-classical NA-MD in a straightforward and efficient way that can be easily applied to quantum dynamics of condensed phase chemical systems.     

Hybrid quantum/classical molecular dynamics for a proton transfer reaction coupled to a dissipative bath

A hybrid quantum/classical molecular dynamics approach is applied to a proton transfer reaction represented by a symmetric double well system coupled to a dissipative bath. In this approach, the proton is treated quantum mechanically and all bath modes are treated classically. The transition state theory rate constant is obtained from the potential of mean force, which is generated along a collective reaction coordinate with umbrella sampling techniques. The transmission coefficient, which accounts for dynamical recrossings of the dividing surface, is calculated with a reactive flux approach combined with the molecular dynamics with quantum transitions surface hopping method. The hybrid quantum/classical results agree well with numerically exact results in the spatial-diffusion-controlled regime, which is most relevant for proton transfer in proteins. This hybrid quantum/classical approach has already been shown to be computationally practical for studying proton transfer in large biological systems. These results have important implications for future applications to hydrogen transfer reactions in solution and proteins.     

Generalized CC-TDSCF and LCSA: The system-energy representation

Typical (sub)system-bath quantum dynamical problems are often investigated by means of (approximate) reduced equations of motion. Wavepacket approaches to the dynamics of the whole system have gained momentum in recent years and there is hope that properly designed approximations to the wavefunction will allow one to correctly describe the subsystem evolution. The continuous-configuration time-dependent self-consistent field (CC-TDSCF) and local coherent-state approximation (LCSA) methods, for instance, use a simple Hartree product of bath single-particle-functions for each discrete variable representation (DVR) state introduced in the Hilbert space of the subsystem. Here we focus on the above two methods and replace the DVR states with the eigenstates of the subsystem Hamiltonian, i.e., we adopt an energy-local representation for the subsystem. We find that stable and semiquantitative results are obtained for a number of dissipative problems, at the same (small) computational cost of the original methods. Furthermore, we find that both methods give very similar results, thus suggesting that coherent-states are well suited to describe (local) bath states. As a whole, present results highlight the importance of the system basis-set in the selected-multiconfiguration expansion of the wavefunction. They suggest that accurate and yet computationally cheap methods may be simply obtained from CC-TDSCF/LCSA by letting the subsystem states be variationally optimized.