Bath of harmonic oscillators

Although Eqs. (4-1) and (4-2) have identical expressions as that of the classical rate constant, there is no variational upper bound in the QTST rate constant because the quantum transmission coefficient Yq may be either greater than or less than one. There is no practical procedure to compute the quantum transmission coefficient Yq- For a model reaction with a parabolic barrier along the reaction coordinate coupled to a bath of harmonic oscillators, the quantum transmission... 

The strategy, usually adopted to achieve a theoretical description of this complex dynamics, is to describe the influence of the solvent environment on the electron-transfer reaction within linear response theory [5, 26, 196, 197] as linear coupling to a bath of harmonic oscillators. Within this model, all properties of the bath enter through a single function called the spectral density [5, 168]... 

Niunerical algorithms for solving the GLE are readily available. Only recently, Hershkovitz has developed a fast and efficient 4th order Runge-Kutta algorithm. Memory friction does not present any special problem, especially when expanded in terms of exponentials, since then the GLE can be represented as a finite set of memoiy-less coupled Langevin equations. " Alternatively (see also the next subsection), one can represent the GLE in terms of its Hamiltonian equivalent and use a suitable discretization such that the problem becomes equivalent to that of motion of the reaction coordinate coupled to a finite discrete bath of harmonic oscillators. ... 

Q vibration is not directly coupled to the bath of harmonic oscillators. This assumption is similar to the approach employed by Silbey and Suarez who used a tunneling splitting that depends on the oscillating transfer distance Q in their spin-boson Hamiltonian. Borgis and Hynes, too, have made this assumption in the context of Marcus theory. 

Another striking example of entanglement-mediated dynamics was presented in Ref. [94]. There we considered a two-level system that first reaches the state of equilibrium with a bath of harmonic oscillators, a state in which large... 

Hereafter we put /ig = 1. Below we express our results in terms of the statistical properties (correlators) of the environment s noise, X(t). Depending on the physical situation at hand, one can choose to model the environment via a bath of harmonic oscillators [6, 3]. In this case the generalized coordinate of the reservoir is defined as X = ]T)Awhere xi are the coordinate operators of the oscillators and Aj are the respective couplings. Eq. 2 is then referred to as the spin-boson Hamiltonian [8]. Another example of a reservoir could be a spin bath [11] 5. However, in our analysis below we do not specify the type of the environment. We will only assume that the reservoir gives rise to markovian evolution on the time scales of interest. More specifically, the evolution is markovian at time scales longer than a certain characteristic time rc, determined by the environment 6. We assume that rc is shorter than the dissipative time scales introduced by the environment, such as the dephasing or relaxation times and the inverse Lamb shift (the scale of the shortest of which we denote as Tdiss, tc [Pg.14]

All of the above methods are easily extended to more degrees of freedom. There are also approaches that rely on a certain form of multidimensional Hamiltonian which should be represented as some system coupled to a bath of harmonic oscillators, as described in Section 2.2. In Chapter 5 we see that integration over these bath... 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

In order to describe the state of the solvent, we represent it as a bath of harmonic oscillators, which interact linearly with the reactant. The corresponding Hamiltonian is written in the form ... 

Here we discuss in detail a model for measurement-induced decay modification in a multilevel system. The system with energies frwn, 1 < n < N, is coupled to a zero-temperature bath of harmonic oscillators with frequencies uj. The corresponding Hamiltonian, in the rotating-wave approximation, is... 

We consider a two-level system coupled to a bath of harmonic oscillators that will be referred to as a boson field. Two variations of this model, which differ from each other by the basis used to describe the two-level system, are frequently encountered. In one, the basis is made of the eigenstates of the two-state Hamiltonian that describes the isolated system. The full Hamiltonian is then written... 

We note that the reaction coordinate is coupled to an infinite bath of harmonic oscillators, which represent the bulk protein, and to a protein promoting vibration. For each mathematical implementation, we here choose the zero of promoting vibration coupling to be in the well rather than at the barrier top, but this is arbitrary. We point out that we can tune this model to allow for both sequential and concerted hydrogen-electron transfer. Sequential transfer is found with a very high transfer rate, and concerted with a lower one. 

We address ourselves here to the case of an asymmetric two-level system (TLS) coupled linearly to a bath of harmonic oscillators, the spin-boson system. This system has been studied extensively as the prototype of a quantum system in a dissipative environment [13, 14, 20, 45-47, 103-... 

Another use for standard models is as a target. It is important to determine at what point the model breaks down and whether that point is significant in realistic chemical dynamics. Some of the more important developments in the tests of Grote-Hynes theory have been in the application of variational transition state theory (VTST) to models of solution reaction dynamics. The origin of the use of VTST in solution dynamics is in the observation that the GLE can be equivalently formulated in Hamiltonian terms by a reaction coordinate coupled to a bath of harmonic oscillators. It has been shown by van der... 

The Golden Rule formula (9.5) for the mean rate constant assumes the Unear response regime of solvent polarization and is completely equivalent in this sense to the result predicted by the spin-boson model, where a two-state electronic system is coupled to a thermal bath of harmonic oscillators with the spectral density of relaxation J(o)) [38,71]. One should keep in mind that the actual coordinates of the solvent are not necessarily harmonic, but if the collective solvent polarization foUows the Unear response, the system can be effectively represented by a set of harmonic oscillators with the spectral density derived from the linear response function [39,182]. Another important point we would like to mention is that the Golden Rule expression is in fact equivalent [183] to the so-called noninteracting blip approximation [71] often used in the context of the spin-boson model. The perturbation theory can be readily applied to... 

The stochastic mean-field [20] (SMF) method simultaneously resolves the following two major issues with NA MD. First, decoherence effects within the quantum subsystem that take place due to its interaction with an environment are included. Second, decoherence naturally leads to the asymptotic branching of NA trajectories. That is, the implementation of the decoherence effect in the SMF approach automatically resolves the branching problem. By extending the ordinary quantum-classical MF approximation, the SMF approach accounts for the quantum features of the environment in the Lindblad formulation. The Lindblad formulation is exact for a bath of harmonic oscillators and is an approximation for other types of solvents. While the quantum nature of the environment is treated by SMF within an approximation, its classical properties are included exactly by classical MD with a true Hamiltonian. 

The dissipative transverse-field Ising chain consists of a one-dimensional transverse-field Ising model, as discussed in the first example above, with each spin coupled to a heat bath of harmonic oscillators. The Hamiltonian reads... 

To this date, no stable simulation methods are known which are successful at obtaining quantum dynamical properties of arbitrary many-particle systems over long times. However, significant progress has been made recently in the special case where a low-dimensional nonlinear system is coupled to a dissipative bath of harmonic oscillators. The system-bath model can often provide a realistic description of the effects of common condensed phase environments on the observable dynamics of the microscopic system of interest. A typical example is that of an impurity in a crystalline solid, where the harmonic bath arises naturally from the small-amplitude lattice vibrations. The harmonic picture is often relevant even in situations where the motion of individual solvent atoms is very anhaimonic in such cases validity of the linear response approximation can lead to Gaussian behavior of appropriate effective modes by virtue of the central limit theorem. ... 

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