Spin-boson model system

In contrast to the subsystem representation, the adiabatic basis depends on the environmental coordinates. As such, one obtains a physically intuitive description in terms of classical trajectories along Born-Oppenheimer surfaces. A variety of systems have been studied using QCL dynamics in this basis. These include the reaction rate and the kinetic isotope effect of proton transfer in a polar condensed phase solvent and a cluster [29-33], vibrational energy relaxation of a hydrogen bonded complex in a polar liquid [34], photodissociation of F2 [35], dynamical analysis of vibrational frequency shifts in a Xe fluid [36], and the spin-boson model [37,38], which is of particular importance as exact quantum results are available for comparison. 

The last term involves derivatives with respect to both mapping and environmental variables. Its contribution is difficult to compute. Calculations on the spin-boson model have shown that even if the last term is neglected, excellent agreement with the exact results for a wide range of system parameters is obtained [53]. 

The main goal in the development of mixed quantum classical methods has as its focus the treatment of large, complex, many-body quantum systems. While applications to models with many realistic elements have been carried out [10,11], here we test the methods and algorithms on the spin-boson model, which is the standard test case in this field. In particular, we focus on the asymmetric spin-boson model and the calculation of off-diagonal density matrix elements, which present difficulties for some simulation schemes. We show that both of the methods discussed here are able to accurately and efficiently simulate this model. 

In this section we present results using the two approaches described in the previous sections the Trotter factorized QCL (TQCL), and iterative linearized density matrix (ILDM) propagation schemes, to study the spin-boson model consisting of a two level system that is bi-linearly coupled to a bath with Mh harmonic modes. This popular model of a quantum system embedded in an environment is described by the following general hamiltonian ... 

In this section we present some applications of the LAND-map approach for computing time correlation functions and time dependent quantum expectation values for realistic model condensed phase systems. These representative applications demonstrate how the methodology can be implemented in general and provide challenging tests of the approach. The first test application is the spin-boson model where exact results are known from numerical path integral calculations [59-62]. The second system we study is a fully atomistic model for excess electronic transport in metal - molten salt solutions. Here the potentials are sufficiently reliable that findings from our calculations can be compared with experimental results. 

Here we apply the LAND-map approach to compute of the time dependent average population difference, A t) = az t)), between the spin states of a spin-boson model. Here az = [ 1)(1 — 2)(2 ]. Within the limits of linear response theory, this model describes the dissipative dynamics of a two level system coupled to an environment [59,63-65]. The environment is represented by an infinite set of harmonic oscillators, linearly coupled to the quantum subsystem. The characteristics of the system-bath coupling are completely described by the spectral density J(w). In the following, we shall restrict ourselves to the case of an Ohmic spectral density... 

The prominence of these quantum dynamical models is also exemplified by the abundance of theoretical pictures based on the spin-boson model—a two (more generally a few) level system coupled to one or many harmonic oscillators. Simple examples are an atom (well characterized at room temperature by its ground and first excited states, that is, a two-level system) interacting with the radiation field (a collection of harmonic modes) or an electron spin interacting with the phonon modes of a surrounding lattice, however this model has found many other applications in a variety of physical and chemical phenomena (and their extensions into the biological world) such as atoms and molecules interacting with the radiation field, polaron formation and dynamics in condensed environments. 

Fig. 12.1 The spin-boson model for a two-level molecule coupled to a system of harmonic oscillators.

Equations (12.28) and (12.29) describe different spin-boson models that are commonly used to describe the dynamics of a two-level system interacting with a boson bath. Two comments are in order ... 

In the rest of this chapter we focus on the pure vibrational relaxation problem, that is, on the relaxation of an oscillator (usually modeled as harmonic) coupled to its thermal environment. From the theoretical point of view this problem supplements the spin boson model considered in Chapter 12. Indeed, these are the N — 2 and the N = oo limits of an A level system. For a harmonic oscillator there is an added feature, that these states are equally spaced. In this case individual... 

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 single-set formulation is to be preferred when the different electronic states have a similar form. The well known spin-boson model is in this category. When the motion on the different electronic states is very different, the multi-set formulation becomes the appropriate choice. The smaller number of SPFs needed for convergence then over-compensates the overhead of the multi-set formulation. For vibronically coupled systems, we have always chosen the multi-set formulation. 

To demonstrate the accuracy of eqn (12.26) together with eqn (12.30) and its boundary condition eqn (12.31), called the quantum Kramers-like theory, we use a symmetric spin-boson model as a concrete system. In this benchmark model, the nuclear vibrational motions are characterized by the Ohmic spectral density ... 

The spin-boson model can be generalized to any number of electronic states - two being the most common. It represents a minimal model for ET reactions. Many theoretical models of ET systems, including Marcus theory, are essentially variants of this model. 

Using bosonization procedures and real-time path integral techniques, the quantum conductance of a rigid (i.e., no lattice distortion) ID quantum wire has been computed. Analytical theories predict that, in the presence of even a single defect, the conductance of a strictly ID quantum wire would vanish at 0 K. Bosonization of this system yields an equivalent spin-boson model with an infinite number of tight-binding states, which turns out to be the same model as for CTCs discussed in... 

Abstract Photoinduced processes in extended molecular systems are often ultrafast and involve strong electron-vibration (vibronic) coupling effects which necessitate a non-perturbative treatment. In the approach presented here, high-dimensional vibrational subspaces are expressed in terms of effective modes, and hierarchical chains of such modes which sequentially resolve the dynamics as a function of time. This permits introducing systematic reduction procedures, both for discretized vibrational distributions and for continuous distributions characterized by spectral densities. In the latter case, a sequence of spectral densities is obtained from a Mori/Rubin-type continued fraction representation. The approach is suitable to describe nonadiabatic processes at conical intersections, excitation energy transfer in molecular aggregates, and related transport phenomena that can be described by generalized spin-boson models. 

We consider a class of generalized spin-boson models, assuming that a system-bath partitioning is appropriate to describe the high-dimensional system of interest. 

Even though there appear many variables and parameters in the equations above, ultimately the spin-boson model, as advertised in [13], is characterized completely by a well defined average property of the system, the spectral function J oj)... 

The key new aspect of our investigation is two-fold first, we base all model parameters on molecular dynamics simulations second, the spin-boson model allows one to account for a very large number of vibrations quantum mechanically. We have demonstrated that the spin-boson model is well suited to describe the coupling between protein motion and electron transfer in biological redox systems. The model, through the spectral function can be matched to correlation functions of the redox energy... 

Although the systems discussed so far exhibit fairly complex vibrational and diabatic electronic relaxation dynamics, their adiabatic population dynamics is relatively simple and thus is well-suited for a SH description. To provide a challenge for the SH method, we next consider various spin-boson-type models, which may give rise to a quite comphcated adiabatic population dynamics. 

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

We have presented some of the most recent developments in the computation and modeling of quantum phenomena in condensed phased systems in terms of the quantum-classical Liouville equation. In this approach we consider situations where the dynamics of the environment can be treated as if it were almost classical. This description introduces certain non-classical features into the dynamics, such as classical evolution on the mean of two adiabatic surfaces. Decoherence is naturally incorporated into the description of the dynamics. Although the theory involves several levels of approximation, QCL dynamics performs extremely well when compared to exact quantum calculations for some important benchmark tests such as the spin-boson system. Consequently, QCL dynamics is an accurate theory to explore the dynamics of many quantum condensed phase systems. 

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