Spin-boson models

There is a vast field in chemistry where the spin-boson model can serve practical purposes, namely, the exchange reactions of proton transfer in condensed media [Borgis et al. 1989 Suarez and Silbey 1991a Borgis and Hynes 1991 Morillo et al. 1989 Morillo and Cukier 1990]. 

Thus the promoting vibrations reduce the Franck-Condon factor itself, which is not reflected in the spin-boson model (5.55), (5.67). As an illustration, three-dimensional trajectories for various interrelations between symmetric (Ws) and antisymmetric (oja) vibration frequencies, and odo are shown in fig. 33. 

In order to get a first impression on the performance of the QC Liouville approach, it is instructive to start with a simple one-mode spin-boson model, that is. Model IVa [205]. In what follows, the QCL calculations used the first-order Trotter scheme (61) with a time step 8r = 0.05 fs. If not noted otherwise, we have employed the momentum-jump approximation (59) and the initial number of random walkers employed was N = 50 000. 

Although the classical mapping formulation yields the correct quantum-mechanical level density in the special case of a one-mode spin-boson model, the classical approximation deteriorates for mulhdimensional problems, since the classical oscillators may transfer their ZPE. As a hrst example. Fig. 21a compares Nc E) as obtained for Model I in the limiting cases y = 0 and 1 (thin solid lines) to the exact quantum-mechanical density N E) (thick line). The classical level density is seen to be either much higher (for y = 1) or much lower (for y = 0) than the quantum result. Since the integral level density can be... 

S2 conical intersection in pyrazine as well as several spin-boson models, we discuss advantages and problems of this semiclassical method. 

Indeed, consider the spin-boson model, with the Hamiltonian... 

Wilhelm, F. K., Kleff, S., and von Delft, J. (2004). The spin-boson model with a structured environment A comparison of approaches. Chem. Phys., 296 345. 

There is a vast field in chemistry where the spin-boson model can serve practical purposes, namely, proton exchange reactions in condensed media [Borgis and Hynes, 1991 Borgis et al., 1989 Morillo et al., 1989 Morillo and Cukier, 1990 Suarez and Silbey, 1991], The early approaches to this model used a perturbative expansion for weak coupling [Silbey and Harris, 1983], Generally speaking, perturbation theory allows one to consider a TLS coupled to an arbitrary bath via the term ftrz, where / is an operator that acts on the bath variables. The equations of motion in the Heisenberg representation for the operators, daldt = i[H, ], have the form... 

Thus, the promoting vibrations reduce the Franck-Condon factor itself, which is not reflected in the spin-boson model of (5.56) and (5.68). As an illustration, three-dimensional trajectories for various interrelations between symmetric (vibration frequencies and w0 are shown in Figure 5.2 When both vibrations have high frequencies, wa,s w0 the transition proceeds along the MEP (curve 1). In the opposite case of low frequencies, tua s < a>0, the tunneling occurs in the barrier, which is lowered and shortened by the symmetrically coupled vibration qs, so that the position of the antisymmetrically coupled... 

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. 

Simulations using this algorithm [40] and the Trotter-based scheme [38] are able to reproduce the exact quantum results for the spin-boson model, verifying its utility. 

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. 

The chapter is organized as follows The quantum-classical Liouville dynamics scheme is first outlined and a rigorous surface hopping trajectory algorithm for its implementation is presented. The iterative linearized density matrix propagation approach is then described and an approach for its implementation is presented. In the Model Simulations section the comparable performance of the two methods is documented for the generalized spin-boson model and numerical convergence issues are mentioned. In the Conclusions we review the perspectives of this study. 

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 ... 

As noted earlier, the fundamental equations of the QCL dynamics approach are exact for this model, however, in order to implement these equations in the approach detailed in section 2 the momentum jump approximation of Eq.(14) is made in addition to the Trotter factorization of Eq.(12). Both approximations become more accurate as the size of the time step 5 is reduced. Consequently, the results presented below primarily serve as tests of the validity and utility of the momentum-jump approximation. For a discussion of other simulation schemes for QCL dynamics see Ref. [21] in this volume. The linearized approximate propagator is not exact for the spin-boson model. However when used as a short time approximation for iteration as outlined in section 3 the approach can be made accurate with a sufficient number of iterations [37]. 

The asymmetric spin boson model presents a significantly more challenging non-adiabatic condensed phase test problem due to the asymmetry in forces from the different surfaces. Approximate mean field methods, for example, will fail to reliably capture the effects of these different forces on the dynamics. 

Fig. 2 B(t) = (az)(t) versus time for the asymmetric spin-boson model with (3 = 25, = 0.13 and Q = 0.4, e = 0.4. (Top) Comparison of exact quantum results (filled circles), ILDM simulations (small open circles), and QCL dynamics (filled triangles). Both ILDM and QCL simulations were carried out for an ensemble of 2 X 106 trajectories and no filters are employed. (Bottom) Convergence of TQCL dynamics with ensemble size 2 X 104 (filled squares) and 1 x 106 (filled triangles). Exact quantum results (filled circles). A filter parameter of Z = 500 is used for these calculations. ...

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 ... 

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