Spin—boson system

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. 

D. Mac Kernan, G. Ciccotti, and R. Kapral. Surface-hopping dynamics of a spin-boson system. J. Chem. Phys., 116(6) 2346-2353, 2002. 

D. McKernan, R. Kapral, and G. Ciccotti (2002) Surface-Hopping Dynamics of a Spin-Boson System. J. Chem. Phys. 116, pp. 2346-2353... 

In Fig. 4 we show results for a spin-boson system at low temperature and large Kondo parameter where the linearization approximation is expected to do most poorly. Indeed, in this situation we see a significant discrepancy between the results of our linearized calculations and exact values. The linearized path integral approximation overemphasizes the effect of the friction, and underestimates the importance of the coherent dynamics. Thus the exact result oscillates around zero while the linearized approximate result is overdamped and shows slow incoherent decay. 

The coupling to the boson bath can change this in a dramatic way because initial levels of the combined spin-boson system are coupled to a continuum of other levels. Indeed Fig. 12.1 can be redrawn in order to display this feature, as seen in Fig. 12.4. Two continuous manifolds of states are seen, seating on level 1 and 2, that encompass the states l,v ) = 1) na Iv ) and 2,v) = 2) Ha Pa) with zero-order energies TTpv and 2.v, respectively, where... 

In the rest of this section we give an in-depth description of how the optimized blocking is done for spin-boson systems. To make the algorithm more accessible to those readers who wish to implement similar methods, we have provided all the necessary details in this section. For those readers who do not wish to go through the mathematics in detail, we give a very concise overview of the essential ideas now. 

Second, canonical transformation methods may be employed to diagonalize the system-bath Hamiltonian partially by a transformation to new ( dressed ) coordinates. Such methods have been in wide use in solid-state physics for some time, and a large repertoire of transformations for different situations has been developed [101]. In the case of a linearly coupled harmonic bath, the natural transformation is to adopt coordinates in which the oscillators are displaced adiabatically as a function of the system coordinates. This approach, known in solid-state physics as the small-polaron transformation [102], has been used widely and successfully in many contexts. In particular, Harris and Silbey demonstrated that many important features of the spin-boson system can be captured analytically using a variationally optimized small-polaron transformation [45-47]. As we show below, the effectiveness of this technique can be broadened considerably when a collective bath coordinate is first included in the system directly. 

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-... 

Figure 5. Normalized solvation free energy of a two-oscillator spin-boson system as a function of the bath cutoff frequency,

J uj) can be assumed to be a smooth function determined by few parameters. These parameters can be determined from a classical molecular dynamics simulation. Once one knows J uj) and V, one can calculate all the properties for the spin-boson system. As a matter of fact, the parameter V is not very important it appears only in a prefactor which multiplies the electron transfer rate. The simple dependence results from an application of Fermi s golden rule, an approximation which appears to be valid in case of electron transfer [7j. 

J uj) in the spin-boson system can be characterized in molecular dynamics simulations through the energy-energy correlation function, as discussed in [7, 8] ... 

To show how problematic the interaction between quantum and classical systems actually is, we consider the interaction between a two-level system and an oscillator. This is commonly known as the spin-boson system, but we take a different approach here than that commonly found in the literature. The proton can be in two states the keto state K) and the enol state E). The unperturbed Hamiltonian is given by... 

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