System-bath model

Suppose the reactive polyatomic molecule of interest can undergo uni-molecular reaction to form several products, and we imagine carrying out a constrained reaction path analysis for each of the product channels. To carry out the analysis of a particular constrained reaction path, Zhao and Rice adopted a system-bath model [74] in which the reaction path coordinate delines the system and all other coordinates constitute the bath. The use of this representation permits the elimination of the bath coordinates, which then increases the efficiency of calculation of the optimal control field for motion along the reaction coordinate. 

However, Eq. (4.1) has another advantage in that it directly connects to the system-bath models used in condensed phase dynamics [38]. Here the reactive coordinates and the substrate modes comprise the relevant system and the bath, respectively. Larger molecules may provide their own bath and Eq. (4.1) can be used to calculate an ah initio system-bath Hamiltonian and microscopic relaxation and dephasing rates [33]. 

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. ... 

In order to study the decoherence effect, we examined the time evolution of a single spin coupled by exchange interaction to an environment of interacting spin bath modeled by the XY-Hamiltonian. The Hamiltonian for such a system is given by [104]... 

Quantum-dynamical modeling of ultrafast processes in complex molecular systems multiconfigurational system-bath dynamics using Gaussian wavepackets... 

The Hamiltonian underlying the calculations of Fig. 1 is based on a diabatic model potential [7-9] of system-bath type, H = HS + HB, where the system and bath parts are coupled via the conical intersection. Here, Hs refers to a four-mode core composed of three tuning type modes Q. Q, Qz and a coupling mode Q4,... 

The simple class of models just discussed is of interest because it is possible to characterize the decay of correlations rather completely. However, these models are rather far from reality since they take no account of interparticle forces. A next step in our examination of the decay of initial correlations is to find an interacting system of comparable simplicity whose dynamics permit us to calculate at least some of the quantities that were calculated for the noninteracting systems. One model for which reasonably complete results can be derived is that of an infinite chain of harmonic oscillators in which initial correlations in momentum are imposed. Since the dynamics of the system can be calculated exactly, one can, in principle, study the decay of correlations due solely to internal interactions (as opposed to interactions with an external heat bath). We will not discuss the most general form of initial correlations but restrict our attention to those in which the initial positions and momenta have a Gaussian distribution so that two-particle correlations characterize the initial distribution completely. Let the displacement of oscillator j from its equilibrium position be denoted by qj and let the momentum of oscillator j be pj. On the assumption that the mass of each oscillator is equal to 1, the momentum is related to displacement by pj =. We shall study... 

In Refs. [55, 79], the truncation at the level of Heg has been tested for several molecular systems exhibiting an ultrafast dynamics at Coin s, and it was found that this approximation can give remarkably good results in reproducing the short-time dynamics. This is especially the case if a system-bath perspective is appropriate, and the effective-mode transformation is only applied to a set of weakly coupled bath modes [55,72]. In that case, the system Hamiltonian can take a more complicated form than given by the LVC model. 

Figure 2. Flow Sheet of the HPLC Unit for Rapid Protein Analysis. A binary gradient system with gradient controller, Model 21500 pumps,Model 2152 controller, Pharmacia, Piscataway, NJ, U.S.A. (A), Variable wavelength UV-Visible detector, Model LC 95, Perkin Elmer, Norwalk, CT, U.S.A (B), Constant temperature circulating bath, Model DL-8 Haake Buchler, Saddlebrook, NJ, USA (C), Heat exchangers (D), Sampling valve,...

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... 

These results leave several basic questions open How to derive a non-Markovian master equation (ME) for arbitrary time-dependent driving and modulation of a thermally relaxing two-level system Would the two-level system (TLS) model hold at all for modulation rates, that are comparable to the TLS transition frequency u)a (between its states e) and g)) which may invalidate the standard rotating-wave approximation (RWA), [to hen-Tannoudji 1992] Would temperature effects, which are known to incur upward g) —> e) transitions, [Lifshitz 1980], further complicate the dynamics and perhaps hinder the suppression of decay How to control decay in an efficient, optimal fashion We address these questions by outlining the derivation of a ME of a TLS that is coupled to an arbitrary bath and is driven by an arbitrary time-dependent field. 

The spectral density (see also Sections (7-5.2) and (8-2.5)) plays a prominent role in models of thermal relaxation that use harmonic oscillators description of the thermal environment and where the system-bath coupling is taken linear in the bath coordinates and/or momenta. We will see (an explicit example is given in Section 8.2.5) that /(co) characterizes the dynamics of the thermal environment as seen by the relaxing system, and consequently determines the relaxation behavior of the system itself. Two simple models for this function are often used ... 

Choosing a physically motivated representation is useful in developing physically guided approximation schemes. A commonly used approximation for the model (12,4)—(12.6) is to disregard tenns with j j in the system-bath interaction (12.5b). The overall Hamiltonian then takes the fonn... 

There are three important issues to consider in the numerical solution of the Redfield equation. The first is the evaluation of the Redfield tensor matrix elements I ,To obtain these matrix elements, it is necessary to have a representation of the system-bath coupling operator and of the bath Hamiltonian. Two fundamental types of models are used. First, the system-bath coupling can be described using stochastic fluctuation operators, without reference to a microscopic model. In this case, the correlation functions appearing in the formulas for parame-... 

Before going on to consider more complicated systems, we review here some of the basic behavior of a two-state quantum system in the presence of a fast stochastic bath. This highly simplified bath model is useful because it allows qualitatively meaningful results to be obtained from a density matrix calculation when bath correlation functions are not available in fact, the bath coupling to any given system operator is reduced to a scalar. In the case of the two-level system, analytic results for the density matrix dynamics are easily obtained, and these provide an important reference point for discussing more complicated systems, both because it is often possible to isolate important parts of more complicated systems as effective two-level systems and because many aspects of the dynamics of multilevel systems appear already at this level. An earlier discussion of the two-level system can be found in Ref. 80. The more... 

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