System-bath coupling correlation functions

Such correlation functions are often encountered in treatments of systems coupled to their thennal environment, where the mode 1 for the system-bath interaction is taken as a product of A or B with a system variable. In such treatments the coefficients Cj reflect the distribution of the system-bath coupling among the different modes. In classical mechanics these functions can be easily evaluated explicitly from the definition (6.6) by using the general solution of the harmonic oscillator equations of motion... 

The Redfield tensor S is defined in terms of stationary correlation functions of the system-bath coupling operator, V, evolving under the bath Hamiltonian, Thus the dynamics of the bath are retained in Eq. (9), the only assumptions being that the bath is in thermal equilibrium and that its dynamics are independent of the state of the system beyond some correlation time, t, short compared to the rate of change of cr. The tensor element R,, / can be written [26, 42]... 

To evaluate the correlation functions in Eqs. (12) and (13), it is usual to complete the separation of the system and bath by decomposing the system-bath coupling into a sum of products of pure system and bath operators. This allows the correlation functions of the system-bath coupling to be replaced, without loss of generality, by correlation functions of bath operators alone, evolving under the uncoupled bath Hamiltonian. Moreover, as we have previously pointed out [39,40], this decomposition of the system-bath coupling make it possible to write the Redfield equation in a highly compact form, without explicit reference to the Redfield tensor at all. 

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-... 

Reaction processes in condensed phase are a major challenge. To describe reactions in solution, one either has to resort to classical mechanics or has to include the coupling to the enviroment in a quantum simulation. Flux correlation functions provide a suitable theoretical framework for this purpose. However, the description of system-bath couplings in realistic quantum systems still is a serious theoretical problem and only first steps in this direction have been taken. 

The kinetic coefficients R((o) that appear in the relaxation operator are given by Fourier-Laplace transforms Rab,cd o)) = dxMab,cd(j)e of the coupling correlation functions M t). These fimctions are defined by Eq. (10.138) and satisfy the symmetry property (10.139). In the more general case where the system-bath coupling is given by (10.122), these functions are given by Eq. (10.143) with the symmetry property (10.144). 

Within a permrbative approach, which is second order in the system-bath coupling, the entire bath dynamics enters into the system dynamics via the complex bath correlation function, defined by... 

By using special forms of the so-called spectral density J(w) it is possible to treat memory effects in QMEs. The spectral density J(w) contains information on the frequencies of the environmental modes and their coupling to the system. Tanimura and coworkers [18,20,26] were the first to do calculations along the lines described here using spectral densities of Drude shape. This spectral densities lead to bath correlation functions with purely exponential... 

The third alternative is to use the classical correlation functions to define an equivalent quantum mechanical harmonic bath. This approach was pioneered by Warshel as the dispersed polaron method [67, 68]. More recently, this idea has been used in studies of electron transfer systems in solution [64] and in the photosynthetic reaction center [65,69] (see also Ref. 70). This approach is based on the realization that the spectral density describing a linearly coupled harmonic bath [Eq. (29)] can be obtained by cosine transformation of the classical time-correlation function of the bath operator [Eq. (28)]. Comparing the classical correlation function for the linearly coupled harmonic bath [Eqs. (25) and (26)],... 

The strength of the bath coupling to each system variable is described by the coupling constants / and, because they enter at second order, the rate constant for the dissipation process arising from each term in Eq. (38) will be proportional to f I- The only important properties of the F t) are their autocorrelation and cross-correlation functions, (FJfi)F t)) and F (0)Fi,(t)), which enter the definition of the Redfield tensor in Eq. (18). These, like the classical correlation functions discussed earlier, do not satisfy the detailed-balance relation and must be corrected in the same way. It is convenient, but not necessary, that the variables be chosen to be independent, so that the cross-correlation functions vanish. 

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... 

The Zusman equation (ZE)/ due mainly to its physically insightful picture on solvation dynamics, is (at least used to be) one of the most commonly used approaches in the study of quantum transfer processes. In this approach, the electronic system degrees of freedom are coupled to a collective bath coordinate that is assumed to be diffusive. The only approximation involved is the classical high temperature treatment of bath. To account for the dynamic Stokes shift, the standard ZE includes also the imaginary part of bath correlation function. This part does not depend on temperature and is therefore exact in the diffusion regime. 

Recently, a QUAPI procedure was developed suitable for evaluating the full flux correlation function in the case of a one-dimensional quantum system coupled to a dissipative harmonic bath and applied to obtain accurate quantum mechanical reaction rates for a symmetric double well potential coupled to a generic environment. These calculations confirmed the ability of analytical approximations to provide a nearly quantitative picture of such processes in the activated regime, where the reaction rate displays a Kramers turnover as a function of solvent friction and quantum corrections are small or moderate, They also emphasized the significance of dynamical effects not captured in quantum transition state models, in particular under small dissipation conditions where imaginary time calculations can overestimate or even underestimate the reaction rate. These behaviors are summarized in Figure 7. 

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