Mori continued fraction

To make MCT calculations simpler, the solvent dynamic quantities are sometimes modeled in such a way that the self-consistency is avoided. This is where the viscoelastic models (VEMs) play an important role. The VEM is usually expressed as a Mori continued fraction where the frequencies are de-... 

F(q, t) is obtained from its Laplace transform form F(q,z)- By using the following well-known Mori continued-fraction expansion and truncating at second order, the viscoelastic expression for F(q, z) can be written as [16, 21, 22]... 

The expansion parameters of the Mori continued fraction are therefore given by... 

When aiming at a more satisfactory simulation of the exact results, we need to include more steps of the Mori continued fraction. This can be interpreted as a reduced model with more than one virtual particle. We have found that an accurate simulation of the exact result (i.e., a curve completely overlapping that of Fig. 2) in the linear case requires as many as 10 virtual particles. 

It is of interest here to compare one of these methods with Mori s continued fraction representation for spectral densities.40 Consider the case of Lorentzian broadening of the spectral density (which describes the response to a damped harmonic perturbation, Section III-A). If we set... 

The same model also can be obtained using the continued fraction representation.4 Following Mori,5 the Laplace transform of the velocity... 

The memory function equation for the time-correlation function of a dynamical operator Ut can be cast into the form of a continued fraction as was first pointed out by Mori.43 We prove this in a different way than Mori. In order to proceed it is necessary to define the set of memory functions K0 t),. .., Kn t). .., such that... 

Although the Zwanzig and Mori techniques are closely related and, from a purely formal point of view, completely equivalent, the elegant properties of the Mori theory such as the generalized fluctuation-dissipation theorem imply the physical system under study to be linear, whereas this is not necessary in the Zwanzig approach. This is the main reason we shall be able to face nonlinear problems within the context of a Fokker-Planck approach (see also the discussion of the next section). An illuminating approach of this kind can be found in a paper by Zwanzig and Bixon, which has also to be considered an earlier example of the continued fraction technique iq>plied to a non-Hermitian case. This method has also been fruitfully applied to the field of polymer dynamics. 

The work of other authors cannot be clearly classified as belonging to one of these three main families. A quantum-statistical theory of longitudinal magnetic relaxation based on a continued fraction expansion has been given by Sauermann, who also pointed out the equivalence between his continued fraction approach (based on the Mori scdar product) and the [AA AT] Pad6 approximants. 

This expression is precisely analogous to that obtained by the original Mori theory, in which the Hermitian nature of F makes purely imaginaiy and purely real. In the present approach, both X and A are complex quantities, resulting in faster convergence than in ref. 33, where only the last step of the chain has a complex X introduced to simulate (i.e., take account of) the rest of the continued fraction. The real part of X represent the dissipative difTusional terms. 

We have thus shown that (z) can be expressed in a continued fraction form, whose expression can be given in terms of the moments s . The only theoretical tool we used to arrive at this important result is the generalized version of the celebrated Mori theory. We now have the problem of evaluating the parameters s to obtain the spectra of interest to EPR spectroscopy. This can be done as follows. First, let us define the nth-order state... 

In this section we shall detail the analytical derivation of the absorption spectrum for a magnetic species in a triplet spin state tumbling in a viscous disordered liquid. This an ytical structure consists of the sum of product of continued fractions. The standard Mori structure of Eq. (2.19) is recovered in the absence of orienting potentials. 

Mori, H.A Continued-fraction representation of the time correlation functions. Prog. Theor. Phys., 1965, 34, No. 3, p. 399 116. 

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. 

As a result of the transformation from the original Hamiltonian equations 15.1-15.4 to the effective-mode Hamiltonian equations 15.6-15.11, the spectral density has to be re-written in terms of the transformed quantities. As shown in Ref. [32], J (o) then takes a continued fraction form which is close to the results obtained in Mori theory [29-31] or the Rubin model [12,46]. 

Based upon the effective-mode construction, a systematic approximation procedure for the environment can be formulated in terms of a series of coarse-grained spectral densities [32,33]. These spectral densities are generated from successive orders of a truncated chain model with Markovian closure. Analytical expressions can be given in terms of Mori type continued fractions. Assuming that an - a priori arbitrarily complicated - reference spectral density can be obtained independently, e.g., from experiments or classical simulations, one can thus (1) extract those features of the spectral density that determine the interaction with the subsystem... 

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