Time correlation functions spectral density

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

The virial expansion of the time correlation functions is possible for times smaller than the mean time x between collisions. Accordingly, the spectral profiles may be expanded in powers of density, for angular frequencies much greater than the reciprocal mean time between collisions, co 1/r. Since at low density the mean time between collisions is inversely proportional to density, lower densities permit a meaningful virial expansion for a greater portion of the spectral profiles. 

Prof. Fleming, the expressions you are using for the nonlinear response function may be derived using the second-order cumulant expansion and do not require the use of the instantaneous normal-mode model. The relevant information (the spectral density) is related to the two-time correlation function of the electronic gap (for resonant spectroscopy) and of the electronic polarizability (for off-resonant spectroscopy). You may choose to interpret the Fourier components of the spectral density as instantaneous oscillators, but this is not necessary. The instantaneous normal mode provides a physical picture whose validity needs to be verified. Does it give new predictions beyond the second-order cumulant approach The main difficulty with this model is that the modes only exist for a time scale comparable to their frequencies. In glasses, they live much longer and the picture may be more justified than in liquids. 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

The persistence of the fluctuating local fields before being averaged out by molecular motion, and hence their effectiveness in causing relaxation, is described by a time-correlation function (TCF). Because the TCF embodies all the information about mechanisms and rates of motion, obtaining this function is the crucial point for a quantitative interpretation of relaxation data. As will be seen later, the spectral-density and time-correlation functions are Fourier-transform pairs, interrelating motional frequencies (spectral density, frequency domain) and motional rates (TCF, time domain). 

Investigating molecular dynamics using NMR, in contrast to DS and LS, involves the application of several conceptually quite different techniques. For example, in spin-lattice relaxation studies one is concerned with familiar time correlation functions that are probed as spectral density point by point (Section II.D.2). In the case of line-shape analysis, usually a two-pulse echo sequence is applied, and the... 

In a theoretical treatment, it is necessary to make approximations in the derivation of the spectral densities (Appendix A.2 - equation (A7)), that is, the Fourier transforms of time correlation functions of perturbations used to express the nuclear spin relaxation times. These theories have been tested against experiments and their limitations have been examined under varying conditions. The advantage of MD simulations to evaluate the theoretical models is the realism of the description and that many approximations in the theoretical model can be tested separately. Because of the conceptual differences between theories and the arbitrariness in their parameterization, it is often not possible discriminate between... 

Figure 4. Time correlation function and spectral density for fluctuations in the dihedral angle, x (Figure I). Top in solution bottom under vacuum. The functions shown dotted at the top are obtained from a Langevin equation ("see text).

The spectral characteristics of the scattered light depend on the time scales characterizing the motions of the scatterers. These relationships are discussed in Chapter 3. The quantities measured in light-scattering experiments are the time-correlation function of either the scattered field or the scattered intensity (or their spectral densities). Consequently, time-correlation functions and their spectral densities are central to an understanding of light scattering. They are, therefore, discussed at the outset in in Chapter 2. 

The spectral density (or power spectrum) IA(a>) of a time-correlation function is defined as11... 

This quantity plays an important role in much of what follows. In fact, as we shall see, what is sometimes measured in light scattering is the spectral density of the electric field of the scattered light. Let us dwell for a moment on some properties of these functions. Fourier inversion of Eq. (2.4.1) leads to an expression for the time-correlation function in terms of the spectral density. 

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)],... 

An alternative way to obtain the spectral density is by numerical simulation. It is possible, at least in principle, to include the intramolecular modes in this case, although it is rarely done [198]. A standard approach [33-36,41] utilizes molecular dynamics (MD) trajectories to compute the classical real time correlation function of the reaction coordinate from which the spectral density is calculated by the cosine transformation [classical limit of Eq. (9.3)]. The correspondence between the quantum and the classical densities of states via J(co) is a key for the evaluation of the quantum rate constant, that is, one can use the quantum expression for /Cj2 with the classically computed J(co). This is true only for a purely harmonic system [199]. Real solvent modes are anharmonic, although the response may well be linear. The spectral density of the harmonic system is temperature independent. For real nonlinear systems, J co) can strongly depend on temperature [200]. Thus, in a classical simulation one cannot assess equilibrium quantum populations correctly, which may result in serious errors in the computed high-frequency part of the spectrum. Song and Marcus [37] compared the results of several simulations for water available at that time in the literature [34,201] with experimental data [190]. The comparison was not in favor of those simulations. In particular, they failed to predict... 

Figure 9.3. The simulated normalized spectral density function, J(cd) = J(a>)/ oda> ct) V(ct>), for water with immersed donor and acceptor molecules of radii 3.5 A at different separation (from [41c] with permission. Copyright (1997) by the American Institute of Physics), as compared against the experimental results (circles, data taken from [202a]). The corresponding time correlation function is shown in Figure 9.4.

The time-dependent power spectral density is given by the Fourier transform of the correlation function [161] ... 

NMR relaxation and its field dependence are a very important source of experimental information on dynamics of molecular motions. This information is conveyed through spectral density functions, which in turn are related to time-correlation functions (TCFs), fundamental quantities in the theory of liquid state. In most cases, characterizing the molecular dynamics through NMR relaxation studies requires the identification of the relaxation mechanism (for example the dipole-dipole interaction between a pair of spins) and models for the spectral densities/correlation functions." During the period covered by this review, such model development was concerned with both small molecules and large molecules of biological interest, mainly proteins. 

Thus, the time correlation function is the Fourier transform of the spectral density... 

To derive an explicit form of the spectral dependence /(g, ui) (Equation 38), information on the correlation of density fluctuations in two different points at different instants of time is required. If the distance between these points and the wavelength is of the same order of magnitude, fluctuations must correlate in time, i.e. the density fluctuation at the first point at t = 0 may propagate or spread to the second point for time i. In this connection, the time correlation function can be derived in terms of the hydrodynamic equations and Onsager s principle. 

Figure 17.2.5 (A) Time correlation function, (B) power spectral density function, and (C) probability density function of the time series in Figure 17.2.4C. Adapted with permission from reference (8).

An IR spectrum reflects the Fourier transform of the molecular dipole moment. The absorption coefficient, a(P) measured by IR spectroscopy is given by Eq. (6), where the infrared spectral density is the Fourier transform of the time-correlation function for the dipole moment [11] ... 

Rather than measuring the time correlation function of the scattering amplitude, one can also use a monochromator and determine the spectral density. As stated by the Wiener-Chinchin theorem, the spectral density of a fluctuating quantity and its time correlation function are related by Fourier-transformations. Applied to our case we may write... 

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