Correlation functions spectra

A. Molecular Reorientation Correlation Function, Spectrum, and Susceptibility... 

In the case where x and y are the same, C (r) is called an autocorrelation function, if they are different, it is called a cross-correlation function. For an autocorrelation function, the initial value at t = to is 1, and it approaches 0 as t oo. How fast it approaches 0 is measured by the relaxation time. The Fourier transforms of such correlation functions are often related to experimentally observed spectra, the far infrared spectrum of a solvent, for example, is the Foiuier transform of the dipole autocorrelation function. ... 

It is a well known fact, called the Wiener-Khintchine Theorem [gardi85], that the correlation function and power spectrum are Fourier Transforms of one another ... 

Of course, knowledge of the entire spectrum does provide more information. If the shape of the wings of G (co) is established correctly, then not only the value of tj but also angular momentum correlation function Kj(t) may be determined. Thus, in order to obtain full information from the optical spectra of liquids, it is necessary to use their periphery as well as the central Lorentzian part of the spectrum. In terms of correlation functions this means that the initial non-exponential relaxation, which characterizes the system s behaviour during free rotation, is of no less importance than its long-time exponential behaviour. Therefore, we pay special attention to how dynamic effects may be taken into account in the theory of orientational relaxation. 

This is a direct generalization of the Hubbard relation (2.27) to the case ft) 0. It is actually an algorithm for extraction of a wide spectral component which forms the pedestal. Bi-Lorentzian spectrum (2.54) may serve as an example of the above algorithm realization. Using its correlation function (2.53) in (2.72), we find TV in addition to G(. 

In Eq. (3.66) the sign + is chosen to provide the decay in time of the spectrum correlation function. When the approximate solution (3.66) is used for the back iterations in Eq. (3.58) from bN = 1 + bN up to ho and subsequent calculation of ao(co) the error does not accumulate. This was proved by comparison of approximate numerical calculations of limiting cases 2 and 3 with exact formulae (3.61) and (3.62). 

To compare this result with that obtained within perturbation theory [273, 279], one must additionally assume the perturbation correlation function to be exponential, as in [273, 279, 280]. In this case, the purely rotational spectrum [273, 279] and that obtained with Eq. (7.71) coincide, if the co-dependence of the / operator is neglected and ([Pg.247]

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

In homogeneous turbulence, the velocity spectrum tensor is related to the spatial correlation function defined in (2.20) through the following Fourier transform pair ... 

This relation shows that for homogeneous turbulence, working in terms of the two-point spatial correlation function or in terms of the velocity spectrum tensor is entirely equivalent. In the turbulence literature, models formulated in terms of the velocity spectrum tensor are referred to as spectral models (for further details, see McComb (1990) or Lesieur (1997)). 

The need to add new random variables defined in terms of derivatives of the random fields is simply a manifestation of the lack of two-point information. While it is possible to develop a two-point PDF approach, inevitably it will suffer from the lack of three-point information. Moreover, the two-point PDF approach will be computationally intractable for practical applications. A less ambitious approach that will still provide the length-scale information missing in the one-point PDF can be formulated in terms of the scalar spatial correlation function and scalar energy spectrum described next. 

In general, the scalar Taylor microscale will be a function of the Schmidt number. However, for fully developed turbulent flows,18 l.,p L and /, Sc 1/2Xg. Thus, a model for non-equilibrium scalar mixing could be formulated in terms of a dynamic model for Xassociated with working in terms of the scalar spatial correlation function, a simpler approach is to work with the scalar energy spectrum defined next. 

Adapted Filter-Diagonalization Calculation of Vibrational Spectrum of Planar Acetylene from Correlation Functions. 

Theoretical studies, based entirely on the local tetrahedral pentamer model of the liquid 47> but including isotropic expansion and contraction with the distribution of 00 separations matched to the observed 00 correlation function, cannot completely account for the observed OH stretching spectrum 90>. The addition of some weak hydrogen bonds improves the predicted spectrum 90>, but still leaves the widths of the band contours largely unaccounted for. It seems likely that inclusion of 000 angle distribution effects (i.e. bent hydrogen bonds as well as small dispersion about 109.5°) will improve the agreement between predicted observed spectra. 

Here is the Rouse time - the longest time in the relaxation spectrum - and W is the elementary Rouse rate. The correlation function x(p,t) x p,0)) of the normal coordinates is finally obtained by ... 

Q-dependent Rouse rates were obtained by fitting each spectrum separately. For a comparison with the BS data (see below) the obtained values for Wf (Q) were transformed into average relaxation times for the Rouse self-correlation function by (r (Q))=18TiQ V[ Wf (Q)] (Eq. 3.18). 

The spectrum of scattered light contains dynamical information related to translational and internal motions of polymer chains. In the self-beating mode, the intensity-intensity time correlation function can be expressed (ID) as... 

We illustrate the general expressions (4.189)-(4.191) for a typical non-Markovian Lorentzian bath spectrum, that is, an exponentially decaying correlation function d>(t) = being the correlation (memory) time. 

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