Spectral function general derivation

Third, the expression for the spectral function pertinent to the HO model is derived in detail using the ACF method. Some general results given in GT and VIG (and also in Section II) are confirmed by calculations, in which an undamped harmonic law of motion of the bounded charged particles is used explicitly. The complex susceptibility, depending on a type of a collision model,... 

The paper [50] was written, when our general linear response-theory (ACF method) was still in progress, so the derivation of the spectral function described in Ref. 50 is more specialized than that given, for example, in VIG and GT. 

A general approach (VIG, GT) to a linear-response analytical theory, which is used in our work, is viewed briefly in Section V.B. In Section V.C we consider the main features of the hat-curved model and present the formulae for its dipolar autocorrelator—that is, for the spectral function (SF) L(z). (Until Section V.E we avoid details of the derivation of this spectral function L). Being combined with the formulas, given in Section V.B, this correlator enables us to calculate the wideband spectra in liquids of interest. In Section V.D our theory is applied to polar fluids and the results obtained will be summarized and discussed. 

Validity of our formulas for the resonance lines, which express the complex susceptibility through the spectral function, could be confirmed as follows. We have obtained an exact coincidence of the equations (353), (370), (371), which were (i) directly calculated here in terms of the harmonic oscillator model and (ii) derived in GT and VIG (see also Section II, A.6) by using a general linear-response theory. 

Statistics of the Estimates. The mean and the variance of the sample estimates of the coherence function ate derived in Ref. 4, and I only reproduce the final results here. In general, the cross-spectral density is evaluated by doing the Fourier transform of a windowed (using a lag window) sequence of cross-correlation estimates. The choice of the smoothing window therefore determines the variance in the estimates of cross-spectral density (numerator in the expression of coherence). The variance of the smoothed coherence estimator is given by ... 

Equation (9.32) is also useful to the extent it suggests die general way in which various spectral properties may be computed. The energy of a system represented by a wave function is computed as the expectation value of the Hamiltonian operator. So, differentiation of the energy with respect to a perturbation is equivalent to differentiation of the expectation value of the Hamiltonian. In the case of first derivatives, if the energy of the system is minimized with respect to the coefficients defining die wave function, the Hellmann-Feynman theorem of quantum mechanics allows us to write... 

To determine the interplay between the spectral properties, both boundary conditions, we return to Weyl s theory [32]. The key quantity in Weyl s extension of the Sturm-Liouville problem to the singular case is the m-function or ra-matrix [32-36]. To define this quantity, we need the so-called Green s formula that essentially relates the volume integral over the product of two general solutions of Eq. (1), u and v with eigenvalue X and the Wronskian between the two solutions for more details, see Appendix C. The formulas are derived so that it immediately conforms to appropriate coordinate separation into the... 

Phase 2 - data preprocessing. There are many ways to process spectral data prior to multivariate image reconstruction and there is no ideal method that can be generally applied to all types of tissue. It is usual practice to correct the baseline to account for nonspecific matrix absorptions and scattering induced by the physical or bulk properties of the dehydrated tissue. One possible procedure is to fit a polynomial function to a preselected set of minima points and zero the baseline to these minima points. However, this type of fit can introduce artifacts because baseline variation can be so extreme that one set of baseline points may not account for all types of baseline variation. A more acceptable way to correct spectral baselines is to use the derivatives of the spectra. This can only be achieved if the S/N of the individual spectra is high and if an appropriate smoothing factor is introduced to reduce noise in the derivatized spectra. Derivatives serve two purposes they minimize broad... 

Chemical kinetics also plays a basic role in the study of the nature of catalytic activity. Studies of the catalyst and reactants in the absence of appreciable over-all reaction, such as studies of the electronic properties of catalytic solids or optical studies of adsorbed molecular species can provide valuable information about these materials. In most cases, however, kinetic data are ultimately needed to establish the relation and relevance of any information derived from such studies to the catalytic reaction itself. For example, a particular adsorbed species may be observed and studied by a spectral technique yet it need not play any essential role in the catalytic reaction since adsorption is a more general phenomenon than catalytic activity. On the other hand, kinetics studies can provide information about the variation, as a function of experimental conditions, of the relative number of adsorbed species that play a basic role in the reaction. Consequently, such information may make it possible to identify which, if any, of the adsorbed species studied by the use of a direct analytical technique are relevant to the reaction. As another example, when studies are made of the solid state properties of a given catalytic solid, the question as to which, if any, of these properties are related to catalytic activity must ultimately be answered in terms of consistency with the observed behavior of the reaction system. 

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