Correlation spectra formal development

F.l Formal Development of Correlation Spectra F.1.1 Dynamic Spectrum... 

The said allows us to understand the importance of the kinetic approach developed for the first time by Waite and Leibfried [21, 22]. In essence, as is seen from Fig. 1.15 and Fig. 1.26, their approach to the simplest A + B —0 reaction does not differ from the Smoluchowski one However, coincidence of the two mathematical formalisms in this particular case does not mean that theories are basically identical. Indeed, the Waite-Leibfried equations are derived as some approximation of the exact kinetic equations due to a simplified treatment of the fluctuational spectrum a complete set of the joint correlation functions x(rJ) for kinds of particles is replaced by the only function xab (a t) describing the correlation of chemically reacting dissimilar particles. Second, the equation defining the correlation function X = Xab(aO is linearized in the function x(rJ)- This is analogous to the... 

Until now we have discussed only elementary methods for determining correlation functions, based on ad hoc models. In this chapter a powerful formalism for computing time-correlation functions is presented. As a by-product of this formalism several useful theorems emerge which result from symmetry considerations. Moreover some of the assumptions made in Chapter 10 are shown to be valid. Throughout this chapter we treat classical systems. The methods developed here can also be applied to quantum systems. This is shown in Appendix 11.A. The formalism of this chapter is applied in Chapter 12 to the calculation of the depolarized spectrum. 

In order to unify, in fhe spirit of quantum defect theory, the treatment of discrefe and confinuous spectra in the presence of discrete Rydberg and valence states and of resonances, Komninos and Nicolaides [82, 83] developed K-mafrix-based Cl formalism that includes the bound states and the Rydberg series, and where the state-specific correlated wavefunc-tions (of the multi-state o) can be obtained by the methods of the SSA. The validity and practicality of fhis unified Cl approach was first demonstrated with the He P° Rydberg series of resonances very close to the n = 2 threshold [76], and subsequently in advanced and detailed computations in the fine-structure spectrum of A1 using fhe Breit-Pauli Hamiltonian [84, 85], which were later verified by experiment (See the references in Ref. [85]). 

To increase interpretability, the dynamic IR spectra are snbjected to mathematical cross-correlation to prodnce two different types of 2D1R correlation spectra, or two-dimensional correlation maps. These maps, in which the. r- and y- axes are independent wavenumber axes (vi, V2), show the relative proportions of in-phase (synchronous) and ont-of-phase (asynchronons) response (Figs. 3.51 and 3.52). Initially, the mathematical formalism was based on the complex Fourier transformation of dynamic spectra [277]. To simplify the computational difficulties, the Hilbert pansform approach was developed [280], which produces two-dimensional correlation maps from a set of dynamic spectra as follows. First, the average spectrum y(v) is subtracted from each spectrum in the set, y(v, Pj) = y v, Pj) — y(v), where Pj is the dynamic parameter. Then, the synchronous spectrum, 5 (vi,V2), and the asynchronous spectrum, A(vi,V2), are calculated as... 

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