Frequency dependence correlation factor

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators are needed. 

The CCS, CC2, CCSD, CC3 hierarchy has been designed specially for the calculation of frequency-dependent properties. In this hierarchy, a systematic improvement in the description of the dynamic electron correlation is obtained at each level. For example, comparing CCS, CC2, CCSD, CC3 with FCI singlet and triplet excitation energies showed that the errors decreased by about a factor 3 at each level in the coupled cluster hierarchy [18]. The CC3 error was as small as 0.016 eV and the accuracy of the CC3 excitation energies was comparable to the one of the CCSDT model [18]. 

In Eqs. (4)-(7) S is the electron spin quantum number, jh the proton nuclear magnetogyric ratio, g and p the electronic g factor and Bohr magneton, respectively. r//is the distance between the metal ion and the protons of the coordinated water molecules, (Oh and cos the proton and electron Larmor frequencies, respectively, and Xr is the reorientational correlation time. The longitudinal and transverse electron spin relaxation times, Tig and T2g, are frequency dependent according to Eqs. (6) and (7), and characterized by the correlation time of the modulation of the zero-field splitting (x ) and the mean-square zero-field-splitting energy (A. The limits and the approximations inherent to the equations above are discussed in detail in the previous two chapters. 

Calculational procedure of all the dynamic variables appearing in the above expressions—namely, the dynamic structure factor F(q,t) and its inertial part, Fo(q,t), and the self-dynamic structure factor Fs(q,t) and its inertial part, Fq (q, t) —is similar to that in three-dimensional systems, simply because the expressions for these quantities remains the same except for the terms that include the dimensionality. Cv(t) is calculated so that it is fully consistent with the frequency-dependent friction. In order to calculate either VACF or diffusion coefficient, we need the two-particle direct correlation function, c(x), and the radial distribution function, g(x). Here x denotes the separation between the centers of two LJ rods. In order to make the calculations robust, we have used the g(x) obtained from simulations. 

A simple theory of the concentration dependence of viscosity has recently been developed by using the mode coupling theory expression of viscosity [197]. The slow variables chosen are the center of mass density and the charge density. The final expressions have essentially the same form as discussed in Section X the structure factors now involve the intermolecular correlations among the polyelectrolyte rods. Numerical calculation shows that the theory can explain the plateau in the concentration dependence of the viscosity, if one takes into account the anisotropy in the motion of the rod-like polymers. The problem, however, is far from complete. We are also not aware of any study of the frequency-dependent properties. Work on this problem is under progress [198]. 

The line forms, described by Eqs. (371) and (373), are illustrated in Figs. 45 and 46. In the first one (Fig. 45) we compare the loss (a) and absorption (b) for the isothermal, Gross, and Lorentz lines see solid, dashed, and dashed-and-dotted curves, respectively. These curves are calculated in a vicinity of the resonance point x = 1. In Figure 45c we show the frequency dependences of a real part of the susceptibility the three curves are extended also to a low-frequency region. The collisions frequency y and the correlation factor g are fixed in Fig. 45 (y = 0.4, g = 2.5). 

Figure 45. Frequency dependence of an imaginary (a) and of real (c) parts of the susceptibility (b) absorption coefficient versus x. All quantities are nondimensional. Calculation for the isothermal (solid curves), Gross (dashed curves), and Lorentz (dashed-and-dotted curves) lines. The normalized collision frequency, y, is 0.4, and the correlation factor, g, is 2.5.

Figure 46. Evolution of frequency dependencies of loss (a, c) and of real part of the susceptibility (b, d) stipulated by change of the correlation factor g (nondimensional quantities). In Figs, (a, b) g = 2.5 and in Figs, (c, d) g — 2. Isothermal line (solid curves) and Gross line (dashed curves). Curves 1, 3 for y = 0.4 and curves 2, 4 for y = 0.8.

From the point of view of the solvent influenee, there are three features of an electron spin resonance (ESR) speetrum of interest for an organic radical measured in solution the gf-factor of the radical, the isotropie hyperfine splitting (HFS) constant a of any nucleus with nonzero spin in the moleeule, and the widths of the various lines in the spectrum [2, 183-186, 390]. The g -faetor determines the magnetic field at which the unpaired electron of the free radieal will resonate at the fixed frequency of the ESR spectrometer (usually 9.5 GHz). The isotropie HFS constants are related to the distribution of the Ti-electron spin density (also ealled spin population) of r-radicals. Line-width effects are correlated with temperature-dependent dynamic processes such as internal rotations and electron-transfer reaetions. Some reviews on organic radicals in solution are given in reference [390]. 

The final application we discuss is one where the maximum entropy formalism is used not only to fit the spectrum but also to extract new results. Specifically we discuss the determination of the time cross-correlation function, Cf, t) (Eq. (43)), which is the Fourier transform of the Raman scattering amplitude a/((Tu) (Eq. (44)) when what is measured is the Raman scattering cross section afi(m) a/((Tii) 2. The problem is that the experiment does not appear to determine the phase of the amplitude. The application proceeds in two stages (i) Representing the Raman spectrum as one of maximal entropy, using as constraints the Fourier transform of the observed spectrum. At the end of this stage one has a parametrization of a/,( nr) 2 whose accuracy can be determined by how well it fits the observed frequency dependence, (ii) The fact that the Raman spectrum can be written as a square modulus as in Eq. (97) implies that it can be uniquely factorized into a minimum phase function... 

The most obvious experimental manifestations of interionic correlation are found in the Haven ratio (deduced from diffusion and conductivity measurement, see Chapter 4), in the static structure factor S(Q) (deduced from partial occupation factors measured by X-ray or neutron diffraction) and from the dynamical properties (S Q, co), quasi- and inelastic-neutron scattering, frequency dependent conductivity) and e(co) dielectric relaxation. 

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

In addition to the dipole-dipole relaxation processes, which depend on the strength and frequency of the fluctuating magnetic fields around the nuclei, there are other factors that affect nOe (a) the intrinsic nature of the nuclei I and S, (b) the internuclear distance (r,s) between them, and (c) the rate of tumbling of the relevant segment of the molecule in which the nuclei 1 and S are present (i.e., the effective molecular correlation time, Tf). 

---