Linear correlation function

Figure 8.22. Testing the first-zero method for the determination of the linear crystallinity, V[, from the linear correlation function, yi (x/Lapp) with Lapp being the position of the first maximum in yi (x) (not shown here - but cf. Fig. 8.21). Model tested Paracrystalline stacking statistics with Gaussian thickness distributions. The interval of forbidden zeroes is shown. An additional horizontal non-linear axis permits to determine the linear crystallinity directly. A corresponding vertical axis shows the variation of the classical valley-depth method ... 

Another critical advance responsible for the success of MP2-R12 is the introduction of nonlinear correlation functions, in particular, the Slater-type correlation function, 1— exp(—yr 2), of Ten-no [29]. It is asymptotically linear in ru near the coalescence point and, hence, satisfies the cusp condition in the leading order. Unlike the linear correlation function, the Slater-type correlation function... 

Goderis, B., Reynaers, H.. Koch, H.M.J., and Malhot, V.B.F. (1999) Use of SAXS and linear correlation functions for the determination of the crystallinity and morphology of semi-crystalline polymers. Application to linear polyethylene. J. Polym. Sci. Part B Polym. Phys., 37,1715-1738. 

Radiation probes such as neutrons, x-rays and visible light are used to see the structure of physical systems tlirough elastic scattering experunents. Inelastic scattering experiments measure both the structural and dynamical correlations that exist in a physical system. For a system which is in thennodynamic equilibrium, the molecular dynamics create spatio-temporal correlations which are the manifestation of themial fluctuations around the equilibrium state. For a condensed phase system, dynamical correlations are intimately linked to its structure. For systems in equilibrium, linear response tiieory is an appropriate framework to use to inquire on the spatio-temporal correlations resulting from thennodynamic fluctuations. Appropriate response and correlation functions emerge naturally in this framework, and the role of theory is to understand these correlation fiinctions from first principles. This is the subject of section A3.3.2. 

The applicability of the two-parameter equation and the constants devised by Brown to electrophilic aromatic substitutions was tested by plotting values of the partial rate factors for a reaction against the appropriate substituent constants. It was maintained that such comparisons yielded satisfactory linear correlations for the results of many electrophilic substitutions, the slopes of the correlations giving the values of the reaction constants. If the existence of linear free energy relationships in electrophilic aromatic substitutions were not in dispute, the above procedure would suffice, and the precision of the correlation would measure the usefulness of the p+cr+ equation. However, a point at issue was whether the effect of a substituent could be represented by a constant, or whether its nature depended on the specific reaction. To investigate the effect of a particular substituent in different reactions, the values for the various reactions of the logarithms of the partial rate factors for the substituent were plotted against the p+ values of the reactions. This procedure should show more readily whether the effect of a substituent depends on the reaction, in which case deviations from a hnear relationship would occur. It was concluded that any variation in substituent effects was random, and not a function of electron demand by the electrophile. ... 

According to a kinetic study which included (56), (56a) and some oxaziridines derived from aliphatic aldehydes, hydrolysis follows exactly first order kinetics in 4M HCIO4. Proton catalysis was observed, and there is a linear correlation with Hammett s Ho function. Since only protonated molecules are hydrolyzed, basicities of oxaziridines ranging from pii A = +0.13 to -1.81 were found from the acidity rate profile. Hydrolysis rates were 1.49X 10 min for (56) and 43.4x 10 min for (56a) (7UCS(B)778). O-Protonation is assumed to occur, followed by polar C—O bond cleavage. The question of the place of protonation is independent of the predominant IV-protonation observed spectroscopically under equilibrium conditions all protonated species are thermodynamically equivalent. 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

A linear correlation of log rate coefficient with acidity function HQ was considered to have been obtained in the dedeuteration of deuterobenzene in aqueous sulphuric acid (Table 121) the slope of the plot being given as 1.36461. However, rigorous examination of the data shows the plot to be a curve which is amplified... 

An exceptionally badly reported kinetic study in which a linear correlation of rate coefficient with acidity function was claimed was that of Mackor et al. 11, who studied the dedeuteration of benzene and some alkylbenzenes in sulphuric acid-trifluoroacetic acid at 25 °C. Rates were given only in the form of a log rate coefficient versus —H0 plot and rate coefficients and entropies of activation (measured relative to p-xylene) together with heats of activation (determined over a temperature range which was not quoted) were also given (Table 129). However,... 

Figure 14. Plot of the potential of zero charge, Ea=0 (from Table 26), against the work function,

It is an experimental fact that the capacitance of an electrode in a given solvent is a function of the nature of the metal. This was pointed out by Frumkin et al,333 and has been discussed several times in the literature.7 349 94 99 999 Trasatti34 901 showed that the reciprocal of the differential capacitance at a = 0 is linearly correlated with the strength of the metal-water interaction. The reader is referred to the original papers for a detailed discussion. 

According to Eq. (2.13), the spectra we are interested in are given by a Fourier transform of the orientational correlation functions of the corresponding order Similarly to what was done in Chapter 3, the correlation functions for linear and spherical molecules may be represented as a superposition of the partial (marginal) components... 

Kluk E., Monkos K., Pasterny K., Zerda T. A means to obtain angular velocity correlation functions from angular position correlation functions of molecules in liquid. Part I. General discussion and its application to linear and spherical top molecules, Acta Physica Polonica A 56, 109-16 (1979). 

Fractions rejected by 1.0 KDa (C>i) and permeated through 0.5 KDa (C0 5) membranes were also subjected to TLC analysis. In Figure 3 are reported the values relevant to the various spots detected in the two samples as a function of an arbitrary polymerisation degree (DP). The good linear correlation between these parameters allows to hypothesise a difference of one monomer units between the subsequent spots [32]. Consequently, C0.5 would correspond to the monomer, Cj to an homologous series fi om the monomer to the hexamer. 

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