Calculating correlation

Figure Bl.9.13. Time-resolved SAXS profiles diirmg isothennal crystallization (230 °C) of PET (the first 48 scans were collected with 5 seconds scan time, the last 52 scans were collected with 30 seconds scan time) calculated correlation fiinctions j(r) (nonnalized by the invariant 0 and lamellar morphological variables...

There have also been methods designed for use with perturbation theory and MCSCF calculations. Correlation effects are necessary for certain technically difficult molecules, such as CO, N2, HCN, F2, and N2O. 

The numerical value of hardness obtained by MNDO-level calculations correlates with the stability of aromatic compounds. The correlation can be extended to a wider range of compounds, including heterocyclic compounds, when hardness is determined experimentally on the basis of molar reffactivity. The relatively large HOMO-LUMO gap also indicates the absence of relatively high-energy, reactive electrons, in agreement with the reduced reactivity of aromatic compounds toward electrophilic reagents. 

The IR bands in a number of nickel complexes of triaryl formazans have been assigned by Arnold and Schiele.415 A similar assignment of the electronic bands has been carried out.414 LCAO-MO calculations correlate well with these assignments417 and have been extended to include both inner ligand transitions as well as charge transfer bands and d—d transitions.418 EPR spectra have been used to study the nature of bonding in copper complexes of heterocyclic-containing formazans.419 Metal formazan complexes have also been studied by electrochemistry.283,398 420-422... 

Nevertheless, the T1 and NOE values are found to be unchanged from the solution to the gel of water content 75% and 53%. As discussed already in Section 2.1, it is very difficult to analyse such relaxation parameters in terms of a single correlation time. The calculated correlation times are found to give consistent values among those obtained from the Ti, T2, and NOE values with the width parameter of p = 18 (PVP solution) and p = 16 (PVP gel) and b = 1000. As the degree of cross-linking is increased, a discrepancy in the correlation times still occurs (a factor of 3 at 75%) but is much improved by this treatment. 

The seven ways (r, through r7) for calculating correlation as the square root of the ratio of the explained variation over the total variation between X (concentration of analyte data) and Y (measured data) are described using many notational forms. For example, many software packages provide built-in functions capable of calculating the coefficient of correlation directly from a pair of X and Y vectors as given by rx (Equation 59-7). 

Ab initio calculations, correlations of molecules geometries, spectroscopic data with chemical properties, and quanta-tive structure-activity relationship have been conducted for both furazans and furoxans. Thus, the structures of 1,2,5-oxadiazole (furazan) and benzo[c][l,2,5]oxadiazole (benzofurazan) have been calculated by ab initio and Becke3-LYP... 

The purpose of calculating correlation and linear regression is to allow unknown values of x (the concentration of the analyte in an unknown sample) to be predicted from a measured value of y (the instrument response). The easiest way of doing this is to invert the equation for the linear regression given above, i.e. ... 

Numerous model calculations correlating aqueous VPIE s using simple harmonic or pseudo-harmonic cell models have been reported (see Fig. 5.8 and Table 5.8 for an ultra-simple version). Such calculations show the importance of the librational hydrogen bonded modes and the stretch-libration interaction in determining VPIE for D or T substitution. 

The correlation constants are displayed in an easy-to-use tabular format which is especially suited for rapid engineering usage with the personal computer or hand calculator. Correlation and experimental results are in favorable agreement. 

The calculated correlation by k2lk will be misleading. We have seen in Section 4.6 that the measured correlation would be determined not by the genuine cooperativi-ties (fc2c/ ic k2j/k j) but by the separation between the two pairs of binding constants, ( i, shown in Section 4.6 that the extent... 

We note that in spite of the large differences in the two sets of results reported in Table 5.3, the binding isotherms computed with these three sets of results were almost indistinguishable on the scale of Fig. 5.19. The most important differences between the calculated correlations and those reported by Senear et al. are, first, there is a large negative cooperativity between sites a and c, while Senear et al. assumed from the outset that no long-range cooperativity exists, and second, the triplet correlation is not additive, i.e., neither Sj nor 62 is zero, while Senear et al. assumed from the outset that 82 = 0. 

Akaganeite particles The first inflection point corresponds to a correlation time larger at 37°C (2.28 ns) than at 5°C (1.76 ns). This temperature dependence is typical of an electronic relaxation time (of Fe " "). The difference between the calculated correlation time and the theoretical relaxation time... 

Table II lists the chemical and physical measuranents that produce the feature space. A list of complaints as coded from the conminications frcm the peculation is also given. Application of the inter-feature correlation calculation, CORREL, on the chemical and physical features listed here, results in a limitation of the number of features without sacrificing too much information. Apart frcm the stability parameter, representing the meteorological conditions, seme chemical constituents of polluted air are found to be of importance in describing the situation (see Table III).

Heat of formation and density calculations correlate so well with performance parameter like detonation velocity that chemists have a good idea of the performance of an energetic material before its synthesis and testing. The pyrazolo[4,3-c]pyrazoles DNPP (9) and LLM-119 (10) were predicted to exhibit performances equal to 85 % and 104 % relative to that of HMX. 

Table 14 Comparison of calculated correlation energies with empirical correlation energies and post-Hartree-Fock energies...

The factor loadings also represent correlations between factors and variables and may be used to obtain the correlations between variables in the factor model. These calculated correlations should be close to the observed correlations. 

We note that /B- 0 if vB>vA, and /B- 1 if vBpowerful methods have been devised to calculate correlation factors. A survey can be found in [A. R. Allnatt, A. B. Lidiard (1993)]. Historically, the so-called five-frequency model introduced by [A. B. Lidiard (1956) R.E. Howard, A. B. Lidiard (1964)] played an important role in the understanding and quantitative treatment of correlation effects and is still widely used today. 

A comparision of the calculated correlation length for the infinite system with distinctive L used in computer simulations permits us to understand a nature of the instability of results observed in many statistical simulations. 

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