Optimum correlation equations

Further studies of the 2-substituted 5-nitroanilines, conducted by Kier and coworkers, searched for a linear combination of structural variables that describes a line, plane, or surface that separates the molecule classes in the optimum manner. They found that sweetness correlated very well with the substituent polarizability-constants for the 2-substituent, implicating the involvement of the 2-substituents in dispersive-binding interactions at the receptor. This is in agreement with the results of Hansch " and McFarland. The correlation equation was not, however, reported. 

Optimum Correlation Equations. Table XI gives the final best-fit Hansch relationships which mathematically describe the pre-emergence herbicidal activity of the 3- and 4-substituted TFMS compounds under all conditions examined in this study. The results and conclusions reached... 

Referring to Equation 4b, it is evident that there is a statistically significant parabolic relationship between apara and x for the 4-substituted TFMS derivatives. Both x2 and x terms are necessary in the optimum correlation equation, as evidenced by the fact that the percent correlation (r2) jumps from 2% in Equation 4a to 90% in Equation 4b upon stepwise addition of the linear term in x- No simple linear correlation between apara and x was found. Fits of the form o-para = f°r ex ... 

Calculate the optimum reflux ratio, Ro, from the Van Winkle and Todd correlation, (Equations 6.27.5 to 6.27.7). 

Each of the two descriptors Foct and AE appear in the correlation equation (eq. 9) with parabolic dependence. This means that each descriptor possesses an optimum value which yields a maximum, or minimum, contribution to determining TD50 These optimum values can be determined by simple partial differentiation of eq. (9),... 

Using the mixture ratio in this equation, they wrote the following correlation for optimum mixing ... 

Optimum capacity utilization ia the two dimensions of time and equipment are cmcial to the overall performance, and miming a fine chemicals company has been described as gap management. Attempts have been made to develop adequate equations for describiag the correlation between... 

In these equations, Dmax is the larger of the summed values of STERIMOL parameters, Bj, for the opposite pair 68). It expresses the maximum total width of substituents. The coefficients of the ct° terms in Eqs. 37 to 39 were virtually equal to that in Eq. 40. This means that the a° terms essentially represent the hydrolytic reactivity of an ester itself and are virtually independent of cyclodextrin catalysis. The catalytic effect of cyclodextrin is only involved in the Dmax term. Interestingly, the coefficient of Draax was negative in Eq. 37 and positive in Eq. 38. This fact indicates that bulky substituents at the meta position are favorable, while those at the para position unfavorable, for the rate acceleration in the (S-cyclodextrin catalysis. Similar results have been obtained for a-cyclodextrin catalysis, but not for (S-cyclodextrin catalysis, by Silipo and Hansch described above. Equation 39 suggests the existence of an optimum diameter for the proper fit of m-substituents in the cavity of a-cyclodextrin. The optimum Dmax value was estimated from Eq. 39 as 4.4 A, which is approximately equivalent to the diameter of the a-cyclodextrin cavity. The situation is shown in Fig. 8. A similar parabolic relationship would be obtained for (5-cyclodextrin catalysis, too, if the correlation analysis involved phenyl acetates with such bulky substituents that they cannot be included within the (5-cyclodextrin cavity. 

Unlike CLS, ILS does not require that we provide concentration values for all of the components present. In equation [47], we are not trying to account for all of the absorbances in the spectra. Instead, the formulation allows us to pick up only that portion of the spectral absorbance that correlates well to the concentrations. If ILS is able to do so well with degraded spectra, imagine how much better we might do if we can find a more optimum way of reducing the dimensionality of the spectra than simply summing them into bins. That is precisely what PCR and PLS will do for us. 

Partitioning into the CNS will be important for hallucinogens, as for any drug that acts centrally. Correlation between 1-octanol/water partition coefficients and human activity has been reported (13). Regression analysis of log human activity on log P yielded a parabolic fit with an optimum at log P 3.14. The derived equation accounted for only 62% of the variance but included compounds with a variety of substitution patterns and, presumably, qualitative differences in activity. 

Martin and coworkers described an application of optimization to an existing tower separating propane and propylene. The lighter component (propylene) is more valuable than propane. For example, propylene and propane in the overhead product were both valued at 0.20/lb (a small amount of propane was allowable in the overhead), but propane in the bottoms was worth 0.12/lb and propylene 0.09/lb. The overhead stream had to be at least 95 percent propylene. Based on the data in Table E12.4A, we will determine the optimum reflux ratio for this column using derivations provided by McAvoy (personal communication, 1985). He employed correlations for column performance (operating equations) developed by Eduljee (1975). 

HF is the simplest of the ab initio methods, named after the fact that they provide approximate solutions to the electronic Schrodinger equation without the use of empirical parameters. More accurate, correlated, ab initio methods use an approximate form for the wavefunction that goes beyond the single Slater determinant used in HF theory, in that the wavefunction is approximated instead as a combination or mixture of several Slater determinants corresponding to different occupation patterns (or configurations) of the electrons in the molecular orbitals. When an optimum mixture of all possible configurations of the electrons is used, one obtains an exact solution to the electronic Schrodinger equation. This is, however, not computationally tractable. 

It should be noticed that from the activity data measured at a certain fixed pH, the correlations of equivalent quality are obtained both for the neutral and ionized forms. There are interrelations a priori between Equations 32 and 33 such as p — p = pA and c — c = pH — pKAstd, where pA is the Hammett reaction constant for the ionization equilibrium and pKA8td is the value of a standard compound. Thus, it is only possible to predict the molecular form responsible for the activity by comparing Equations 32 and 33 derived from data obtained at various pH. Moreover, the optimum value of log KA, log KA°, for the apparent activity of ionizable congeners can be derived by setting the derivative of either Equation 32 or 33 equal to zero as shown in Equations 44 and 45. 

The great similarity of Equations 37 and 38 highlights the common mechanism of action of two superficially different types of amides. In each of the above examples, compounds having log P values below the optimum of log P0 were studied. Hence, addition of a term in (log P)2 did not result in an improved correlation in either case. The confidence intervals on the intercepts of Equations 37 and 38 are tighter than those on eight sets of parabolic equations correlating hypnotic activity of barbiturates in a variety of animals (22). However, for six of the parabolic equations with moderately good confidence intervals, a mean intercept... 

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