Empirical R values

The identity of the empirical r value for the solvolysis transition state with that for the corresponding gas-phase cation indicates that the structures of the transition states in benzylic solvolyses should be essentially constant and that... 

These theoretical considerations reveal that the empirical r values are intimately related to the theoretical indices of the structures of benzylic carbocations derived from resonance theory. The coefficient r in the Y-T equation can thus be replaced as a first approximation by a set of theoretical quantities, e.g. increment of bond orders (PAr,a)H as in (35) or sum of charge populations in the aryl ring (2 for the parent carbocations (X = H) as... 

FIGURE 1.13 The van der Waals interaction energy profile as a fnnction of the distance, r, between the centers of two atoms. The energy was calcnlated nsing the empirical equation U= B/r — A/r. (Values for the parameters B = 11.5 X 10 kJnm /mol and A = 5.96 X 10 kJnmVtiiol for the interaction between two carbon atoms are from Levitt, M., Journal of Molecular Biology... 

No extensive comparison with experiment to test the values in Table IV will be made. The close agreement between the purely theoretical and the experimental results in the case of helium and neon allows one to place confidence in the R values for ions with these structures and the same remark applies with less force in the case of the argon structure, where only a small empirical correction was introduced. It is interesting to note that the theoretical values 3-57 and 6-15 for the rubidium and the caesium ion agree very well with the experimental ones, 3-56 and 6-17 (Table III), which were not used at all in the evaluation of the empirical corrections for these structures. Finally, we may mention that our values agree in general with those of Fajans and WulfE.i obtained by them from the experimental R values for salt solutions by the application of only the simplest theoretical considerations. 

The A//het(R-R ) value, whieh is derived from Arnett s empirical linear (28) and (29), might be used as an index of feasibility of salt formation. The values calculated by use of (28) or (29) for the combination of [2 ] with [1+], [24+], [26+], [28+] and [40+] are not less than zero but in the range of 10-18 kcalmoP. Tliese A//het(R-R ) values seem to be too large for salt formation. Therefore, the salt formation might be controlled not only by electronic factors, but also by steric hindrance to bond formation. 

In the Bom like approaches to solvation energy, the electrostatic potential of the ion appears as the basic variable of the theory. From Eq (1), it may be seen that if we have accurate electron densities at hand, the electrostatic potential strongly depends on the ionic radius r. The choice of suitable ionic radii usually introduces some arbitrariness in the calculation of AESolv there is no a physical criterium to justify the use of empirical rA values coming from different sources [15-16],... 

Obviously whenever binding energy data are fit to Eq. (9), the empirically determined values of k and l automatically take into account the atomic relaxation energy, ntr. Equation 9 gives very good correlations for molecules which have similar structures, presumably because of the automatic accounting for R>ntr and the fact that similar molecules have similar R>W values and that - w is therefore absorbed into the constant /. 

The procedure adopted to portray the scope and utility of a linear free-energy relationship for aromatic substitution involves first a determination of the p-values for the reactions. These parameters are evaluated by plotting the values of log (k/ka) for a series of substituted benzenes against the values based on the solvolysis studies (Section IV, B). The resultant slope of the line is p, the reaction constant. The procedure is then reversed to assess the reliability and validity of the Extended Selectivity Treatment. In this approach the log ( K/ H) observations for a single substituent are plotted against p for a variety of reactions. This method assays the linear or non-linear response of each substituent to variations in the selectivity of the reagents and conditions. Unfortunately, insufficient data are available to allow the assignment of p for many reactions. It is more practical in these cases to adopt the Selectivity Factor S as a substitute for p and revert to the more empirical Selectivity Treatment for an examination of the behavior of the substituents. 

Nakata et al., 1996, 1999). TTie agreement between the theoretical dihedral angles 0caic and the empirical 0expi of the twisted benzyhc cations confirms that the observed decrease in the r value should be ascribed to a loss of resonance interaction caused by deviation from coplanarity of the carbocation centre and the benzene rr-system. 

For a large-scale application of electrostatic potentials — comparisons among large sets of molecules and investigations on big molecules — it would be desirable to be able to resort to semi-empirical wave functions, which can be computed a good deal faster than ab initio SCF ones. It is necessary, however, to ensure that the reliability of the W (r) values is not too much affected by going over to approximate wave functions. 

Zien, A., R. Zimmer, and T. Lengauer, Empirical p-values for Threading scores, in ISMB1999. 1999. Heidelberg. 

The defects of the SHM arise from the fact that it treats only n electrons, and these only very approximately. The basic Hiickel method described here has been augmented in an attempt to handle non- r substituents, e.g. alkyl groups, halogen groups, etc., and heteroatoms instead of carbon. This has been done by treating the substituents as jr centers and embodying empirically altered values of a and so that in the Fock matrix values other than -1 and 0 appear. However, the values of these modified parameters that have been employed vary considerably [51], which tends to diminish one s confidence in their rehabihty. 

The r value is commonly employed to quantify the degree of association between predicted values (from either a physics-based or empirical model) and observed values from eqn (9.1). The endpoints could be as diverse as estimates of affinity from 3-dimensional protein-ligand complexes, to estimates of solubility from a quantitative structure-activity model. The coefficient of variation (r ) expresses the fraction of the variation in the observed values that is explained by the predicted values, or more generally the fraction of the variation in the y-data that is explained by the x-data. 

TaUe XIV summarizes the comparisons made to date between experi-mqital and calculated parameters involved in these correlations ... 

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