Mean value, polarizability

The next step towards increasing the accuracy in estimating molecular properties is to use different contributions for atoms in different hybridi2ation states. This simple extension is sufficient to reproduce mean molecular polarizabilities to within 1-3 % of the experimental value. The estimation of mean molecular polarizabilities from atomic refractions has a long history, dating back to around 1911 [7], Miller and Sav-chik were the first to propose a method that considered atom hybridization in which each atom is characterized by its state of atomic hybridization [8]. They derived a formula for calculating these contributions on the basis of a theoretical interpretation of variational perturbation results and on the basis of molecular orbital theory. 

Table 7-1 lists some comparisons between experimental mean molecular polarizabilities and those estimated by Eq. (6). In this scheme, the estimation of mean molecular polarizability for acetic add needs five values, values for sp -C, for sp -C, for sp -O, for sp -O, and for a hydrogen atom. 

Table 7,1. Experimental mean molecular polarizabilities and values calculated by Eq. (6).

In many chemical applications, however, it would be more interesting to know how polarizability can stabilize a charge introduced into a molecule. Thus, rather than the global molecular property, mean molecular polarizability, a local, site-specific value for polarizability is needed. 

The first ab initio smdy of an interaction polarizability was that of O Brien et al. (1973) on a pair of helium atoms. They obtained /0(r) for the range r = 3.5ao through lOao- The experimentally determined value of is negative, which suggests that the incremental mean pair polarizability must be negative around the minimum in the potential curve. 

Hogeveen138 measured apparent acidity constants of substituted /i-phenylthio-, /J-phenylsulfinyl- and / -phenylsulfonyl-acrylic acids (cis and trans) in 50% v/v ethanol. The p values for transmission through SCH=CH, SOCH=CH and S02CH=CH were 0.531, 0.389 and 0.320 respectively for the cis acids and 0.652, 0.282 and 0.331 for the trans acids. These results were discussed in considerable detail and compared with those of pertinent related systems. Little importance was attached to the small differences between p values for cis/trans isomers, and the relative transmissions were taken as the mean p values 0.59,0.34 and 0.33. The superior transmission of SCH=CH was attributed to greater polarizability. The values for pKJtrans) — pKJcis) of isomeric acids were also discussed. For the sulfonyl acids this is almost constant at 0.1 unit for the sulfinyl adds there is some variation about a mean value of 0.26 unit, and for the thio adds there is also some variation about a mean value of — 0.15 unit. The differing behavior of the three systems in this respect was explained in terms of hypothetical conformations and electrostatic interactions therein. 

Fig. 1. BLYP/uncDZ mean dipole polarizability of the mercury atom as a function of frequency. All values in atomic units. SR+SO refers to calculations based on the Dirac-Coulomb Hamiltonians, whereas SR refers to calculations in which all spin-orbit interaction has been eliminated.

Nonequilibrium solvation model for the electric dipole polarizability. b Result corrected for local field effects. c Ref.[27], mean value for T between 283.1 5 and 293.1 5 K. 

Fig. 16.2 Convergence of the average value of the mean dipole polarizability of liquid Ar using Model 1, with respect to the number of MC structures included in the quantum calculation.

Fig. 16.3 Histogram and normal distribution of calculated values for the dipole polarizability of liquid Ar, using Model 1. Average polarizability is (a) = 11.59 ao3 and statistical error, v=0.03 ao3. Also shown (below) are the individually calculated mean dipole polarizabilities of the different configurations used...

This invariant can be thought of as measuring the average deviation of the polarizability from its mean value, and is therefore known as the anisotropy of n. 

As structural criteria of aromaticity, indexes N= + b (where R are bond lengths, and a and b are parameters characteristic for a given pair of atoms) and Julg s parameter J = 1 — 225/ S,s(l —d Jdf (where n is the number of peripheral bonds rs, are their lengths, and d is their mean length) have been applied. The less the value SaA and closer to 1 the value J, the greater is the aromaticity of a heterocycle. Polarizability exaltations F = M — M , where M and M arc the mean dipole polarizability and the mean atomic or group polarizability, respectively. 

Numerical values of linear and non-linear polarizabilities of atoms and molecules calculated theoretically and determined from experimental data. Only mean values are given a = (ou + a%i + 033)/s, b = (6u3 + 6333 +... 

The interaction of the carbon dioxide molecule with the sieve includes electrostatic, induction, dispersion, and repulsion contributions. The CO2 molecule was assumed to be capable of free rotation, so that the directional interactions could be averaged over all orientations using a Boltzmann weighting factor (JJ) this causes the electrostatic ion-quadrupole interaction to depend on the temperature. Mean values were used for the polarizability (a), the diamagnetic susceptibility (x), and the equilibrium radius of the CO2 molecule. Using vector summation for the total electric field at the CO2 molecule, the total potential, c(r), at a given position r is given by ... 

The diamagnetic susceptibility of carbonyl dibromide has never been measured. It has been estimated, however, from the computed value of the mean molecular polarizability (89.24 X 10"25 cm3) the mass diamagnetic susceptibility, Xm> estimated as... 

The values of w, at which the matrix fl — w2I becomes singular, are very useful because they can be combined with the sum-over-states expression for the mean dynamic polarizability ... 

It was found that the value of the factor F decreases in the series Li+, Na+, K+, Rb+, Cs+ and attains the mean values 1.45, 1.36, 1.31, 1.29, and 1.26, respectively. This can be explained by the dependence of this factor on the size and polarizability on ions. For lithium salts, the value of the factor F is higher than /2. This is not surprising since in this case the effect of the anion-anion repulsion should be considered. Similarly, the... 

We must add a remark with regard to polarisability. In what precedes we have taken account only of the mean value of a over all directions—a procedure which, in the case of a gas, whose molecules ban rotate freely, is certainly permissible as a first approximation. But by suitable experiments we can also determine the anisotropy of the polarizability, and so also form for ourselves a picture of the anisotropy of the electron cloud. We have already mentioned (p. 230) that the polarizability is a tensor, and can be represented by the so-called ellipsoid of polarization (see fig. 3). This has the following... 

In a homogeneous liquid or gas, the molecules are randomly orientated, and we must consider the polarizability components averaged over all molecular orientations. The results are expressed in terms of two quantities a mean value) and y (anisotropy) ... 

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