Reciprocal polarizabilities

Figure 3.9 C44 elastic moduli vs. reciprocal polarizabilities for prototype alkali halide crystals.

Figure 9.1 Shows the linear correlation between hardness and reciprocal polarizability for 11 alkali halides. The polarizability data are from Ruffa (1963), and the hardness data from Sirdeshmukh et al. (1995).

The connection between hardness and a measure of the ease with which a crystal can change its shape is its reciprocal polarizability (as shown above). The softer the crystal, the greater its polarizability. The hardnesses have the dimensions of pressure, and the polarizabilities are derived from the dielectric constants through the Clausius-Mosottii equation. That is ... 

It is shown that the stabilities of solids can be related to Parr s physical hardness parameter for solids, and that this is proportional to Pearson s chemical hardness parameter for molecules. For sp-bonded metals, the bulk moduli correlate with the chemical hardness density (CffD), and for covalently bonded crystals, the octahedral shear moduli correlate with CHD. By analogy with molecules, the chemical hardness is related to the gap in the spectrum of bonding energies. This is verified for the Group IV elements and the isoelec-tronic III-V compounds. Since polarization requires excitation of the valence electrons, polarizability is related to band-gaps, and thence to chemical hardness and elastic moduli. Another measure of stability is indentation hardness, and it is shown that this correlates linearly with reciprocal polarizability. Finally, it is shown that theoretical values of critical transformation pressures correlate linearly with indentation hardness numbers, so the latter are a good measure of phase stability. 

The good agreement obtained between the measured and calculated d.m. (taking into account the N-methyl group d.m.) is evidence of a slight reciprocal polarizable interaction between the imidazole and pyridine rings. This confirms the adequacy of the chosen method used for the calculation of the resultant d.m. for IbP. 

A similar linear dependence is found for their reciprocal molecular volumes which are proportional to their polarizabilities, a. Thus H is expected to be proportional to 1/a and indeed it is (Dimitrov and Komatsu, 2002). See Figure 11.5. Furthermore, since their shear moduli C44 are proportional to 1/a (Gilman, 1997), the graph also indicates that their hardnesses are proportional to their shear moduli (Singh et al., 2007). 

Methods for determining permanent dipole moments and polarizabilities can be arbitrarily divided into two groups. The first is based on measuring bulk phase electrical properties of vapors, liquids, or solutions as functions of field strength, temperature, concentration, etc. following methods proposed by Debye and elaborated by Onsager. In the older Debye approach the isotope effects on the dielectric constant and thence the bulk polarization, AP, are plotted vs. reciprocal temperature and the isotope effect on the polarizability and permanent dipole moment recovered from the intercept and slope, respectively, using Equation 12.5. 

Pearson35,36 and Parr and co-workers366 c developed the principle of maximum hardness, which states that reacting molecules will arrange their electrons so as to be as hard as possible. Chemical equilibrium, then, is the state of maximum hardness. Soft donors prefer soft acceptors because both partners can increase their hardness by reacting with one another—the shared electrons flow to become less polarizable. To implement this theory quantitatively, Pearson et al. introduced scales of absolute hardness rj and its reciprocal, softness a ... 

Fro. IX-1.—Values of the ratio of polarisation P to field strength E for hydrogen chloride gas, as a function of the reciprocal of the absolute temperature. The slope of the line is a measure of the permanent electric dipole moment of the molecules, and the intercept of the line is a measure of the temperature-independent polarizability of the molecules. 

There is therefore a correlation between the rate and the reciprocal of the self-polarizability, due to this accidental first-order correspondence between it and the reactivity number. The correlation does not in any way support the physical model on which it was originally based. 

In the language of reciprocal space, nonlocal metal response refers to the dependence of the metal dielectric constant on the wavevector k of the various plane waves into which any probing electric fields can be decomposed. Such an effect is often mentioned in reports on SERS, but it is usually neglected. One of the oldest papers addressing the importance of nonlocal effects on the polarizability of an adsorbed molecule is the article by Antoniewicz, who studied the static polarizability of a polarizable point dipole close to a linearized Thomas-Fermi metal [63], The static dielectric constant eTF(k) of such a model metal can be written as ... 

Electrostatic Terms. The electrostatic energy of a lattice of atoms of zero polarizability may be calculated exactly by the method of Bertaut, provided the position and charge of each atom in the structure are known, This method involves the infinite sum in reciprocal space... 

Here a is the polarizability of the molecule. An oscillating dipole emits electromagnetic waves in all directions with electric field proportional to the acceleration of charges (d p/dt ) and decaying reciprocally with the distance r from the molecule. For a detector located in the horizontal yz plane at distance r = /y + from the origin (see Fig. 1.23) the scattered wave has electric field... 

For pure liquids, the Debye equation suggests that the molar polarization should be a linear function of the reciprocal temperature. Furthermore, one should be able to analyze relative permittivity data for a polar liquid like water as a function of temperature to obtain the dipole moment and polarizability from the slope and intercept, respectively. In fact, if one constructs such a plot using data for a polar solvent, one obtains results which are unreasonable on the basis of known values of p and ocp from gas phase measurements. The reason for the failure of the Debye model in liquids is the fact that it neglects the field due to dipoles in the immediate vicinity of a given molecule. However, it provides a reasonable description of the dielectric properties of dilute polar gases. In liquids, relatively strong forces, both electrostatic and chemical, determine the relative orientation of the molecules in the system, and lead to an error in the estimation of the orientational component of the molar polarization. 

With optical techniques, vibrational dynamics are probed on spatial scales much greater than molecular sizes, or unit cell dimensions in crystals, commonly encountered. The scale is directly related to the wavelength of the incident radiation (in the range fl om 1 to 10(X) p,m in the infrared or about 0.5 p,m for Raman). Oscillators at very short distances, compared to the wavelength, are excited exclusively in phase. For molecules, only overall variations of the dipole moment or polarizability tensor can be probed. In crystals, only a very thin slice of reciprocal space about the centte of the BriUouin zone (k 0) can be probed. This corresponds to in-phase vibrations of a virtuaUy infinite number of unit cells. With optical techniques, band intensities are largely determined by symmetry-related selection rules, although these rales hold only in the harmonic approximation. 

Dielectric constants Liquid fraction at equilibrium Initial liquid fraction in a foam liquid fraction in the dry portion of a foam Polarizability of water Inverse of dimensionless surface viscosity Surface viscosity Reciprocal Debye length Decay length in Eq. (24)... 

For D D , the reciprocals of these correlation times are raised by an amount [(D /D ) — 1] as indicated in Eq. (7.55). As noted previously, K rriL, ttim) values at a particular temperature are computed from (P2) and (P4). The fourth-rank order parameter P4) cannot be directly measmed from a NMR spectrmn, but may be derived from measurements of the mean square value of a second-rank quantity [7.19-7.22]. In the Raman scattering technique [7.21], the second-rank molecular quantity is the differential polarizability tensor of a localized Raman mode. In fluorescence depolarization [7.19], the average of the product of the absorption and emission tensors is used to determine (P4). Since there is a lack of experimental determination of (P4) in liquid crystals, this may be calculated based on the Maier-Saupe potential... 

On the other hand the description of basis stacking provided by the empirical potential study of Poltev and Shulyupina was quite successful [22]. This unexpected agreement between modem quantum-chemical and empirical potential results can be easily understood. The most demanding part of a quantum-chemical treatment of base stacking is a proper description of intermolecular electron correlation which is responsible for the dispersion attraction. Semi-empirical methods have never succeeded in including this contribution, and it is still not within the reach of DFT techniques. The use of the MP2 method is the minimal requirement. On the other hand, the dispersion attraction is a rather isotropic contribution. It can be well described by the simple empirical London dispersion energy proportional to polarizabilities and the sixth power of the reciprocal interatomic distance. 

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