Mean polarizability

Intensities of Raman transitions are proportional to R and therefore, from Equation (6.13), to (da/dx)g. Since a is a tensor property we cannot illustrate easily its variation with x instead we use the mean polarizability a, where... 

Figure 6.2 Variation of the mean polarizability c/. with intemuclear distance r in a diatomic molecule...

Here, auir) is the mean polarizability of a pair of species separated by distance r, while pn is their electric dipole moment and U the potential energy, p o is the permanent electric dipole moment for the two species at infinity. 

A comparison of MP2/6-31G structural parameters of 1,2-oxazole 19 (isox-azole) and 1,3-oxazole 20 with microwave data is provided by Kassimi et al. (Scheme 16) [96JPC8752]. The general agreement is excellent. The same authors investigated dipole moments, quadrupole moments, octopole moments, and dipole polarizabilities of 19 and 20 together with several oxadiazoles and oxatriazoles [96JPC8752, 99JPC(A) 10009]. For the mean polarizability of these species, they found the approximative formula... 

In this approximation the mean polarizability a is given in atomic units and h is the number of hydrogen atoms. They founda mean absolute error of 1.2% and... 

Here (Oj is the excitation energy ErE0 and the sum runs over all excited states I of the system. From equation (5-37) we immediately see that the dynamic mean polarizability a(co) diverges for tOj=co, i. e has poles at the electronic excitation energies 0)j. The residues fj are the corresponding oscillator strengths. Translated into the Kohn-Sham scheme, the exact linear response can be expressed as the linear density response of a non-interacting... 

Table 12-2. Experimental values and deviation from experiment of the R0 H bond distance, the symmetric (vs) and the antisymmetric (vas) stretching frequency [cm-1], the dipole moment [D], and the mean polarizability [A3] of the water molecule. The aug-cc-pVTZ basis set is used throughout. 

Miller and Savchik (12) have given Equation 1 for estimating the mean polarizability, a, of a molecule, where N is the total number of electrons in the molecule, and x.is a polarizability contribution for each atom i, characteristic of the atom type and its hybridization state. 

The optimized value of the electrostatic misfit coefficient, a, in Eq. (6.1) is 20% smaller than the simple electrostatic estimate including a mean polarizability correction. Considering the number of approximations included in the electrostatic misfit picture, this is a satisfactory agreement. 

The Lorentz-Lorenz equation can be used to express the components of the refractive index tensor in terms of the polarizability tensor. Recognizing that the birefringence normalized by the mean refractive index is normally very small, ( A/i / 1), it is assumed that Aa /a 1, where the mean polarizability is a = (al + 2oc2)/3 and the polarizability anisotropy is Aa = a1-a2. It is expected that the macroscopic refractive... 

Equation (7.34) is the leading term in a near field expansion of the electric field generated from a radiating dipole. Substituting equations (7.34) and (7.33) into equation (7.32), and multiplying the result by the mean polarizability ot- of segment i, the following... 

The mean-polarizability approximation, discussed in detail by Agranovitch,16 presents the same advantages (simplicity, arbitrary concentrations, etc.), and the same limitations as the average-locator approximation in particular, this theory provides two bands of persistence behavior for all values of the parameters. This may be checked on the example of a cubic crystal, where the local field has a very simple form The modes of the mixed crystal are given by... 

Figure 4.22. Polariton solutions for a 3D mixed crystal in the mean-polarizability approximation (4.117). In strong local field (A3), one obtains a resonance of the virtual crystal cAwA + cBwB another solution, strongly shifted, exists at low frequencies. On the contrary, in weak local fields (A,), the frequencies of the pure A and B crystals, slightly shifted, are solutions. We note that for cB - 0, one of the solutions tends, for any strength of the local field A, to ojb, which is the frequency of B unshifted by the interaction with the lattice A.

In appraising the average accuracy one must bear in mind that experimentally the mean polarizabilities are usually obtained from the refractive index n (at 5893 A, the sodium D-line) and the Lorenz-Lorentz equation (with M molecular weight, p macroscopic density, Vav Avogadro s number) ... 

In the discussion of Raman and ROA theory an important concept is that of the invariants. The values of invariants, the simplest example of which is the mean polarizability a, do not alter if the principal axes of the molecule are rotated. 

---