Polarizability mean, approximation

As argued above, this result is found to work best for substances in which both the 1,1 and 2,2 forces are either London or dipole-dipole. Even the case of one molecule with a permanent dipole moment interacting with a molecule which has only polarizability and no permanent dipole moment-such species interact by permanent dipole-induced dipole attraction-is not satisfactorily approximated by Eq. (8.46). In this context the like dissolves like rule means like with respect to the origin of intermolecular forces. 

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... 

While Onsager s formula has been widely used, there have also been numerous efforts to improve and generalize it. An obvious matter for concern is the cavity. The results are very sensitive to its size, since Eqs. (33) and (35) contain the radius raised to the third power. Within the spherical approximation, the radius can be obtained from the molar volume, as determined by some empirical means, for example from the density, the molar refraction, polarizability, gas viscosity, etc.90 However the volumes obtained by such methods can differ considerably. The shape of the cavity is also an important issue. Ideally, it should be that of the molecule, and the latter should completely fill the cavity. Even if the second condition is not satisfied, as by a point dipole, at least the shape of the cavity should be more realistic most molecules are not well represented by spheres. There was accordingly, already some time ago, considerable interest in progressing to more suitable cavities, such as spheroids91 92 and ellipsoids,93 using appropriate coordinate systems. Such shapes... 

The notion of homogeneity is not absolute all substances are inhomogeneous upon sufficiently close inspection. Thus, the description of the interaction of an electromagnetic wave with any medium by means of a spatially uniform dielectric function is ultimately statistical, and its validity requires that the constituents—whatever their nature—be small compared with the wavelength. It is for this reason that the optical properties of media usually considered to be homogeneous—pure liquids, for example—are adequately described to first approximation by a dielectric function. There is no sharp distinction between such molecular media and those composed of small particles each of which contains sufficiently many molecules that they can be individually assigned a bulk dielectric function we may consider the particles to be giant molecules with polarizabilities determined by their composition and shape. 

Expressions for the optical anisotropy AT of Kuhn s random link (an equivalent to the stress-optical coefficient) of stereo-irregular and multirepeat polymers are derived on the basis of the additivity principle of bond polarizabilities and the RIS approximation for rotations about skeletal bonds. Expressions for the unperturbed mean-square end-to-end distance , which are required in the calculation of Ar, are also obtained. 

The infinite sum may be avoided by means of the closure method 2 I >< Pm = 1. An average energy is chosen for all the excited states, which enables the polarizability to be written in terms of a constant denominator and matrix elements for the ground-state wavefunction only. This is only a rough approximation which, though widely applied in approximate calculations, is of little use for more than an order of magnitude estimate of a. 

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. 

Being a pragmatic, adopting sometimes very rough approximations, required simplicity, efficacy, rapidity, but that does not mean that Barriol sacrificed the scientist s principles on the altar of the results. On the contrary, one finds here the occasion to point out that this theoretical chemist at work was theoretical in the most classical sense of the word The one who thinks about the bases, who works on the foundations. Every time Barriol followed his intuition, every time he adopted a rough, simple, and far-from-reality model, he justified his choice and showed, a posteriori, why it worked. It has been shown above how the non-acceptable results for polarizability, when extended from methane to hydrochloric acid, constituted an occasion to reexamine the precedently admitted hypothesis and more precisely, the theoretical significance of what was considered as a charge. Let us examine two other examples. 

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 a mean field and linear approximation, the induced dipole moments are related to the wave function through Eq. (13-14) and we introduce the polarizability and the electric fields in place of the induced moments by inserting this equation once into Eq. (13-18). Taking the expectation value of Eq. (13-8) we obtain the following expression for the QM/MM energy [24,54]... 

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