Anisotropic Bond Polarizabilities

The principal polarizabilities of molecules can be analysed in terms of anisotropic bond polarizabilities. A suggestion that bonds may have different polarizabilities along their lengths and in the two perpendicular transverse directions was first made qualitatively by Meyer and Otter-bein (1931) and pursued quantitatively by Sachsse (1935), Wang (1939), and Denbigh (1940). If a bond X—Y has principal polarizabilities LY, Y and 6yY (L = longitudinal, T = transverse, V = vertical), then for a molecule XY2, with an angle YXY of 20°, we have 

Further, from the average polarizability of X—Y, via an equation analogous to (22), we have the sum (6YY + 6 Y + by Y). In fact, however, when 

1 Chantry and Plane (1960) have recently suggested that measurements of absolute Raman intensities could supply the required additional information provided the treatment by Volkenstein (1941, 1960) be accepted as valid. From data on CH4 and CCI4 fcL and bT are deduced for the C—H bond as 0-0858 and 0-0546 and for C—Cl as 0-376 and 0-204 respectively in the units used in Table 22. 

Fevre (1955, review) except where otherwise noted. 

Reference has already been made (p. 31) to the limited success of attempts to connect bond refractivities with bond lengths. Denbigh, using data available prior to 1940, suggested that it was the longitudinal, rather than the mean, polarizability that seemed to be related to the intercentre distances rA B in a link A—B. He proposed formulae of the type 

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Weber, H-J. Anisotropic bond polarizabilities in birefringent crystals. Acta Cryst. A44, 320-326 (1988). 

An important addition to the model was the inclusion of virtual particles representative of lone pairs on hydrogen bond acceptors [60], Their inclusion was motivated by the inability of the atom-based electrostatic model to treat interactions with water as a function of orientation. By distributing the atomic charges on to lone pairs it was possible to reproduce QM interaction energies as a function of orientation. The addition of lone pairs may be considered analogous to the use of atomic dipoles on such atoms. In the model, the polarizability is still maintained on the parent atom. In addition, anisotropic atomic polarizability, as described in Eq. (9-28), is included on hydrogen bond acceptors [65], Its inclusion allows for reproduction of QM polarization response as a function of orientation around S, O and N atoms and it facilitates reproduction of QM interaction energies with ions as a function of orientation. 

These equations use Cartesian tensor notation in which a repeated Greek suffix denotes summation over the three components, and where ay7 is the third-rank antisymmetric unit tensor. For a molecule composed entirely of idealized axially symmetric bonds, for which [3 (G )2 = /3(A)2 and aG1 = 0 [13, 15], a simple bond polarizability theory shows that ROA is generated exclusively by anisotropic scattering, and the CID expressions then reduce to [13]... 

Apart from a normal coordinate analysis, rather less information is required to perform an atom dipole interaction calculation than a bond polarizability calculation of ROA specifically, the derivatives of the local isotropic atomic polarizabilities with respect to the internal coordinates, which can be decuded from ordinary Raman intensities or else transferred from other (simpler) molecules. The anisotropic parts of the polarizability derivatives are generated automatically by the dipole-dipole interaction function. 

The additive contribution of the individual bonds to the anisotropic potential draws attention to the rule of the additivity of bond polarizabilities [122]. This strongly suggests that the average orientation of the substituted benzenes is directly related to their principal polarizabilities and that the anisotropic solute-solvent interaction is determined by London dispersion forces. Considering dispersion forces (in dipolar approximation) one obtains the following expression for the... 

In anisotropic materials, the electronic bonds may have different polarizabilities for different directions (you may think of different, orientation-dependent spring constants for the electronic harmonic oscillator). Remembering that only the E-vector of the light interacts with the electrons, we may use polarized light to test the polarizability of the material in different directions, lno is one of the most important electro-optic materials and we use it as an example. The common notations are shown in Figure 4.7. If the E-vector is in plane with the surface of the crystal, the wave is called a te wave. In this example, the te wave would experience the ordinary index na of LiNbOs (nG 2.20). If we rotate the polarization by 90°, the E-ve ctor will be vertical to the surface and the wave is called tm. In lno, it will experience the extraordinary index ne 2.29. Therefore these two differently polarized waves will propagate with different phase velocities v c/n. In the example of Figure 4.7, the te mode is faster than the tm mode. 

The so-called polarizable continuum model (PCM) offers a unified and well sound framework for the evaluation of all these contributions both for isotropic and anisotropic solutions. In PCM, the solute molecule (possibly supplemented by some strongly bound solvent molecules, to include short-range effects such as hydrogen bonds) is embedded in a cavity formed by the envelope of spheres centered on the solute atoms. The procedures to assign the atomic radii and to form the cavity have been described in detail together with effective classical approaches for evaluating K vand ,... 

The induction term is calculated using atomic and bond anisotropic polarizabilities to reproduce molecular polarizability, but fitted with isotropic atomic polarizabilities. 

Define a locally anisotropic property as a property whose value is highly sensitive to whether each atom or bond is in the chain backbone or in a side group. By contrast, while the value of a globally anisotropic property may be very sensitive to the overall orientation of the polymer chains, it is somewhat less sensitive to the precise location of any given atom or bond in the structure. For example, the refractive index is a globally anisotropic property. The stress-optic coefficient, whose value depends both on the refractive index and on the difference in the polarizability of a polymer chain segment parallel and perpendicular to the chain, is a locally... 

The b s quoted (from Mortensen and Smith, 1960) represent the two bonds as being more anisotropic than they appear in Table 22, but Ziircher comments that the Mortensen-Smith polarizabilities should be taken with caution. He stresses that his own calculations have an approximate character and involve quantities and equations of varying precision. Nevertheless their interest lies in the indication that the directions of greatest diamagnetic susceptibility in bonds should be perpendicular to those of greatest electronic polarizability and vice-versa. Such an inverse relationship is in accord with empirical (Bothner-By and Naar-Colin, 1958) and theoretical (Pople, 1957) deductions to date rough predictions of the shielding of a proton by a remote bond... 

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