Ellipsoid of polarizability

The polarizability of a body can be mathematically described by an ellipsoid of polarizability possessing three orthogonal semi-axes fij, 2> 63. By the equation x lb ) + y fb + z jh%)—, points... 

The polarizability can be described by an ellipsoid of polarizability P This ellipsoid is a surface such that the distance to the surface from the origin in any direction is proportional to in that direction. The ellipsoid of polarizability will generally possess all of the symmetry elements of the molecule. If a rotational or vibrational motion modulates the polarizability the value of the polarizability in some direction varies periodically as the motion occurs. This corresponds to changes in the shape or orientation of the ellipsoid of polarizability. 

There are two contributions to the polarizability of a molecule the distortion of the electronic wave function and the distortion of the nuclear framework. The major contribution is from the electrons, and can be considered to be the sum of contributions from the individual electrons. The contributions of the inner-shell electrons are nearly independent of orientation and these contributions can be ignored. The polarizability of electrons in a bond parallel to the bond direction is different from the polarizability perpendicular to that bond. As a diatomic molecule or linear polyatomic molecule rotates, the components of the polarizability in fixed directions are modulated (fluctuate periodically) as the ellipsoid of polarizability rotates. The rotation of a diatomic or linear polyatomic molecule will be Raman active (produce a Raman spectram). In a nonlinear polyatomic molecule, the polarizabilities of the individual bonds add vectorially to make up the total polarizability. If the molecule is a symmetric top, the total polarizability is the same in all directions and the ellipsoid of polarizability is a sphere. A spherical top molecule has no rotational Raman spectmm. Symmetric tops and asymmetric tops have anisotropic polarizabilities and produce rotational Raman spectra. 

For a polyatomic molecule the polarizability P in a principal direction of the polarizability ellipsoid of the molecule is given by... 

The oriented test ellipsoid T may be chosen to represent an external electromagnetic field, or the main direction of a cavity of an enzyme molecule, or a polarizability ellipsoid of a molecule, or an alignment on the surface of a catalyst, or some other internal or external constraint [199]. 

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... 

At least three observational equations are necessary (unless, from symmetry, a polarizability ellipsoid of revolution, with two equal semi-axes, can be safely foreseen, e.g. as with HX, CH3X, CHX3, etc.). The first of these is provided by (22), whereby the sum 61 + 62+63 may be extracted from the electronic polarization FP of a molecule ... 

Such axes are called principal axes of polarizability. In terms of the polarizability ellipsoid, the vibration is Raman-active if the polarizability ellipsoid changes in size, shape, or orientation during the vibration. 

Fig. 1-8. Changes of polarizability ellipsoids during the normal vibrations of C03 and HjO.

Principal a, os, of inertia, 284 of polarizability ellipsoid, 44 Principal moments of inertia, 284 Principal values, of moments of inertia, 284... 

Here A. is a geometric factor reflecting the ellipsoid of rotation of the ion pair to be obtained from its dimensions and f. is the reaction field, expressions for which having been reported by Barthel et al. [39], /<. and are the dipole moment and polarizability of the dipolar species, and g. is the dipole orientation correlation factor, generally set as = 1. Eventually the concentrations of the ion pair species, c., are used for the calculation of the partial association equilibrium quotients, K, K, and and the overall constant which may be compared with values obtained by other methods. 

The spherical shell model can only account for tire major shell closings. For open shell clusters, ellipsoidal distortions occur [47], leading to subshell closings which account for the fine stmctures in figure C1.1.2(a ). The electron shell model is one of tire most successful models emerging from cluster physics. The electron shell effects are observed in many physical properties of tire simple metal clusters, including tlieir ionization potentials, electron affinities, polarizabilities and collective excitations [34]. 

The polarizability ellipsoid rotates with the molecule at a frequency say, and the radiation sees the polarizability changing at twice the frequency of rotation since, as can be seen from Figure 5.14, the ellipsoid appears the same for a rotation by n radians about any of the cartesian axes. The variation of a with rotation is given by... 

The polarizability tensor may therefore be defined by a set of nine components which reduce in number to six because the tensor is symmetric. The physical significance of molecular polarizability is often explained in terms of the polarizability ellipsoid which is defined by the equation ... 

With the new coordinate system only the three diagonal components axx, ayy, and olzz referred to as principal values of a are nonzero. The halfaxes of the ellipsoid are a 2, ay]/2, and aj1/2. If the polarizability ellipsoid... 

Conceptually, the self-consistent reaction field (SCRF) model is the simplest method for inclusion of environment implicitly in the semi-empirical Hamiltonian24, and has been the subject of several detailed reviews24,25,66. In SCRF calculations, the QM system of interest (solute) is placed into a cavity within a polarizable medium of dielectric constant e (Fig. 2.2). For ease of computation, the cavity is assumed to be spherical and have a radius ro, although expressions similar to those outlined below have been developed for ellipsoidal cavities67. Using ideas from classical electrostatics, we can show that the interaction potential can be expressed as a function of the charge and multipole moments of the solute. For ease... 

From the viewpoint of quantum mechanics, the polarization process cannot be continuous, but must involve a quantized transition from one state to another. Also, the transition must involve a change in the shape of the initial spherical charge distribution to an elongated shape (ellipsoidal). Thus an s-type wave function must become a p-type (or higher order) function. This requires an excitation energy call it A. Straightforward perturbation theory, applied to the Schroedinger aquation, then yields a simple expression for the polarizability (Atkins and Friedman, 1997) ... 

As in infrared spectroscopy, not all vibrations are observable. A vibration is Raman active if it changes the polarizability of the molecule. This requires in general that the molecule changes its shape. For example, the vibration of a hypothetical spherical molecule between the extremes of a disk-shaped and a cigar-shaped ellipsoid would be Raman active. We recall that the selection rule for infrared spectroscopy was that a dipole moment must change during the vibration. As a consequence the stretch vibrations of for example H2 (4160.2 cm"1), N2 (2330.7 cm-1) and 02 (1554.7 cm"1) are observed in Raman spectroscopy but not in infrared. The two techniques thus complement each other, in particular for highly symmetrical molecules. 

These problems were partially solved through the inclusion of multipole expansions in ellipsoidal cavities [23] or through the use of the polarizable continuum method... 

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... 

---