Ellipsoid, polarizability

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... 

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

The pure-rotational Raman spectrum of a polyatomic molecule provides information on the moments of inertia, hence allowing a structural determination. For a molecule to exhibit a pure-rotational Raman spectrum, the polarizability must be anisotropic that is, the polarizability ellipsoid must not be a sphere. As noted in Section 5.2, a spherical top has a spherical polarizability ellipsoid, and so gives no pure-rotational Raman spectrum. Symmetric and asymmetric tops have asymmetric polarizabilities. The structures of several nonpolar molecules (which cannot be studied by microwave spectroscopy) have been determined from their pure-rotational Raman spectra these include F2, C2H4, and C6H6. 

Figure 1-15 Changes in polarizability ellipsoids during vibration of CO2 molecule.

Figure 1-17 Changes in polarizability ellipsoid during normal vibrations of H2O molecule.

Suppose that a tetrahedral molecule such as CCI4 is irradiated by plane polarized light (E ). Then, the induced dipole (Section 1.7) also oscillates in the same yz-plane. If the molecule is performing the totally symmetric vibration, the polarizability ellipsoid is always sphere-like namely, the molecule is polarized equally in every direction. Under such a circumstance, I (Iy) = 0 since the oscillating dipole emitting the radiation is confined to the xz-plane. Thus, pp = 0. Such a vibration is called polarized (abbreviated as p). In liquids and solutions, molecules take random orientations. Yet this conclusion holds since the polarizability ellipsoid is spherical throughout the totally symmetric vibration. 

If the molecule is performing a non-totally symmetric vibration, the polarizability ellipsoid changes its shape from a sphere to an ellipsoid during the... 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a - w w) is a symmetric second-rank tensor like Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, and If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like /J is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of /3 (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of /3 is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)-(44). 

As we see, the parameter 1 results from Langevin reorientation of the polarizability ellipsoid and is always positive. The second of the above parameters, 2, corresponds to Bom s term in the Kerr effect and can be positive or negative, depending on the electric structure of the molecule. The third, the Debye parameter 3, has no counterpart in other phenomena of molecular orientation, and is specific to the non-linear dielectric behaviour of dipolar substances. 

General Treatment of Fluctuational Processes. The previous treatment is good only as long as we deal with strongly dipolar substances and all other polarizational effects remain negligible. In the majority of substances, besides reorientation of permanent dipoles, one has to consider reorientation of the polarizability ellipsoids as well as statistical-fluctuational processes. In calculating the electric polarization (277), one has to include the term accounting for linear distortional polarizability of the dielectric (non-linear polarizabilities are dealt with below) ... 

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]. 

The ellipsoid described by the above equation conforms to the shape of the electron clouds surrounding the entire molecule. The ellipsoid may be further regarded as representing the polarizability because vibrational modes which induce a change in the size or shape of the polarizability ellipsoid are considered to be Raman active modes. 

Figure 9.13 illustrates the change of the polarizability ellipsoid in normal mode vibrations of the H2O molecule. Its iq, i 2 and m modes are all Raman active because the ellipsoid changes size, shape or orientation as shown in Figure 9.13. We should interpret the ellipsoid change with caution. Some modes with apparent ellipsoid change with vibrations are not Raman active. For example, the change of the polarizability ellipsoid in CO2 vibrations (Figure 9.14) does not mean Raman active in all modes. The symmetric stretching model, v, of CO2 is Raman active because the ellipsoid size changes during vibration. However, i>2 and m of CO2 are...

Figure 9.13 Normal modes of H2O vibrations and changes in polarizability ellipsoids. Vj, V2 and v3 are all Raman active. (Reproduced with permission from J.R. Ferraro, K. Nakamoto, and C.W. Brown, Introductory Raman Spectroscopy, 2nd ed., Academic Press, San Diego. 2003 Elsevier B.V.)...

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 ... 

In the general case, when fi acts at known finite angles with the 1,2, and 3 directions, the components /xx, /x2, and /x3 must be inserted into (26), and extraction of the b, s from (22), (26), and (27) becomes more tedious. Some typical polarizability ellipsoids, specified by their principal semiaxes, and drawn from values, are shown in Table 21. On p. 291... 

Changes in the polarizability tensor can be visualized if we draw a polarizability ellipsoid by plotting 1 / oc in every direction from the origin. This gives a three-dimensional surface such as shown below ... 

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. 

---