Rotational polarizability tensor

The second term in Eqs. (9.75) and (9.76), die rotational atomic polarizability tensor reflects the contribution of molecular translation and rigid-body rotation to ax- The inclusion of the six external molecular coordinates in those equations - the diree translations Xy and X2, and the three rotations p, Py and P2, completes die set of molecular coordinates up to 3N. In diis vray polarizability dmivatives are transformed into quantities corresponding to a space-fixed Cartesian coordinate system. As already pointed out in section 4.1, the great advantage of such a step is that the imensity parameters defined in terms of a space-fixed coordinate system are independent on isotopic substitutions provided the symmetry of the molecule is preserved. This will be illustrated with an example in the succeeding section. By analogy with Eq. (9.77), die rotational polarizability tensor can be represented as... 

The atomic polarizability tensor ax for SO2 evaluated by employing Eq. (9.76) is shown as follows together with the vibrational and rotational polarizability tensors (in units A ) ... 

The atomic polarizability tensor of an atom a is defined by expression (9.73). Arranged in a row all ax tensors form the polarizability tensor of the molecule, ax [E<1-(9.74)]. Its elements can be obtained via relations (9.75) and (9.76). As was discussed in Section 9.IV, atomic polarizability tensors are sum of two arrays (i) vibrational and (ii) rotational polarizability tensors [Eq. (9.79)]. The elements of atomic polarizability tensors are interconnected by the dependency conditions (9.84) and (9.85). The presence of such relations hampers die physical interpretation of these quantities. 

The components of the translation and rotation vectors are given as Tx> Ty, T and RX Ry, Rz, respectively. The components of the polarizability tensor appear as linear combinations such as axx + (xyy> etc, that have the symmetry of the indicated irreducible representation. 

The theory of strain birefringence is elaborated in terms of the RIS model as applied to vinyl polymer chains. Additivity of the polarizability tensors for constituent groups is assumed. Stress-birefringence coefficients are calculated for PP and for PS. Statistical weight parameters which affect the Incidences of various rotational states are varied over ranges consistent with other evidence. The effects of these variations are explored in detail for isotactic and syndlotact/c chains. 

Over the last years we have explored several advanced techniques for high-resolution rotational coherence spectroscopy (RCS [1]) in order to study the structures of molecules and clusters in the gas phase [2]. We have provided spectroscopic examples demonstrating (i) mass-selectivity (Fig. 1, [3]), (ii) that the rotational constants of the ground and electronic excited states can be obtained independently with high precision (lO MO"5, [4]), (iii) that the transition dipole moment alignment, (iv) centrifugal distortion constants, and (v) information on the polarizability tensor can be obtained (Fig.l, [5]). Here we review results pertaining to points (i), (ii), (iv) and (v) [2,3,5],... 

The method of fs DFWM spectroscopy and our theoretical model for the spectral simulation is discussed in our second contribution in this volume. The experimental setup has been detailed in a former publication [5], Here, we would like to highlight some special features of this technique with emphasis on the possibility to obtain, besides the rotational constants, centrifugal distortion constants (CDs) and information on the polarizability tensor (PT). 

In inelastic Raman scattering a photon loses (or gains) one quantum of rotational or vibrational energy to (or from) the molecule. The process involves the electric field of the radiation inducing an electric dipole in the molecule and so depends on the polarizability tensor of the molecule. (A (second-order) tensor is a physical quantity with nine components.) The induced electric dipole D is proportional to the electric field E ... 

The polarizability tensor, a, introduced in section 4.1.2, is a measure of the facility of the electron distribution to distortion by an imposed electric field. The structure of the electron distribution will generally be anisotropic, giving rise to intrinsic birefringence. This optical anisotropy reflects the average electron distribution whereas vibrational and rotational modes of the molecules making up a sample will cause the polarizability to fluctuate in time. These modes are discrete, and considering a particular vibrational frequency, vk, the oscillating polarizability can be modeled as... 

For oriented samples, the rotation of the plane-polarized light becomes a tensor - that is, the optical rotation becomes directionally dependent - and includes a contribution from the electric dipole-electric quadrupole polarizability tensor, which is traceless and thus vanishes for freely rotating molecules [30], The term arising from these quadrupolar interactions can be expressed as [30]... 

To see why this is the case, we first consider the portion of the response that arises from llsm. According to Equation (10), we can express (nsm(t) nsm(0)> in terms of derivatives of llsm with respect to the molecular coordinates. Since in the absence of intermolecular interactions the polarizability tensor of an individual molecule is translationally invariant, FIsm is sensitive only to orientational motions. Since the trace is a linear function of the elements of n, the trace of the derivative of a tensor is equal to the derivative of the trace of a tensor. Note, however, that the trace of a tensor is rotationally invariant. Thus, the trace of any derivative of with respect to an orientational coordinate must be zero. As a result, nsm cannot contribute to isotropic scattering, either on its own or in combination with flDID. On the other hand, although the anisotropy is also rotationally invariant, it is not a linear function of the elements of 11. The anisotropy of the derivative of a tensor therefore need not be zero, and nsm can contribute to anisotropic scattering. 

Let us use these selection rules for investigating the main features of the Rayleigh and pure rotational Raman scattering by spherical-top molecules in the lowest vibronic states (Ogurtsov et al., 1978). The polarizability tensors dif2i and can be expanded into components of irreducible tensor operators that in cubic groups transform as E, T2, and T, respectively. Here the behavior of the operators dir 71 with respect to time reversal 0 has to be taken into consideration. To do this, we use the explicit form of the operator djj(a>) in Cartesian coordinates ... 

The intensities of the spectral lines and the depolarization coefficients are functions of the reduced matrix elements of the polarizability tensor calculated by vibronic functions. In order to estimate the possibility of observation of the pure rotational Raman spectra under consideration, one has to consider in more detail the polarizability operator. Its components belonging to the line y of representation f can be presented in the form of a power series with respect to the displacement qriri active in the Jahn-Teller effect (the other components can be neglected as not active in the pure rotational Raman spectrum under consideration) ... 

If a bond has a threefold for higher rotation axis, its polarizability tensor can be written as follows in terms of components referred to principal axes ... 

B(t) is the scattering amplitude of a particle, which depends on the particle polarizability at given orientation. B(t) changes with time due to reorientation of the particle. If the scatterers are spherical, B(t) is constant and Cg(x) = 1. Note that Cg(x) does not depend on the scattering angle and can be calculated if the polarizability tensor and the rotational diffusion tensor of the particles are known. The calculation of CJi q,x) requires averaging of the translational diffusion tensor of the particle over all possible orientations to obtain the averaged translational diffusion coefficient. 

It is important to note that the first term on the right-hand side of Eq. (7.2.6a) involves the isotropic part of the polarizability tensor and is hence independent of the rotations. This term would appear even for spherical molecules (when / = 0), whereas the other terms would be zero. This term gives rise to isotropic scattering and we consequently define... 

Where XYZ stands for a specified Cartesian coordinate frame. Thus once a Cartesian coordinate frame is chosen, the nine spherical components can be determined using Eq. (7.4.1) from the nine Cartesian components. Clearly the spherical components and the Cartesian components change if the coordinate axes are rotated. Suppose we know the values of the Cartesian components of the polarizability tensor in a coordinate frame rigidly fixed within the molecule6 (the body-fixed frame OXY Z). Then the problem confronting us is to determine the Cartesian components of the polarizability tensor in a coordinate system rigidly fixed in the laboratory (the laboratory frame OX Y Z ). The relative orientations of the molecular and laboratory-fixed... 

The frame (B) was chosen such that the rotational diffusion tensor is diagonal. In general, the polarizability tensor a will not be diagonal in the same body fixed frame that diagonalizes. In the special case when a and are simultaneously diagonalized in the frame (B) that is, when the molecule is a true symmetric top, then aij(B) = 0 for i = j. Referring back to Eq. (7.4.1) we see that in this eventuality azz(B) = a and axx(B) = ccyy(B) = a , and... 

Equation (7.5.22) applies rigorously if the Z axis is a four fold or more symmetry axis of rotation. The more general result given in Eq. (7.5.19) holds if the polarizability tensor does not have cylindrical symmetry about the Z axis in the body-fixed coordinate system while the rotational diffusion tensor does. 

The results simplify considerably if the body-fixed axis system is a principal axis system for the polarizability tensor as well as for the rotational diffusion tensor. In this case Ag = At = A5 = 0 in Eq. (7.5.27). Then the spectrum consists of only two Lorentzians. Many asymmetric diffusors do have enough symmetry to rigorously satisfy this condition-for instance, planar molecules with at least one two fold rotation axis in the molecular plane. Others may have these axes so close together that Ag = A4 = A5 = 0 and the spectrum effectively consists of only two Lorentzians. In any particular application, it must be kept in mind that the spectrum might very well be the five-Lorentzian form given by the Fourier transform of Eq. (7.5.25). 

As the molecule rotates, the vector u reorients and the tensor a changes in time. We note that because aSap is independent of u, this part of the polarizability tensor does not change as the molecule rotates. aSap is said to be rotationally invariant. Pap on the other hand depends on u, so that this part of the polarizability changes as the molecule rotates. 

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