Optical activity tensor

The fundamental scattering mechanism responsible for ROA was discovered by Atkins and Barron (1969), who showed that interference between the waves scattered via the polarizability and optical activity tensors of the molecule yields a dependence of the scattered intensity on the degree of circular polarization of the incident light and to a circular component in the scattered light. Barron and Buckingham (1971) subsequently developed a more definitive version of the theory and introduced a definition of the dimensionless circular intensity difference (CID),... 

Optical activity in light scattering thus arises from interference between the molecular polarizability and optical activity tensors. 

Early theoretical treatments [3,4] of optically active scattering by molecules did little to arouse renewed interest in a measurement of ROA. The decisive cross-terms between the electric dipole-electric dipole polarizability and the optical activity tensor were missed, and effects predicted for the optical activity tensor alone were too small to be practically useful. It was only after the proper cross-terms were identified [5] that new interest in the measurement of ROA accrued, and that the existence of the phenomenon was finally proved [6,7], It was not long before the first measurement of whole ROA spectra was demonstrated [8],... 

Ordinary Raman scattering is determined by derivatives of the electric dipole-electric dipole tensor ae, and ROA by derivatives of cross-products of this tensor with the imaginary part G,e of the electric dipole-magnetic dipole tensor (the optical activity tensor) and the tensor Ae which results from the double contraction of the third rank electric dipole-electric quadrupole tensor Ae with the third rank antisymmetric unit tensor s of Levi-Civita. The electronic property tensors have the form ... 

The occurrence of Raman scattering is connected to the change in polarizability during the transition of the molecule from one vibrational state to the other. Circular polarization ROA arises from interference of the electric dipole electric dipole polarizability tensor with the electric dipole - magnetic dipole and the electric dipole electric quadrupole optical activity tensors. Due to limited space, no rigorous derivation of the theory will be given here, but only the most important results shall be shown. 

In Raman optical activity one encounters three additional invariants the isotropic part of the magnetic dipole optical activity tensor aG, and its anisotropic part, and... 

Similarly to the transition polarizability the vibrational transition optical activity tensors are written as ... 

A central feature in the development of theories of the generation of ROA within chiral molecules has been the realization that the optical activity tensors are origin-dependent. If the origin is moved from O to a point O + a, where a is some constant vector, it is found that32,5)... 

The extension of the bond polarizability theory to ROA is based on the origin-dependence of G p and AaPr Thus using (2.6) the optical activity tensors of the molecule, written as sums of corresponding bond tensors, are... 

As demonstrated elsewhere45,5), if all the bonds in the molecule are axially-symmetric and achiral, the terms in the ROA expressions (4.8) involving intrinsic bond optical activity tensors and A vanish. For this particular case, the bond polarizability expressions (4.8) have been developed into a computational form that enables the ROA to be calculated for any normal model45) but because of their... 

Thus the polarizability tensor of the molecule is written as a sum of local atomic polarizabilities, each modified through dipolar interactions with the electric dipole moments on all the other atoms induced by the electric vector of the incident light wave. Similarly for the local atomic polarizabilities appearing in the origin-dependent parts of the optical activity tensors. But unlike the bond polarizability development, no allowance can be made for intrinsic local optical activity tensors Gj ap and Aj since these now pertain to spherical atoms. We refer to the original articles for the explicit Raman intensity and optical activity expressions generated by the atom dipole interaction theory. 

Thus Eq. (4.7) for the optical activity tensors of the molecule are again employed, but now the summation is over all occupied LMOs and the vectors Ri define the positions of the orbital centroids. Once the wavefunctions are known, the polarizability of the ith LMO and the position of its centroid R( can be determined. The derivatives of a, and Rj with respect to the normal coordinates are calculated using the electric field perturbation approach recently shown to be very effective for the calculation of conventional infrared and Raman intensities61) the required derivatives of R= and a. are determined from the first and second derivatives, respectively, of the gradient of the molecular potential energy with respect to a small applied electric field. One important aspect of this method is that both infrared CD and ROA can be determined from the same conceptual and calculational method, which will enhance the study of the relationship between these two forms of vibrational optical activity. So far, only one ROA calculation using LMO methods has been reported59), and since that was for the model compound NHDT there has been no comparison with experimental data. 

Natural ROA originates in interference between waves scattered via the polarizability and optical activity tensors of the molecule. The relevant experimental quantity is a dimensionless circular intensity difference... 

The appearance of E1-E2 optical activity is restricted to those symmetry groups in which the components of a second rank odd-parity tensor are totally symmetric. As pointed out by Jerphagnon and Chemla, optical activity may be observed even in nonenantiomorphous systems due to the nonpseudoscalar parts of the optical activity tensor—only enantiomorphous crystal classes having a nonvanishing pseudoscalar part. 

Table 4 shows the occurrence of the pseudoscalar, vector, and second rank odd-parity (pseudodeviator) parts of the optical activity tensor in the noncentrosyimnetric crystallographic point groups. 

Table 4 Irreducible components of the optical activity tensor for the noncentrosymmetric crystallographic point groups.

The quantum mechanical expression for the average optical rotation of a molecule was first given by Rosenfeld (1928). In 1937 Kirkwood (1937) and Condon (1937) presented similar derivations. A quantum mechanical expression for the optical activity tensor... 

In the general non-resonant case there are three contributions to the observed ROA spectra aG, the isotropic ROA invariant stemming from the electric dipole-magnetic dipole optical activity tensor, which is also responsible for the anisotropic invariant y, and the anisotropic invariant due to the quadru-pole transition tensor. The anisotropic invariants are also often written as = P(G ) and = P(A). ... 

I = scattered light intensity K = defined by Equation [11] N = number of events NA = numerical aperture R = distance between scattering molecule and observer S = surface area S/N - signal to noise ratio a = half angle of light cone = isotropic Raman invariant aG = isotropic ROA invariant due to the optical activity tensor = anisotropic Raman invariant =j8(G )2 = anisotropic ROA invariant due to the optical activity tensor <5 = f A) = anisotropic ROA invariant due to the quadrupole tensor fiQ = permeability of the vacuum to = angular frequency. 

The first calculations of ROA using ab initio molecular orbital methods have been carried out recently. This has been achieved by starting with expressions for the polarizability and Rayleigh optical activity tensors in the zero-frequency limit of the FFR approximation as... 

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