Optical activity, pseudoscalar

Keywords chiral molecules optical activity pseudoscalars liquids second-order nonlinear optics ... 

Symmetry arguments show that parity-odd, time-even molecular properties which have a non-vanishing isotropic part underlie chirality specific experiments in liquids. In linear optics it is the isotropic part of the optical rotation tensor, G, that gives rise to optical rotation and vibrational optical activity. Pseudoscalars can also arise in nonlinear optics. Similar to tlie optical rotation tensor, the odd-order susceptibilities require magnetic-dipole (electric-quadrupole) transitions to be chirally sensitive. 

As forcefully stated by Pasteur, whatever the precise arrangement of atoms in the molecule, ce qui ne peut etre l objet d un doute is that the chirality of the atomic arrangement is the necessary and sufficient condition for molecular enantiomorphism and for its manifestation in a pseudoscalar property such as optical activity. It cannot be emphasized too strongly that no recourse to structural theory was needed to arrive at this conclusion, which was based purely on a symmetry argument and which F. M. Jaeger has referred to as la loi de Pasteur ... 

In equation 1 each term is called a source of optical activity (SOA). Each SO A describes a particular way of generating optical activity and is built from a number of equivalent elements of optical activity (EOAs). The elements of optical activity are combined in such a manner that the corresponding SOA has the correct (pseudoscalar) transformation property to describe optical rotations. 

Nonlinear optical activity phenomena arise at third-order and include intensity dependent contributions to optical rotation and circular dichroism, as well as a coherent form of Raman optical activity. The third-order observables are - like their linear analogs - pseudoscalars (scalars which change sign under parity) and require electric-dipole as well as magnetic-dipole transitions. Nonlinear optical activity is circular differential. 

This chapter is organized as follows In Section 2 we discuss the general symmetry requirements of chiroptical processes in isotropic media. In particular, we consider linear and nonlinear optical activity and we describe how frequency conversion at second-order (and at fourth-order) is specific to chiral molecules in fluids. In Section 3 we discuss our work on SFG in optically active solutions, and the computation of the SFG pseudoscalar is described in Section 4. Results from recent computations are given in Section 5. Conclusions are drawn in Section 6. 

We have discussed the symmetries of linear optical activity and sum-frequency generation. The former is an odd-order process that requires a nonlocal response tensor in order to be specific to chiral molecules in solution, whereas the latter is an even-order response where the dominant electric-dipolar susceptibility is a probe of chirality. These observations can be extended to pseudoscalars at third-and fourth-order. 

In the case of non-degenerate frequencies, the nonlocal third-order effects may give rise to chiral pump-probe spectroscopies. The only observation of a coherent Raman optical activity process to date is also due to a third-order pseudoscalar. Spiegel and Schneider have observed Raman optical activity in coherent anti-Stokes Raman scattering in a liquid of (-l-)-trans-pinane and report chiral signals that are 10 of the conventional electric-dipolar CARS intensity [23],... 

Figure 7. Dispersion of the chirality specific (chiral) pseudoscalar relative to for the SFG process in optically active l,l -bi-2-naphthol. Configuration Interaction Singles SOS calculation with the cc-pVDZ basis, damping = 3000 cm...

In the optical activity arising from higher-order cross-terms, the effects are in most cases expected to be orientation-dependent. Pseudoscalar terms are the only ones which survive in random orientation (molecules in solution or liquid phase). At the same order of perturbation as El-Ml there is a product of the electric dipole and electric quadrupole transition operators (E1-E2). Since the latter product involves tensors of unequal rank, the result cannot be a pseudoscalar and this term would not, therefore, contribute in random orientation but can be significant for oriented systems with quadrupole-allowed transitions. The E1-E2 mechanism was developed by Buckingham and Dunn and recognized by Barron" as a potential contribution to the visible CD in oriented crystals containing the [Co(en)3] " ion. 

In all XNCD measured so far, it has been found that the predominant contribution to X-ray optical activity is from the E1-E2 mechanism. The reason for this is that the El-Ml contribution depends on the possibility of a significant magnetic dipole transition probability and this is strongly forbidden in core excitations due to the radial orthogonality of core with valence and continuum states. This orthogonality is partially removed due to relaxation of the core-hole excited state, but this is not very effective and in the cases studied so far there is no definite evidence of pseudoscalar XNCD. 

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. 

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