Placzek polarizability theory

These results apply specifically to Rayleigh, or elastic, scattering. For Raman, or inelastic, scattering the same basic CID expressions apply but with the molecular property tensors replaced by corresponding vibrational Raman transition tensors between the initial and final vibrational states nv and rn . In this way a s are replaced by (mv aap(Q) nv), where aQ/3(<3) s are effective polarizability and optical activity operators that depend parametrically on the normal vibrational coordinates Q such that, within the Placzek polarizability theory of the Raman effect [23], ROA intensity depends on products such as (daaf3 / dQ)0 dG af3 / dQ) and (daaf3 / dQ)0 eajS dAlSf / dQ)0. 

The tensors which enter theoretical expressions are transition tensors 7, for a transition between an initial state i and a final state /. The Placzek polarizability theory for vibrational Raman scattering [56], which we use here, is valid in the far from resonance limit, i and / are then vibrational states. If we assume that they differ for normal mode p, then the transition tensors can be written as... 

The separation into a vibrational and an electronic part is implied by the Placzek polarizability theory. The further analysis of vibrational motions has in the past typically been accomplished by calculating the vibrational energy distribution in valence coordinates. For the large-scale skeletal motions often important in ROA, and for relating Raman and ROA scattering cross-sections to the vibrational motions of structural parts of an entity, a different approach is needed. 

B) THE MICROSCOPIC HYPERPOLARIZABILITY IN TERMS OF THE LINEAR POLARIZABILITY THE KRAMERS-HEISENBERG EQUATION AND PLACZEK LINEAR POLARIZABILITY THEORY OF THE RAMAN EFFECT... 

As mentioned above, the basic theory of the Raman effect was developed before its discovery. However, at this time numerical calculations of the intensity of Raman lines were impossible, because these require information on all eigenstates of a scattering system. Placzek (1934) introduced a semi-classical approach in the form of his polarizability theory. This provided a basis for many other theoretical and experimental studies. Physicists and chemists worldwide realized the importance of the Raman effect as a tool for qualitative and quantitative analysis and for the detennination of structure. 

As already pointed out, this description of the Raman effect is based on the polarizability theory (Placzek, 1934) which is valid in a good approximation if the exciting frequency is much higher than the frequency of the vibrational transition // , but lower than the frequency of the transition to the electronic excited state If, on the other hand, is approaching then resonances occur which considerably enhance the intensities of the Raman lines, i.e., the resonance Raman effect. This effect and its applications are described in Sec. 6.1 and also in Secs. 4.2 and 4.8. 

In accordance with Placzek s theory (1934) we can write the real part of the complex transition polarizability as the dynamic vibrational polarizability operator (which is a function of a static configuration Q of nuclei) acting on the vibrational state functions and... 

The bond polarizability theory of conventional Raman intensity is well-established 46,47). The starting point is Placzek s approximation for the vibrational Raman transition polarizability at transparent frequencies48 . On expanding the effective polarizability operator aotp(Q) in the normal vibrational coordinates Qp, the transition polarizability becomes... 

Recalling Eq. (4) that gives the Raman intensity expression for a molecule based on Placzek s polarizability theory, the following gives a more complete expression with regard to the instrumental and surface factors ... 

The original Placzek theory of Raman scattering [30] was in terms of the linear, or first order microscopic polarizability, a (a second rank tensor), not the third order h3q)erpolarizability, y (a fourth rank tensor). The Dirac and Kramers-Heisenberg quantum theory for linear dispersion did account for Raman scattering. It turns out that this link of properties at third order to those at first order works well for the electronically nonresonant Raman processes, but it cannot hold rigorously for the fully (triply) resonant Raman spectroscopies. However, provided one discards the important line shaping phenomenon called pure dephasing , one can show how the third order susceptibility does reduce to the treatment based on the (linear) polarizability tensor [6, 27]. 

Quantum-mechanical expressions for the polarizability and other higher-order molecular response tensors are obtained by taking expectation values of the operator equivalent of the electric dipole moment (2.5) using molecular wavefunctions perturbed by the light wave (2.4). This particular semi-classical approach avoids the complications of formal time-dependent perturbation theory it has a respectable pedigree, being found in Placzek s famous treatise on the Raman effect [9], and also in the books by Born and Huang [lO] and Davydov [ll]. Further details of the particular version outlined here can be found in my own book [12]. 

The Raman intensity, in the Placzek theory, is proportional to the derivatives of the polarizability, and is often described in terms of the coefTicient... 

The feature of the all considered experimental methods is that they allow us to define the values of the molecular polarizabilify only at the equilibrium position of molecular nuclei. To obtain the dependencies of molecular polarizabilities on the mutual location of nuclei in a molecule, the Raman effect can be used The line intensities of Raman spectra depend on the values of polarizability derivatives with respect to the nuclei displacements. The first works to define the polarizability derivatives of molecules have appeared immediately after the creation of the theory of Raman light scattering (Placzek theory of polarizability) [17]. However, the experimental technique of pre-laser period could not obtain the high-quality results. Some experimental results of this period are summarized in [18]. Currently, these data have only a historical interest. Now, laser technologies allow to increase the measurement accuracy and, as a result, significantly improve and revise the pre-laser data. Nevertheless, up to day the experimental data on flie polarizability derivatives of molecules are fragmentary and do not give the impression of systematic studies of the polarizability of molecules as a function of the nuclei coordinates, even for diatomic molecules [19-39]. 

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