Placzek s theory

Placzek s theory (1934) which treats molecules as quantum objects and electromagnetic fields classically, satisfactorily describes the Raman effect on the condition that the exciting frequency differs considerably from the frequencies of electronic as well as of vibrational transitions. 

The term on the right-hand side of this equation represents the microscopic parameters of the sample. According to Placzek s theory (1934), this expression should be independent of the frequency of the exciting radiation. In the absence of a resonance or pre-resonance Raman effect, it would be equal for all exciting lines. The term on the left-hand side normalizes the observed Raman intensity by including the P factor. These values have the dimension cm - sr. ... 

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 absolute intensity of the Raman scattered light can be described mathematically, to a good approximation, by using the equations of Placzek s theory (see Box 8.1). Because it is actually possible to calculate the Raman line intensities, and compare them with observations, Raman spectroscopy can be (and... 

The essential assumption of Placzek s theory for Raman line intensities is that the frequency of the exciting radiation vl is larger than the frequency associated with the energy difference between two ro-vibrational states Vvib,rot> but is less than the frequency associated with an electronically excited energy level Vei- In general, this is easy to realize in the majority of Raman scattering experiments. For the Stokes and anti-Stokes line intensities and one finds... 

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 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 ... 

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]. 

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