QUANTUM THEORY OF RAMAN SCATTERING

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1968

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Ohio State University

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Placzek's polarizability $theory^{1}$ using the concept of the ground state polarizability could not explain the polarized Raman spectrum observed in anthracene $crystal.^{2}$ The correct way of handling the problem of Raman scattering is to consider the polarizability derived directly from dispersion $theory^{3} \begin{equation}\begin{array}{lll}\langle Oj|\alpha_{xy}|Oi\rangle& =&h^{-1} \sum\limits_{n,k}\left[\langle Oj|x|nk\rangle \langle n k|y|Oi\rangle/(\nu_k+\nu_{u}-\nu_{O}-\nu_{j}-\nu)+\right.\\&&\qquad \qquad\left.\langle Oj|x|nk\rangle \langle nk|x|Oi\rangle/(\nu_{k}+\nu_{u}-\nu_{O}-\nu_{i}+\nu)\right]\end{array}\end{equation}$ where i,j, and k are vibrational quantum numbers, and O,n are electronic quantum numbers. $\nu_{k}, \nu_{j}, \nu_{i}, \nu_{o}, \nu_{u}$ are the frequencies of these quantum states. The summation should be carried over all excited states$|nk$). The frequency denominators in Eq. (1) must not be treated as constants, as generally assumed. They are to be expanded into a Taylor's series with respect to $\nu_{k}/\nu^{1}$, where $\nu^{1} = \nu_{n}-\nu_{o}-\nu_{j}-\nu$. If only the first term in Eq. (1) is expanded and the Born-Oppenheimer approximation introduced, we have [FIGURE] $\begin{equation}\langle Oj|\alpha_{xy}|Oi\rangle=h^{-1}\sum_{n} \langle O|x|n\rangle \langle n|y|Ok \rangle \sum \langle Oj|nk\rangle \langle nk|Oi\rangle [\nu^{1-1}-\nu_{k}/\nu^{**2}+\nu_{k^{2}}/\nu^{**3}+\ldots]\end{equation}$ where $\langle O|x|n\rangle$ is the transition moment between electronic states, $\langle Oj|nk\rangle$ is the Franck-Condon integral. The summations over the vibrational levels k in an excited state n has been evaluated to be $\begin{equation}\sum_{k} \nu_{k} \nu \langle Oj|nk\rangle \langle nk|Oi\rangle=Const.x\delta_{i,j\pm p}\end{equation}$ Therefore, the $\nu^{1-1}$ term causes the Rayleigh scattering $i = j$, the second term $\nu_{k}/\nu^{***2}$ causes the $\Delta v = \pm 1$ Raman scatterings, the third term causes the $\Delta v = \pm 2$ Raman scatterings, and so forth. The theory has been developed along this line and the following conclusions have been reached: 1. The absolute intensities of Rayleigh and Raman scatterings can be quantitatively predicted from the electronic spectrum of the scattering system. 2. Under the Born-Oppenheimer approximation, only the totally symmetric vibrations are allowed in Raman spectrum, and only the totally symmetric polarizability tensor elements are nonvanishing. 3. The appearance of non-totally symmetric vibrations in Raman spectrum is due to the breakdown of the Born-Oppenheimer approximation, usually by the strong solvent-solute interactions. These vibrations are thus weak and depolarized. 4. Raman spectrum in a particular molecule resembles closely the fluorescence spectrum, with a slightly different intensity distribution among vibrations. Factors affecting this distribution are pointed out. 5. Selection rules on the rotational-vibrational Raman spectrum can be derived easily.

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$^{1}$ G. Placzek, Handbuch der Radiologie, ed. by E. Marx (Academische Verlag., Leipzig, 1934) Vol. VI, Part 2, pp. 205. $^{2}$ C. H. Ting, the following paper. $^{3}$ H A. Kramers and W.C. Heisenberg, Z. Physik 31, 681 (1925).
Author Institution: Department of Chemistry, National Taiwan University

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