The Kramers-Heisenberg-Dirac Theory

Kramers and Heisenberg [2], who predicted the phenomenon of Raman scattering several years before Raman discovered it experimentally, advanced a semiclas-sical theory in which they treated the scattering molecule quantum mechanically and the radiation field classically. Dirac [3] soon extended the theory to include quantization of the radiatiOTi field, and Placzec, Albrecht and others explored the selection rules for molecules with various symmetries [4, 5]. A theory of the resonance Raman effect based on vibratiOTial wavepackets was developed by Heller, Mathies, Meyers and their colleagues [6-11]. Mukamel [1, 12] presented a comprehensive theory that considered the nonlinear response functions for pathways in LiouvUle space. Having briefly described the pertinent pathways in Liouville space above, we will first develop the Kramers-Heisenberg-Dirac theory by a second-order perturbation approach, and then turn to the wavepacket picture. 

Consider a molecule with groxmd-state wavefunction Fa and excited-state wavefunction Pig, and energies Ea and E]g. When a weak, continuous radiation field with frequency Vg and amplitude Eg[exp(2 riVgt) + exp(—2 rivgt)] is introduced, the coefficient for state k (C ) oscillates with time. We need an expression for 

12 Raman Scattering and Other Multi-photon Processes 

We found in Chap. 10 that the steady-state value can be written 

The intensity of Rayleigh or Raman scattering of nearly monochromatic light should be proportional to the integral of QC over a narrow band of the fi-equency difference Vg — (Eq. 4.8). If we perform this integration for the first term in the braces in Eq. (12.8), the rate of scattering becomes 

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Two-photon excitation was predicted in her doctoral thesis by Maria Goppert-Mayer [88], who recognized that it was a corollary of the Kramers-Heisenberg-Dirac theory of light scattering. It was not observed experimentally until 30 years later, when pulsed ruby lasers finally provided the high photon flux that was required [89]. Goppert-Mayer received the physics Nobel prize in 1963 for imre-lated work on nuclear structure. 

Raman scattering is a two-photon process and must be described by second-order perturbation theory. The cross section for a transition from state ot(Et)> with energy Ei to state of(Ef)) with energy Ef (in the following the indices 0 and 1 will label the lower and the upper electronic state, respectively) is given by the Kramers-Heisenberg-Dirac formula (Kramers and Heisenberg 1925 Dirac 1927 for a sufficiently detailed derivation see, for example, Weissbluth 1978 ch.24)... 

The traditional approach to evaluating RR intensities involves a summation over all unperturbed eigenstates of the resonant electronic state. This is a direct consequence of the quantum-mechanical derivation of the polarizability tensor components employing second order perturbation theory as given by the Kramers-Heisenberg-Dirac (KHD) relation for the transition polarizability tensor ... 

The theory discussed until now is based on the Kramers-Heisenberg-Dirac dispersion relation for the transition polarizability tensor as given in Eq. (6.1-1). The expression shown in this equation describes a steady state scattering process and contains no explicit reference to time. Therefore, the resonance Raman theory which is based on the KHD dispersion relation is sometimes also termed as time-independent theory (Ganz et al., 1990). 

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]. 

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