Kramers-Heisenberg-Dirac

The more conventional, energy domain fonnula for resonance Raman scattering is the expression by Kramers-Heisenberg-Dirac (KHD). The differential cross section for Raman scattering into a solid angle dD can be written in the fomi... 

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 time-dependent formulation of Raman scattering has been introduced by Lee and Heller (1979), Heller, Sundberg, and Tannor (1982), Tannor and Heller (1982), and Myers, Mathies, Tannor, and Heller (1982). Its derivation is strikingly simple. We start from the Kramers-Heisenberg-Dirac formula (14.1) and (14.2) without the nonresonant term and transform it into an integral over time by using the identity... 

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). 

To determine the scattered radiation spectrum of an oscillating molecule under conditions of resonance excitation, we must consider how the polarizability a varies not only with normal modes of vibration but also with frequency of the incident radiation that excites them. For a molecule in a molecular state ) (initial) perturbed by the electromagnetic wave of frequency vq so that it passes into a molecular state I /) (final) while scattering light of frequency vo r (v = V/ - Vg), the matrix elements of a for the vibrational transition k, [oipa]k, are given by the Kramers-Heisenberg-Dirac (KHD) dispersion equation ... 

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. 

Here X(, i(g) denotes a vibrational wavefunction for level 6, of mode i in the ground electronic state. As we discussed in connection with absorption spectra in Chap. 11, Eq. (12.18) makes the wavepacket formalism much more manageable than the Kramers-Heisenberg-Dirac expression for calculating Raman excitation spectra. It is not necessary to sum over all possible combinations of quantum numbers for the different modes. 

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. 

The tfaeoiy of intensities of re onantly>enhanced Raman lines is based on the Kramers-Heisenberg-Dirac dispersion equation [266,267], The problem is analyzed in teims of vibronic interactions in die molecule. The JK matrix component of the molecular polarizability for the gi gf transition in die Raman spectrum can be expressed as follows [256,257]... 

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