Kramers-Heisenberg

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... 

B1.3.2.5 THE MICROSCOPIC HYPERPOLARIZABILITY TENSOR, ORIENTATIONAL AVERAGING, THE KRAMERS-HEISENBERG EXPRESSION AND DEPOLARIZATION RATIOS... 

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

The identity of the Kramers-Heisenberg formula obtained in the two gauges is a celebrated example [22], It should be noted, however, that the popular multipolar Hamiltonian ... 

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... 

Details of the derivation of the harmonic o.scillator dielectric function and of the Kramers-Kronig transformation are described in standard textbooks, such as (Kuzmany, 1990b Kittel, 1976). Eq. 4.8-1 is also well known as the Kramers-Heisenberg dielectric function. The integrated absorption coefficient in Eq. 4.8-5 is very often used in conventional vibronic IR spectroscopy to characterize the concentration of the absorbing species. 

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 ... 

The polarisability tensor is given by a second-order perturbation expression known as the Kramers-Heisenberg dispersion formula (4)... 

To conclude this section, there is one other phenomenon we should like to discuss, viz. the Raman effect. Let it be mentioned beforehand, however, that this is not a revolutionary discovery, like, for example, the discovery of the wave nature of the electron, but an effect which was predicted by the quantum theory (Smekal (1923), Kramers-Heisenberg) some years before it was found experimentally, though it can also be explained within the framework of classical physics (Cabannes (1928), Rocard, Placzek) its great importance rests rather on the facility with which it can be applied to the study of molecules, and on the colossal amount of material relating to it which has been accumulated so quickly. The effect was discovered simultaneously (1928) by Raman in India, and by Landsberg and Mandelstam in Russia. They found that scattered light contains, in addition to the frequency of the incident light, a series of other frequencies. 

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]. 

Consider Raman transitions between thermalized molecular eigenstate g (ground) and molecular eigenstate/ (final). The quantum mechanical expression for responding to colours / and j is the famous (thermalized) Kramers Heisenberg equation [29]... 

Remarkably, the expansion (4.422) may be also obtained by requiring that the Kramers-Heisenberg dispersion relation reduce to the Thomas scattering formula at high energies indeed, through re-writing Eq. (4.422) in the form... 

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. 

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