Transition polarizability

From a theoretical perspective, since the designation of the lab-fixed axes is arbitrary, what is relevant is the relative orientation of the polarizations of the excitation and scattered light. Thus the line shape for excitation light polarized along axis p, and scattered light polarized along axis q (p or q denote X, Y, or Z axes in the lab frame) is called Ipq(co). When p = q this is lyy, and when p q this is IVH. Mixed quantum/classical formulae for Ipq(co) are identical to those for the IR spectmm, except mPi is replaced by apqP which is the pq tensor element of the transition polarizability for chromophore i. Thus we have, for example [6],... 

To obtain Raman spectra one needs the trajectories of the pq tensor elements of the chromophore s transition polarizability. Actually, for the isotropic Raman spectrum one needs only the average transition polarizability. This depends weakly on bath coordinates and this, together with the weak frequency dependence of the position matrix element, was included in our previous calculations [13, 98, 121]. For the VV and VH spectra, others have implemented... 

A number of usually small terms related to the ground-state polarizability, the transition polarizability and the square of the dipole difference have been neglected in (120) and (121). Complete expressions have been given (Liptay, 1974 Wortmann et ai, 1992). Neglecting changes of the polarizability upon excitation, the effective dipole difference in solution in (120), (121) is given by (122). 

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 ... 

Finally we mention, that the 6-term contribution to the transition polarizability may give rise to an asymmetric scattering tensor, i. e. Op 7 a p, leading to the phenomenon... 

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 transition polarizability a can also be enhanced as a result of electronic interaction between the adsorbate and the metal. The most significant mechanism that produces an increase in the apparent value of a involves charge transfer between the metal and the adsorbate. Creighton (1986) has given a very illustrative example how charge transfer may occur from the metal to the adsorbate, which we will describe in the following. 

Ordinary Raman scattering intensity (far from resonance) can be expressed in terms of two invariants the isotropic transition polarizability a, and the anisotropic transition polarizability... 

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... 

Similarly to the transition polarizability the vibrational transition optical activity tensors are written as ... 

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... 

Wavenumber of the incident beam Wavenumber of the capillary wave Backward electron transfer rate constant Forward electron transfer rate constant Apparent ion transfer rate constant Raman transition polarizability... 

Although the complex transition polarizability expressed as a sum of real and imaginary parts analogous to (2.15), it is usually best to express it in the form... 

This general complex transition polarizability has no definite symmetry with respect to permutation of the tensor subscripts, unlike the ordinary polarizability which has a pure symmetric real part (2.13a) and a pure antisymmetric imaginary part (2.13b). 

Gas phase Raman measurements can, of course, resolve individual rotational transitions, so it is necessary to consider rotational-vibrational electronic states in the general transition polarizability (2.18). The classical isotropic averages derived in the previous section provide a useful background because, in accordance with van Vleck s principle of spectroscopic stability [14, a quantum-statistical average over all allowed transitions should yield the... 

It can be seen from the general transition polarizability (2.18) that if the laser frequency coincides with an electronic absorption frequency of the molecule, considerable enhancement of the Raman... 

Many theories have been published which relate the third-order susceptibility to molecular properties. These theories range from full quantum electrodynamics treatments to simple classical models. The simple semi-classical treatment given below is sufficient for our purposes since it will expose the basic physics of the coherent scattering process and will also give us an expression in terms of the conventional spontaneous Raman transition polarizabilities. 

The susceptibility expression (4.17) can be developed further by writing the spontaneous Raman transition polarizability for the vibrational transition as... 

Bennett, S.C. and Wieman, C.E., Measurement of the6S-7S transition polarizability in atomic cesium and an improved test of the standard model, Phys. Rev. Lett., 82, 2484, 1999 Young, R.D., Carlini, R.D., Thomas, A.W., and Roche, J., Testing the standard model by precision measurement of the weak charges of quarks, Phys. Rev. Lett., 99, 122003, 2007. 

These expressions include averaging over all orientations of the chromo-phore with respect to the field and the polarization of the measuring light The factors Uij and bijj in Eqs. (B4.15.2) and (B4.15.3) are elements of the transition polarizability tensor (a) and the transition hyperpolarizability (b), which describe the effects of the external field on the dipole strength of the absorption band. These effects probably are relatively minor in most cases, and are neglected in Eq. (4.64). The transition polarizabifity tensor... 

Raman scattering spectroscopies describe the light-matter interaction caused by the radiation with a frequency that does not correspond to any absorption frequency of the molecule [246]. For example, by employing a radiation in the visible part, it is possible to cause a vibrational transition. A key quantity for Raman spectroscopies is the dynamic transition polarizability, defined as... 

---