Raman transition polarizability

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... 

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... 

Intensities of Raman transitions are proportional to R and therefore, from Equation (6.13), to (da/dx)g. Since a is a tensor property we cannot illustrate easily its variation with x instead we use the mean polarizability a, where... 

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... 

Consideration of the matrix elements m a n of the polarizability shows that the selection rule for a pure-rotational Raman transition of a l2 diatomic molecule is (see Wilson, Decius, and Cross)... 

These results apply specifically to Rayleigh, or elastic, scattering. For Raman, or inelastic, scattering the same basic CID expressions apply but with the molecular property tensors replaced by corresponding vibrational Raman transition tensors between the initial and final vibrational states nv and rn . In this way a s are replaced by (mv aap(Q) nv), where aQ/3(<3) s are effective polarizability and optical activity operators that depend parametrically on the normal vibrational coordinates Q such that, within the Placzek polarizability theory of the Raman effect [23], ROA intensity depends on products such as (daaf3 / dQ)0 dG af3 / dQ) and (daaf3 / dQ)0 eajS dAlSf / dQ)0. 

In contrast, Raman46 absorption involves inelastic scattering, with gain or loss of energy in detail, the Raman transition involves the polarizability... 

It is important to note that the two electric fields that lead to a Raman transition can have different polarizations. Information about how the transition probability is affected by these polarizations is contained within the elements of the many-body polarizability tensor. Since all of the Raman spectroscopies considered here involve two Raman transitions, we must consider the effects of four polarizations overall. In time-domain experiments we are thus interested in the symmetry properties of the third-order response function, R (or equivalently in frequency-domain experiments... 

In the case of a Raman transition, the same symmetry arguments apply, except that the dipole function p. must be replaced by the polarizability tensor elements etc. [see... 

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). 

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... 

Formally, one can think of the Raman transition probability being proportional to the elements of the polarizability tensor of a bound electron as the exciting frequency approaches the resonance frequency, these elements are enhanced in a Lorentz model of the bound electron. A common example of this mechanism is furnished by the ring-breathing (in-plane expansion) modes of porphyrins. Another mechanism, called vibronic enhancement, involves vibrations which couple two electronic excited states. In both mechanisms, the enhancement factors are nearly proportional to the intensities in the absorption spectrum of the adsorbate. 

Here, Na and Nb are populations in the lower and upper states of the Raman transition, da/dq is the rate of change of polarizability with normal coordinate, no and ns are the refractive indices at o)o and 0)s5 r is the Raman linewidth and y the reduced mass. Eq. (1) describes a Stokes Raman laser producing coherent radiation at 0)3. In this brief and simplified description, we may consider that the strong pump and Stokes waves generate a coherent material excitation at 0)r. This oscillation causes variations in the refractive index which then modulate and scatter the incident laser radiation (o)o) thus producing sidebands or many orders of coherent Stokes and anti-Stokes radiation at frequen-... 

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... 

For this reason, the alkali atoms, with their voluminous valence orbitals and closely spaced energy levels, exhibit the largest atomic polarizabilities in the periodic table (Table 1.2), and such atoms have figured prominently in the development of nonlinear optics. These expressions for molecular polarizabilities become useful in Chapter 10, where Raman transition probabilities are discussed. 

We may obtain working selection rules for vibrational Raman transitions by expanding the molecular polarizability tensor in the normal coordinates Qi... 

Allowed vibrational Raman transitions involve only modes that change the molecular polarizability (the right side of Eq. 10.32 would otherwise reduce into E2 (( ) El symmetry-allowed, but very weak. For example, alkali halide molecules consist of pairs of oppositely charged ions whose electronic structures are nearly insensitive to the internuclear separation R for R near R. The alkali halide molecular polarizability... 

Consider the hypothetical case in which the laser frequency co is tuned close to the lowest El-allowed electronic transition in C2H2, so that resonance Raman emission occurs and the polarizability expression (10.32) for the Raman transition probability amplitude is no longer applicable. Which of the vibrational levels listed above can be reached from 0000°0° acetylene by El symmetry-allowed resonance Raman transitions, even though they cannot be reached by conventional Raman transitions ... 

A very rough rule is that a vibrational mode is accessible from the ground state by a Raman transition if the vibrational motion changes the overall size of the molecule, because the size is roughly proportional to the polarizability. By this argument, the breathing mode stretch in BF3 would be Raman-active, and it is. 

CHECKPOINT At this point in our derivation of the potential energy for dispersion, an expression for the polarizability a drops out of the math. The dispersion and the polarizability are two manifestations of the same phenomenon, and scattering (including Raman transitions) can be said to be a third. Our constmction of electronic wavefunctions in Chapters 4 through 7 has been valuable for understanding how molecules are put together, but we find that by itself it is not sufficient to explain how molecules befiave. 

Raman spectroscopy also has selection rules. The gross selection rule for a Raman-active vibration is related to the polarizability of the molecule. Polarizability is a measure of how easily an electric field can induce a dipole moment on an atom or molecule. Vibrations that are Raman-active have a changing polarizability during the course of the vibration. Thus, a changing polarizability is what makes a vibration Raman-active. The quantum-mechanical selection rule, in terms of the change in the vibrational quantum number, is based on a transition moment that is similar to the form of M in equation 14.2. For allowed Raman transitions, the transition moment [a] is written in terms of the polarizability a of the molecule ... 

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