Raman intensities, transition moments

Here, and are total wavefunctions of the m and e states, respectively, and pa is the a component of the electric dipole moment. Te is the band width of the eth state, and the iYe term is called the damping constant. In normal Raman scattering, v0 is chosen so that vo vem Namely, the energy of the incident beam is much smaller than that of an electronic transition. Under these conditions, the Raman intensity is proportional to (vo - vm )4. As vo approaches vem, the denominator of the first term in the brackets of Eq. (1-63)... 

Raman Selection Rules. For polyatomic molecules a number of Stokes Raman bands are observed, each corresponding to an allowed transition between two vibrational energy levels of the molecule. (An allowed transition is one for which the intensity is not uniquely zero owing to symmetry.) As in the case of infrared spectroscopy (see Exp. 38), only the fundamental transitions (corresponding to frequencies v, V2, v, ...) are usually intense enough to be observed, although weak overtone and combination Raman bands are sometimes detected. For molecules with appreciable symmetry, some fundamental transitions may be absent in the Raman and/or infrared spectra. The essential requirement is that the transition moment F (whose square determines the intensity) be nonzero i.e.. 

In Eq. (2), is Ihe pth component of the transition dipole moment for the electronic transition, r) <— i, between initial state / and excited state r is the frequency of this transition and iT is a damping factor related to the lifetime of the excited state 8). Equation (2) shows that the Raman intensity can be increased dramatically when the wavelength of the exciting laser is in resonance with an electronic absorption of the sample the process is referred to as resonance Raman spectroscopy (RRS). 

The dipole polarizability can be used in place of the dipole moment function, and this will lead to Raman intensities. Likewise, one can compute electrical quadrupole and higher multipole transition moments if these are of interest. 

Here Uab is the Raman transition moment, fic is the infrared transition moment, g and V refer to ground and excited vibrational states, coir is the input infrared frequency, coq is the resonance frequency of the adsorbate, and T is a damping factor [8, 14—17]. Thus, the SFG intensity is related to the product of an (anti Stokes) Raman transition and an infrared transition. The SFG intensity is enhanced when the input infrared wavelength coincides with a vibrational mode of the adsorbate and the result of an SFG spectrum corresponds to the vibrational levels of the molecule. This situation is shown schematically in Fig. 5.1. From (5), non-zero SFG intensity will occur only for transitions that are both Raman and IR allowed. This situation occurs only for molecules lacking inversion symmetry [19]. 

The term includes the Raman polariTability tensor elements as well the IR-dipole transition moment The variable N refers to the number of resonant modes on the surface and iT specifies the damping constant or lifetime of the vibration. From Eq. (7.2), it is obvious that signal intensities increase as the term — ft>q) goes to zero as it approaches resonance. 

If we assume co = o), for a nontotally symmetric harmonic oscillator, Franck-Condon contributions to its Raman scattering cross section will vanish since Q = Qg by symmetry for such modes. The only source of Raman intensity is then vibronic coupling. In its simplest form, this mechanism can be described as due to the Q-dependence of the electronic transition moments in Eq. (19) ... 

Figure 9 (Henneker et al., 1978a) shows an example of Raman scattering by a non totally symmetric mode involved in (linear) Herzberg-Teller coupling of two excited states, d> and <1> . It is an elaboration of Fig. 7 in that the result of resonance with both excited states is shown. For convenience, we have assumed that both transitions -> and <5, -> are allowed and have the same transition moment, but different polarization. This assumption makes the Rayleigh profile symmetric relative to the two states, but has no effect on the structure of the fundamental REP since its matrix elements are proportional to the product of the two transition moments. The difference in intensity of the overtones in the and <1> band region reflects the frequency differences cOg — co and cOg — co . Figure 9 is based...

It has been found that the above theory is quite useful in predicting the correct overall behavior of matter in the presence of electromagnetic radiation. However, in order to predict correctly Raman intensities, it has been found necessary to refine the theory by accounting for small deviations in the electronic wave functions with nuclear motion. In the framework of the Herzberg-Teller theory, it is assumed that the corrected electronic wave functions may be obtained by the use of a first-order perturbation expansion as a linear combination of the complete set of zero-order Born-Oppenheimer functions, discussed above. Since here we are mainly interested in the normal Raman effect, we shall consider only corrections to the second term in Eq. (41). If we first examine corrections to the state X, the resulting expression for the derivative of the transition moment with motion along a normal mode is ... 

Another related issue is the computation of the intensities of the peaks in the spectrum. Peak intensities depend on the probability that a particular wavelength photon will be absorbed or Raman-scattered. These probabilities can be computed from the wave function by computing the transition dipole moments. This gives relative peak intensities since the calculation does not include the density of the substance. Some types of transitions turn out to have a zero probability due to the molecules symmetry or the spin of the electrons. This is where spectroscopic selection rules come from. Ah initio methods are the preferred way of computing intensities. Although intensities can be computed using semiempirical methods, they tend to give rather poor accuracy results for many chemical systems. 

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