Raman vibrational transition elements

The subscripts I, m and n represent the molecular inertial axes a, b and c) aim and iM represent the Raman and dipole vibrational transition elements, respectively... 

Vibrational Spectra of Transition Element Compounds Table 9. IR and Raman data (in cm i) for orthorhombic molybdates and tungstates 84)... 

Eq. (35-7)]. For molecules of symmetry, these elements belong to the symmetry species, Fly, and so that the condition for a Raman-active transition is that the product F(i/fj) X include one of these species. Thus, from the Xg ground state of acetylene, Raman transitions to the (10000) Xg, (01000) 2, and (00010) Il levels are allowed and can be used to determine the nj, V2, and V4 fundamental frequencies, respectively. As can be seen in Table 1, these three modes do not produce a dipole change as vibration occurs, and thus these transitions are absent from the infrared spectrum. This is an example of the rule of mutual exclusion, which applies for IR/Raman transitions of molecules with a center of symmetry... 

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. 

We see from (9.12) that initial and final states are connected by two-photon transitions which implies that both states have the same parity. For example, the vibrational transitions in homonuclear diatomic molecules, which are forbidden for single-photon infrared transitions, are accessible to Raman transitions. The matrix elements depend on the symmetry characteristics... 

The changes in molecular polarizability during vibrational transitions determine the intensities of Raman lines. The aj polarizability matrix element for a transition from a vibronic state m to a vibronic state n can be presented as follows [4,254-256]... 

Since the vibrational eigenstates of the ground electronic state constitute an orthonomial basis set, tire off-diagonal matrix elements in equation (B 1.3.14) will vanish unless the ground state electronic polarizability depends on nuclear coordinates. (This is the Raman analogue of the requirement in infrared spectroscopy that, to observe a transition, the electronic dipole moment in the ground electronic state must properly vary with nuclear displacements from... 

Since the vibrational spectra of sulfur allotropes are characteristic for their molecular and crystalline structure, vibrational spectroscopy has become a valuable tool in structural studies besides X-ray diffraction techniques. In particular, Raman spectroscopy on sulfur samples at high pressures is much easier to perform than IR spectroscopical studies due to technical demands (e.g., throughput of the IR beam, spectral range in the far-infrared). On the other hand, application of laser radiation for exciting the Raman spectrum may cause photo-induced structural changes. High-pressure phase transitions and structures of elemental sulfur at high pressures were already discussed in [1]. 

Spectroscopy produces spectra which arise as a result of interaction of electromagnetic radiation with matter. The type of interaction (electronic or nuclear transition, molecular vibration or electron loss) depends upon the wavelength of the radiation (Tab. 7.1). The most widely applied techniques are infrared (IR), Mossbauer, ultraviolet-visible (UV-Vis), and in recent years, various forms ofX-ray absorption fine structure (XAFS) spectroscopy which probe the local structure of the elements. Less widely used techniques are Raman spectroscopy. X-ray photoelectron spectroscopy (XPS), secondary ion imaging mass spectroscopy (SIMS), Auger electron spectroscopy (AES), electron spin resonance (ESR) and nuclear magnetic resonance (NMR) spectroscopy. 

A most recently developed description of the problem interprets the amplitude mode model in terms of a molecular concept of chains with cyclic boundary conditions (Zerbi et al., 1989). An effective conjugation coordinate Qja with a corresponding force constant fja is defined. The vibrational frequencies are calculated as a function of , which plays the same role as A in the amplitude mode model. Accordingly, the Raman intensity is obtained from the respective ja components of individual modes. This new concept, like the amplitude mode model, properly describes the relative intensities of different modes, but the correct line shapes and line intensities for excitation with different laser lines can not be obtained, since transition matrix elements are not evaluated explicitly. 

As written, the CIDs (2.3) and (2.5) apply to Rayleigh scattering. The same expression can be used for Raman optical activity if the property tensors are replaced by corresponding vibrational Raman transition tensors. This enables us to deduce the basic symmetry requirements for natural vibrational ROA 15,5) the same components of aap and G p must span the irreducible representation of the particular normal coordinate of vibration. This can only happen in the chiral point groups C , Dn, O, T, I (which lack improper rotation elements) in which polar and axial tensors of the same rank, such as aaP and G (or e, /SAv6, ) have identical transformation properties. Thus, all the Raman-active vibrations in a chiral molecule should show Raman optical activity. 

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. 

The four normal modes of vibrations of a pyramidal XY3 molecule are shown in Fig. 2.8. All four vibrations are both infrared- and Raman-active. The G and F matrix elements of the pyramidal XY3 molecule are given in Appendix VII. Table 2.3a lists the fundamental frequencies of XH3-type molecules. Several bands marked by an asterisk are split into two by inversion doubling. As is shown in Fig. 2.9, two configurations of the XH3 molecule are equally probable. If the potential barrier between them is small, the molecule may resonate between the two structures. As a result, each vibrational level splits into two levels (positive and negative) [560]. Transitions between levels of different signs are allowed in the infrared spectrum, whereas those between levels of the same sign are allowed in the Raman spectrum. The transition between the two levels at u = 0 is also observed in the microwave region (v = 0.79 cm ). 

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