Specific selection rule

The selection rules illustrated above are general, as they depend only on the symmetry properties of the functions involved. However, more limiting, selection rules depend on the form of the wavefunctions involved. A relatively simple example of the development of specific selection rules is provided by the harmonic oscillator. The solution of this problem in quantum mechanics,... 

From the properties of the two 6-j symbols in Eq. (20) the specific selection rules for the 7c / coupling scheme are derived as ... 

Every atom, from Z = 3, leads to specific radiation that follows specific selection rules. For all the elements, fluorescence appears in the energy range of 40 eV to more than lOOkeV (31 to 0.012 nm). 

The fact that the magnetic interaction Hamiltonians are compound tensor operators can be exploited to derive more specific selection rules than the one given above. Furthermore, as we shall see later, the number of matrix elements between multiplet components that actually have to be computed can be considerably reduced by use of the Wigner-Eckart theorem. 

Consider the Menon-Agarwal approach to the Autler-Townes spectrum of a V-type three-level atom. The atom is composed of two excited states, 1) and 3), and the ground state 2) coupled by transition dipole moments with matrix elements p12 and p32, but with no dipole coupling between the excited states. The excited states are separated in frequency by A. The spontaneous emission rates from 1) and 3) to the ground state 2) are Tj and T2, respectively. The atom is driven by a strong laser field of the Rabi frequency il, coupled solely to the 1) —> 2) transition. This is a crucial assumption, which would be difficult to realize in practice since quantum interference requires almost parallel dipole moments. However, the difficulty can be overcome in atomic systems with specific selection rules for the transition dipole moments, or by applying fields with specific polarization properties [26]. 

Figure 5 shows the SFG vibrational spectra of carbon monoxide obtained at 10 -700 Torr of CO and at 295 K. When the clean Pt(lll) surface was exposed to 10 L (1 L=10 Torr sec) of CO in UHV, two peaks at 1845 cm and 2095 cm were observed which are characteristic of CO adsorbed at bridge and atop sites. LEED revealed that a c(4 X 2) structure was formed in which an equal number of carbon monoxide molecules occupied atop and bridge sites [15]. Such results are in agreement with previous HREELS [16] and reflection-absoiption infrared spectroscopy (RAIRS) [17] studies. ITie much higher relative intensity of atop bonded CO to bridge bonded CO in the SFG spectra is due to the specific selection rule for the SFG process [18]. As mentioned earlier, SFG is a second order, nonlinear optical technique and requires the vibrational mode under investigation to be both IR and Raman active, so that the SFG intensity includes contributions from the Raman polarizability as well as the IR selection mle for the normal mode. 

When a polymer is melted, the molecular conformations become disordered. As a result, the vibrational spectrum is significantly different from the spectrum of the crystalline state. Because of the presence of new conformations in the melt, many new vibrational bands appear in the spectrum, or splitting of the bands observed in crystalline phase disappears. The vibrational modes are broad because of the variety of structures in the melt. Because the structures are disordered, no specific selection rules can be applied. The Raman spectrum of molten POE is presented in Figure 2 (dashed line). The band assignments for the molten state become difficult, not only because of the broadening, but also because of the large number of local structures with small vibrational energy differences that may be present. 

One disadvantage of the EPR spectroscopy lies in the fact that in rigid media the EPR lines are inhomogeneously broadened by the large number of nuclei which interact with the unpaired electron(s). As a consequence, ligand hf couplings are often not resolved in the EPR display. In addition, the specific selection rules which characterize the EPR spectrum generally prevent the detection of nuclear quadrupole interactions ... 

To consider the specific selection rules on electric dipole (El) transitions in one-electron atoms, we evaluate the matrix elements of the pertinent electric dipole moment operator p = Sj e,r,- = Zer — er, where tff and r are the positions of the nucleus and electron referenced to an arbitrary origin in space. Then... 

There is a more specific selection rule that depends on the quantum number for the vibrational state, v. Because the normal modes of vibration are independent of each other (they are orthogonal to each other), this selection rule is applicable to one normal mode at a time. That is, we will consider the application of this new selection rule to each individual normal mode of vibration. The selection rule does not address simultaneous changes in more than one normal mode of vibration. As indicated in section 14.2, such specific selection rules are group-theoretical. The transition moment integral for vibrations is... 

Specific selection rules for atoms and molecules can also be determined using group-theoretical analyses of the functions in equation 15.1, exactly as we did in the previous chapter for allowed IR and Raman vibrational transitions. 

However, this is misleading. Although electronic energy levels are dictated by the principal quantum number, we should remember that a principal quantum shell in a hydrogen atom has other quantum numbers, namely, f and m. If the symmetries of the operator and wavefunctions in equation 15.1 were examined, one would find that it is the angular momentum quantum number that dictates the selection rule. The specific selection rule for allowed electronic transitions in the hydrogen atom (or, for that matter, hydrogen-like atoms) is... 

There are no specific selection rules for such a photo-ionisation process. The kinetic energy KE) of the ejected electrons can be expressed as follows ... 

The specific selection rule for infrared absorption spectra is... 

The specific selection rule for vibrational Raman transitions is the Scune as for infrared transitions Av = +1). The photons that are scattered with a lower wavenumber than that of the incident Hght, the Stokes hnes, cffe those for which Ai =+1. The anti-Stokes lines (for which Ai = —1) cffe less intense them the Stokes lines because very few molecules are in m excited vibrationed state initially. 

CD 7. A specific selection rule is a statement about which changes in quantum number may occur in a transition. 

If the orientation of a molecule could be kept fixed in space relative to the exciting photon beam the photoionization event can be made very specific. Selection rules, more powerful than those based on odd-even symmetry, can be used. One way of experi-... 

Any spectrum can consist of any or all combinations of the aforementioned mode types. The range of possibilities for a given ordered polymer structure is determined solely by the symmetry properties of the polymer. However, the picture may be distorted by the limited spectral sensitivity of the Raman and IR instrumentation (in other words, a Raman or IR vibrational mode can be expected theoretically, but the intensity of the mode is so low that it is not observed experimentally). Instruments yielding high signal-to-noise ratios will minimize this problem. Another complication is the presence of multiple or disordered structures (which have no specific selection rules) that cause deviations from the pure mode selection rule requirements of the perfect chain. The spectral bands that are associated with minor amounts of irregular structures are weak and variable in intensity from sample to sample, which makes their detection possible in some cases. 

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