Rotational energy transitions

Example 14.10 illustrates the use of the rotational transition energy to calculate of the bond length of a diatomic molecule. 

As staggering, Eq. (1), is strongly a differential property, i.e., the fourth-order finite difference of the rotational transition energy derived by using Eqs... 

If the experunental technique has sufficient resolution, and if the molecule is fairly light, the vibronic bands discussed above will be found to have a fine structure due to transitions among rotational levels in the two states. Even when the individual rotational lines caimot be resolved, the overall shape of the vibronic band will be related to the rotational structure and its analysis may help in identifying the vibronic symmetry. The analysis of the band appearance depends on calculation of the rotational energy levels and on the selection rules and relative intensity of different rotational transitions. These both come from the fonn of the rotational wavefunctions and are treated by angnlar momentum theory. It is not possible to do more than mention a simple example here. 

Electron-impact energy-loss spectroscopy (EELS) differs from other electron spectroscopies in that it is possible to observe transitions to states below the first ionization edge electronic transitions to excited states of the neutral, vibrational and even rotational transitions can be observed. This is a consequence of the detected electrons not originating in the sample. Conversely, there is a problem when electron impact induces an ionizing transition. For each such event there are two outgoing electrons. To precisely account for the energy deposited in the target, the two electrons must be measured in coincidence. 

The energies at whieh the rotational transitions oeeur appear to fit the AE = 2B (J+1) formula rather well. The intensities of transitions from level J to level J+1 vary strongly with J primarily beeause the population of moleeules in the absorbing level varies with J. 

In mierowave speetroseopy, the energy of the radiation lies in the range of fraetions of a em-i through several em-i sueh energies are adequate to exeite rotational motions of moleeules but are not high enough to exeite any but the weakest vibrations (e.g., those of weakly bound Van der Waals eomplexes). In rotational transitions, the eleetronie and vibrational states are thus left unehanged by the exeitation proeess henee /ei = /ef and Xvi... 

We have seen in Section 5.2.1.4 that there is a stack of rotational energy levels associated with all vibrational levels. In rotational spectroscopy we observe transitions between rotational energy levels associated with the same vibrational level (usually v = 0). In vibration-rotation spectroscopy we observe transitions between stacks of rotational energy levels associated with two different vibrational levels. These transitions accompany all vibrational transitions but, whereas vibrational transitions may be observed even when the sample is in the liquid or solid phase, the rotational transitions may be observed only in the gas phase at low pressure and usually in an absorption process. 

The range of photon energies (160 to 0.12 kJ/mol (38-0.03 kcal/mol)) within the infrared region corresponds to the energies of vibrational and rotational transitions of individual molecules, of electronic transitions in many semiconductors, and of vibrational transitions in crystalline lattices. Semiconductor electronics and crystal lattice transitions are beyond the scope of this article. 

Molecules vibrate at fundamental frequencies that are usually in the mid-infrared. Some overtone and combination transitions occur at shorter wavelengths. Because infrared photons have enough energy to excite rotational motions also, the ir spectmm of a gas consists of rovibrational bands in which each vibrational transition is accompanied by numerous simultaneous rotational transitions. In condensed phases the rotational stmcture is suppressed, but the vibrational frequencies remain highly specific, and information on the molecular environment can often be deduced from hnewidths, frequency shifts, and additional spectral stmcture owing to phonon (thermal acoustic mode) and lattice effects. 

Figure 4-11. INDQ/SCI-caleulalcd evolution of the transition energies (upper pan) and related intensities (bottom pan) of the lowest two optical transitions of a cofacial dimer formed by two stilbenc molecules separated by 4 A as a function of the dihedral angle between the long molecular axes, when rotating one molecule around the stacking axis and keeping the molecular planes parallel (case IV of Figure 4-10). Open squares (dosed circles) correspond to the S(J - S2 (S0 — S, > transition.

The latter relation results from energy conservation and forbids rotational transitions when translational energy is deficient. The back processes with transfer of the rotational energy to translational energy are unrestricted. As a consequence, the lower limit of integration in Eq. (5.18) equals f — ej at j < j and otherwise it is equal to 0. It is this very difference that leads to an exact relation between off-diagonal elements of the impact operator... 

Fig. 5.4. Rotational transition cross-section for Ar-N2 versus kinetic energy of collision (Kelvins) from [191]. (+) CC calculations of C2-.4 from [211].

Additional experimental verification that molecules of hydrogen in condensed phases are in states approximating those for free molecules is provided by the Raman effect measurements of McLennan and McLeod.13 A comparison of the Raman frequencies found by them and the frequencies corresponding to the rotational transitions / = 0—>/ = 2 and/= 1— / = 3 (Table II) shows that the intermolecular interaction in liquid hydrogen produces only a very small change in these rotational energy levels. 

Our group has continued to examine several other rare Rg- XY systems, and a list of the experimental and theoretical T-shaped and linear binding energies of these complexes is presented in Table 1 [52-55,57]. The general features and characteristics of the spectra for all of the complexes investigated are similar to those observed in the spectra for the He ICl and He Br2 systems. The linear features are observed to higher transition energies than the T-shaped features. In contrast to the rather simple rotational structure of the T-shaped features, the linear features possess much broader and structured... 

Figure 13. Action spectrum of the linear He I Cl complex near the He + I Cl(By = 2) dissociation limit obtained by scanning the excitation laser through the ICl B—X, 2-0 region and monitoring the l Cl E—>X fluorescence induced by the temporally delayed probe laser, which was fixed on the l Cl E—B, 11-2 band head, (a). The transition energy is plotted relative to the I Cl B—X, 2-0 band origin, 17,664.08 cm . Panels (b), (c), and (d) are the rotational product state spectra obtained when fixing the excitation laser on the lines denoted with the corresponding panel letter. The probe laser was scanned through the ICl B—X, 11-2 region. Modified with permission from Ref. [51].

Molecules in the gas phase have rotational freedom, and the vibrational transitions are accompanied by rotational transitions. For a rigid rotor that vibrates as a harmonic oscillator the expression for the available energy levels is ... 

Table 3. Pure rotational and rovibrational transition energies...

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