Rotation energy levels

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. 

The simplest case is a transition in a linear molecule. In this case there is no orbital or spin angular momentum. The total angular momentum, represented by tire quantum number J, is entirely rotational angular momentum. The rotational energy levels of each state approximately fit a simple fomuila ... 

Electronic structure theory describes the motions of the electrons and produces energy surfaces and wavefiinctions. The shapes and geometries of molecules, their electronic, vibrational and rotational energy levels, as well as the interactions of these states with electromagnetic fields lie within the realm of quantum stnicture theory. 

Each and every electronic energy state, labelled k, has a set, labelled L, of vibration/rotation energy levels k,L and wavefiinctions... 

Most infrared spectroscopy of complexes is carried out in tire mid-infrared, which is tire region in which tire monomers usually absorb infrared radiation. Van der Waals complexes can absorb mid-infrared radiation eitlier witli or without simultaneous excitation of intennolecular bending and stretching vibrations. The mid-infrared bands tliat contain tire most infonnation about intennolecular forces are combination bands, in which tire intennolecular vibrations are excited. Such spectra map out tire vibrational and rotational energy levels associated witli monomers in excited vibrational states and, tluis, provide infonnation on interaction potentials involving excited monomers, which may be slightly different from Arose for ground-state molecules. 

The homonuclear rare gas pairs are of special interest as models for intennolecular forces, but they are quite difficult to study spectroscopically. They have no microwave or infrared spectmm. However, their vibration-rotation energy levels can be detennined from their electronic absorjDtion spectra, which he in the vacuum ultraviolet (VUV) region of the spectmm. In the most recent work, Hennan et al [24] have measured vibrational and rotational frequencies to great precision. In the case of Ar-Ar, the results have been incoriDorated into a multiproperty analysis by Aziz [25] to develop a highly accurate pair potential. 

Far-infrared and mid-infrared spectroscopy usually provide the most detailed picture of the vibration-rotation energy levels in the ground electronic state. However, they are not always possible and other spectroscopic methods are also important. 

The expressions for the rotational energy levels (i.e., also involving the end-over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet II electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the isomorfic Hamiltonian was inboduced, which has later been widely used in treating linear molecules (see, e.g., [55]). 

It is antieipated that a eourse dealing with atomie and moleeular speetroseopy will follow the student s mastery of the material eovered in Seetions 1-4. For this reason, beyond these introduetory seetions, this text s emphasis is plaeed on eleetronie stmeture applieations rather than on vibrational and rotational energy levels, whieh are traditionally eovered in eonsiderable detail in speetroseopy eourses. 

The progression of sections leads the reader from the principles of quantum mechanics and several model problems which illustrate these principles and relate to chemical phenomena, through atomic and molecular orbitals, N-electron configurations, states, and term symbols, vibrational and rotational energy levels, photon-induced transitions among various levels, and eventually to computational techniques for treating chemical bonding and reactivity. 

These energy spaeings are of relevanee to mierowave speetroseopy whieh probes the rotational energy levels of moleeules. 

In using symmetry to help simplify moleeular orbital or vibration/rotation energy level ealeulations, the following strategy is followed ... 

Microwave spectroscopy is used to probe transitions between rotational energy levels in mol ecules... 

Question. Using Equation (1.62) calculate, to four significant figures, the rotational energy levels, in joules, for J= 0, 1 and 2 for Then convert these to units of cm. [Use a bond... 

Answer. Using Equation (2.11) for the rotational energy level with 7=4, and the Planck relation between energy and wavenumber of Equation (2.2), we have... 

Figure 5.2 shows a set of rotational energy levels, or, strictly, term values, for the CO molecule. 

Figure 5.6 Rotational energy levels for (a) a prolate and (b) an oblate symmetric rotor...

The rotational energy levels for a prolate and an oblate symmetric rotor are shown schematically in Figure 5.6. Although these present a much more complex picture than those for a linear molecule the fact that the selection mles... 

For a symmetric rotor the modification Eg to the rotational energy levels in an electric field S is larger than in a linear molecule and is given, approximately, by... 

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. 

Figure 6.7(a) illustrates the rotational energy levels associated with two vibrational levels u (upper) and il (lower) between which a vibrational transition is allowed by the Au = 1 selection rule. The rotational selection rule governing transitions between the two stacks of levels is... 

As for diatomic molecules, there are stacks of rotational energy levels associated with all vibrational levels of a polyatomic molecule. The resulting term values S are given by the sum of the rotational and vibrational term values... 

As in Section 5.2.4 on rotational spectra of asymmetric rotors, we do not treat this important group of molecules in any detail, so far as their rotational motion is concerned, because of the great complexity of their rotational energy levels. Nevertheless, however complex the stack associated with the v = 0 level, there is a very similar stack associated with each excited vibrational level. The selection mles for transitions between the rotational stacks of the vibrational levels are also complex but include... 

At low energies, the rotational and vibrational motions of molecules can be considered separately. The simplest model for rotational energy levels is the rigid dumbbell with quantized angular momentum. It has a series of rotational levels having energy... 

There are therefore two ways in which lasers may be used to bring about photon-assisted film formation. If the laser emits radiation in the near-ultra-violet or above, photochemical decomposition occurs in the gas phase and some unabsorbed radiation arrives at the substrate, but this latter should be a minor effect in die thin film formation. This procedure is referred to as photolysis. Alternatively, if the laser emits radiation in the infra-red, and tire photons are only feebly absorbed to raise the rotational energy levels of the gaseous... 

Fourier transform infrared (FTIR) analyzers can be used for industrial applications and m situ measurements in addition to conventional laboratory use. Industrial instruments are transportable, rugged and relatively simple to calibrate and operate. They are capable of analyzing many gas components and determining their concentrations, practically continuously. FTIR analyzers are based on the spectra characterization of infrared light absorbed by transitions in vibrational and rotational energy levels of heteroatomic molecules. 

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