Quantum numbers rotational-vibrational spectroscopy

The emission spectrum observed by high resolution spectroscopy for the A - X vibrational bands [4] has been very well reproduced theoretically for several low-lying vibrational quantum numbers and the spectrum for the A - A n vibrational bands has been theoretically derived for low vibrational quantum numbers to be subjected to further experimental analysis [8]. Related Franck-Condon factors for the latter and former transition bands [8] have also been derived and compared favourably with semi-empirical calculations [25] performed for the former transition bands. Pure rotational, vibrationm and rovibrational transitions appear to be the largest for the X ground state followed by those... 

It should be noted that the rotational spectroscopy of CO confined to a single vibrational level, usually the ground v = 0 level, provides only a limited amount of information about molecular structure. In the field of vibration-rotation spectroscopy, however, CO has been studied extensively and particular attention paid to the variation of the rotational and centrifugal distortion constants with vibrational quantum number. Vibrational transitions involving v up to 37 have been studied with high accuracy [78, 79, 80], and the measurements extended to other isotopic species [81] to test the conventional isotopic relationships. CO is, however, an extremely important and widespread molecule in the interstellar medium. CO distribution maps are now commonplace and with the advent of far-inffared telescopes, it is also an important... 

While the tg structure represents the most well-defined molecular geometry, it is not, unfortunately, one that exists in nature. Real molecules exist in the quantum states of the 3N-6 (or 5) vibrational states with quantum numbers (vj, V2.-..V3N-6 (or 5)). Vj = 0, 1, 2,. Even in the lowest (ground) (0,0...0) vibrational state, the N atoms of the molecule undergo their zero point vibrational motions, oscillating about the equilibrium positions defined by the B-O potential energy surface. It is necessary then to speak of some type of average or effective structures, and to account for the vibrational motions, which vary with vibrational state and isotopic composition. In spectroscopy, a molecule s structural information is carried most straightforwardly by its molecular moments of inertia (or their inverses, the rotational constants), which are determined hy analysis of the pure rotational spectrum or fire resolved rotational structure of vibration-rotation bonds. Thus, the spectroscopic determination of molecular structure boils down to how one uses the rotational constants of a molecule... 

Rotational constants - In molecular spectroscopy, the constants appearing in the expression for the rotational energy levels as a function of the angular momentum quantum numbers. These constants are proportional to the reciprocals of the principal moments of inertia, averaged over the vibrational motion. 

The function fk(R) is a product of a spherical harmonic that describes the rotations of the molecule (J and M stand for the corresponding quantum numbers) and a function that describes the vibrations of the nuclei. The diagram of the energy levels shown in Fig. 6.4 represents the basis of molecular spectroscopy. The diagram may be summarized in the following way ... 

The way the spins of the protons and neutrons in the nucleus combine can lead to nuclei with spin quantum numbers (/), varying from 0 to 6, and isotopes of the same element can have, and usually do have, different spin quantum numbers. Isotopic nuclear spin is important in NMR. For example, has I — 0 and has I = A, so that NMR spectroscopy is not possible with nuclei but is possible with Fortunately for chemists the natural carbon isotopes include 1 % and NMR is a very important tool in chemical structure elucidation [14]. Nuclear spin can also have subtle effects in other spectroscopies [15], and nuclear mass is important in molecular vibrational and rotational spectroscopies. 

One way to directly measure rotational transitions in non-polar molecules such as these is by rotational Raman spectroscopy, which operates on the same principle as other Raman techniques (see Section 6.3). A rotational Raman transition connects initial and final rotational levels within the same vibrational state, so only the rotational quantum number changes. However, this technique is limited in precision by the uncertainties in the photon energies of the incident and scattered light. The scattering intensity increases dramatically with photon energy. 

If vibrational quantum numbers change in a transition, there will be associated changes in rotational quanmm numbers for gases. We discuss the consequences of these changes for vibrational spectroscopy in Section 8.6.2, but note here the essential features governing the generation of rotational strucmre in vibration bands. In all cases, rotation constants can be determined for upper and lower vibrational states, and in favorable cases molecular stmctures can be determined with accuracy comparable with that obtained from pure rotation spectra. 

As with rotational spectroscopy, there are several ways of stating selection rules for spectral transitions involving vibrational states of molecules. There is a gross selection rule, which generalizes the appearance of absorptions or emissions involving vibrational energy levels. There is also a more specific, quantum-number-based selection rule for allowed transitions. Finally, there is a selection rule that can be based on group-theoretical concerns, which were not considered for rotations. 

In all disciplines of molecular spectroscopy, the rotational-vibrational energy is described by a power seri of J(J-l-l) and (u-l-1/2), where J and v are the rotational and vibrational quantum numbers, respectively. As parameters, the Dunham notation Tu or the original spectroscopic notation. .. are in use. 

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