Rotation molecular energy levels

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

Whereas the gas lasers described use energy levels characteristic of individual atoms or ions, laser operation can also employ molecular energy levels. Molecular levels may correspond to vibrations and rotations, in contrast to the electronic energy levels of atomic and ionic species. The energies associated with vibrations and rotations tend to be lower than those of electronic transitions thus the output wavelengths of the molecular lasers tend to He farther into the infrared. 

The first attempt to explain the characteristic properties of molecular spectra in terms of the quantum mechanical equation of motion was undertaken by Born and Oppenheimer. The method presented in their famous paper of 1927 forms the theoretical background of the present analysis. The discussion of vibronic spectra is based on a model that reflects the discovered hierarchy of molecular energy levels. In most cases for molecules, there is a pattern followed in which each electronic state has an infrastructure built of vibrational energy levels, and in turn each vibrational state consists of rotational levels. In accordance with this scheme the total energy, has three distinct components of different orders of magnitude,... 

In the above expressions for C(t), the averaging over initial rotational, vibrational, and electronic states is explicitly shown. There is also an average over the translational motion implicit in all of these expressions. Its role has not (yet) been emphasized because the molecular energy levels, whose spacings yield the characteristic frequencies at which light can be absorbed or emitted, do not depend on translational motion. However, the frequency of the electromagnetic field experienced by moving molecules does depend on the velocities of the molecules, so this issue must now be addressed. 

Here we use the label i to denote a molecular energy level, which may denote at once the specific translational (t), rotational (r), vibrational (u), and electronic (e) energy level of the molecule. From Eq. 8.46 and the definition of the molecular partition function q,... 

Infrared Spectrophotometry (IR). Atoms are in constant motion within molecules, and associated with these motions are molecular energy levels that correspond to the energies of quanta of IR radiation. These motions can be resolved into rotation of the whole molecule in space and into motions corresponding to the vibration of atoms with... 

For molecules, the spectroscopic nomenclature for molecular energy levels and their vibronic and rotational sublevels is messy and very specialized. Already for homonuclear or heteronuclear diatomic molecules a new quantum number shows up, which quantifies the angular momentum along the internuclear axis, but the reader need not be burdened with the associated nomenclature. 

In order to understand molecular energy levels, it is helpftd to partition the kinetic energies of the nuclei and electrons in a molecule into parts which, if possible, separately represent the electronic, vibrational and rotational motions of the molecule. The details of the processes by which this partitioning is achieved are presented in chapter 2. Here we give a summary of the main procedures and results. 

The high-resolution spectroscopy of OH has been perhaps the most important test bed for the development of the theory of the molecular energy levels, both in zero field and in the presence of applied magnetic fields. In this section, we concentrate on the A-doubling and hyperfine structure, as probed by the molecular beam studies. In chapter 9 we discuss the complex theory of the Zeeman effect, and in chapter 10 deal with rotational transitions. Our discussion therefore follows a pattern similar to that adopted for the NO molecule. 

Molecules can exist in any of the allowed electron, vibrational and rotational levels. Thus the overall molecular energy level diagram appears much like that for an atom however, on top of each electron level is a series of vibrational levels and on each of these is a series of rotational levels. The overall appearance is that of a series of bands of very cioseiy spaced energy ieveis. 

Figure 7.1. Molecular energy levels and (a) electronic, (b) rotational, and (c) vibrational transitions.

Rotational-vibrational energy levels fitted to a quadratic-cum-Lorentzian model potential of cylindrical symmetry about the linear unstable equilibrium configuration. Barrier to inversion in the molecular plane 1.10(13) eV (Gilchrist et al ). 

However, a complete set of molecular energy levels needed for calculation of the partition function (Eq. (1.16)) is not available in most cases. The arising problem can be simplified through the approximation that the different types of motion such as vibration, rotation, and electronic excitations are on a different timescale and therefore are unaffected by each other and can be treated as decoupled motions. This leads to a separation of Q into factors that correspond to separate partition functions for electronic excitations, translation, vibration, external molecular rotation, and hindered and free internal rotation ... 

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