Molecules, complex polyatomic, rotation

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

In the case of polyatomic molecules, such as CO2, H2O, O3, etc., the principles discussed above still apply, but the spectra become more complex. Polyatomic molecules do not rotate only about one single axis, but about three mutually perpendicular axes. In addition, the number of vibrational degrees of freedom is also increased. 

The principles discussed for diatomic molecules generally apply to polyatomic molecules, but their spectra are much more complex. For example, instead of considering rotation only about an axis perpendicular to the internuclear axis and passing through the center of mass, for nonlinear molecules, one must think of rotation about three mutually perpendicular axes as shown in Fig. 3-lb Hence we have three rotational constants A, B, and C with respect to these three principal axes. 

Equation (9.38), if restricted to two particles, is identical in form to the radial component of the electronic Schrodinger equation for the hydrogen atom expressed in polar coordinates about the system s center of mass. In the case of the hydrogen atom, solution of the equation is facilitated by the simplicity of the two-particle system. In rotational spectroscopy of polyatomic molecules, the kinetic energy operator is considerably more complex in its construction. For purposes of discussion, we will confine ourselves to two examples that are relatively simple, presented without derivation, and then offer some generalizations therefrom. More advanced treatises on rotational spectroscopy are available to readers hungering for more. 

The more complex rotations and vibrations of polyatomic molecules are subject to the same principles, and distribution laws of the same kind apply, as will be shown in the following section. 

Polyatomic molecules have more complex microwave spectra, but the basic principle is the same any molecule with a dipole moment can absorb microwave radiation. This means, for example, that the only important absorber of microwaves in the air is water (as scientists discovered while developing radar systems during World War II). In fact, microwave spectroscopy became a major field of research after that war, because military requirements had dramatically improved the available technology for microwave generation and detection. A more prosaic use of microwave absorption of water is the microwave oven it works by exciting water rotations, and the tumbling then heats all other components of food. 

For many types of electron spectroscopies there are still comparatively few studies of SOC effects in molecules in contrast to atoms, see, e.g., [1, 2, 3, 4, 5, 6, 7] and references therein. This can probably be referred to complexities in the molecular analysis due to the extra vibrational and rotational degrees of freedom, increased role of many-body interaction, interference and break-down effects in the spectra, but can also be referred to the more difficult nature of the spin-orbit coupling itself in polyatomic species. Modern ab initio formulations, as, e.g., spin-orbit response theory [8] reviewed here, have made such investigations possible using the full Breit-Pauli spin-orbit operator. 

As might be expected, the rotational spectra of polyatomic molecules are more complex than those for diatomic molecules. For example, depending on their structures polyatomic molecules can have as many as three different moments of inertia. Although more complicated, the analysis of the structures of polyatomic molecules using rotational spectra follows the same principles as we have discussed for diatomic molecules. 

Polyatomic molecules have up to three different moments of inertia, corresponding to rotations about three axes (Fig. 20.6). The rotational spectra for nonlinear polyatomic molecules are more complex than the example just illustrated, but their interpretation is carried out in the same way and has enabled chemists to determine with high accuracy the molecular geometries for many small polyatomic molecules. 

The example given above provides a simple illustration of the use of moments of inertia and vibration frequencies to calculate equilibrium constants. The method can, of course, be extended to reactions involving more complex substances. For polyatomic, nonlinear molecules the rotational contributions to the partition functions would be given by equation (16.34), and there would be an appropriate term of the form of (16.30) for each vibrational mode. ... 

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