Vibration and rotation of molecules

A molecule can be visualized as an aggregate of atoms bound together by a balance of mutually attractive and repulsive forces. Individual atoms vibrate with respect to one another while the molecule as a whole rotates about any spatial axis. Both types of motion occur simultaneously, and transitions between pairs of vibration-rotation states create the characteristic patterns of infrared spectra. 

Sharing of valence electrons, that is, electrons in the outer shell among their orbitals, binds atoms into a molecular stmcture. Electronic binding forces create a 

For small displacements from equlibrium a molecule can be regarded as a group of atomic or nucleonic masses linked by springs the atoms behave as a set of coupled harmonic oscillators. Each atom is in a part of the potential that is approximately parabolic and nearly obeys Hooke s law. The potential energy, V, is then 

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Even if one restricts one s attention to vibrations and rotations of molecules, there are a variety of Lie algebras one can use. In some applications, the algebras associated with the harmonic oscillator are used. We mention these briefly in Chapter 1. We prefer, however, even in zeroth order to use algebras associated with anharmonic oscillators. Since an understanding of the algebraic methods requires a comparison with more traditional methods, we present in several parts of the book a direct comparison with both the Dunham expansion and the solution of the Schrodinger equation. 

It is assumed that the reader has previously learned, in undergraduate inorganic or physical chemistry classes, how symmetry arises in molecular shapes and structures and what symmetry elements are (e.g., planes, axes of rotation, centers of inversion, etc.). For the reader who feels, after reading this appendix, that additional background is needed, the texts by Cotton and EWK, as well as most physical chemistry texts can be consulted. We review and teach here only that material that is of direct application to symmetry analysis of molecular orbitals and vibrations and rotations of molecules. We use a specific example, the ammonia molecule, to introduce and illustrate the important aspects of point group symmetry. 

The Vibration and Rotation of Molecules.—The nature of the vibrational motion and the values of the vibrational energy levels of a molecule are determined by the electronic energy function, such as that shown in Figure VII-1. The simplest discussion of the vibrational motion of a diatomic molecule is based upon the approximation of the energy curve in the neighborhood of its minimum by a parabola that is, it is assumed that the force between the atoms of the molecule is proportional to the displacement of the internuclear distance from its equilibrium value r.. This corresponds to the approximate potential function... 

Vibrations and Rotations of Molecules Infrared and Microwave Spectroscopy... 

RAMAN SPECTROSCOPY, which is based on a phenomenon studied by CHANDRASEKHARA VENKATA RAMAN (1888 - 1970) can be used to study the vibration and rotation of molecules. 

Many problems in vibration analysis are difficult to solve using a Cartesian co-ordinate system. These coordinates change in a complex manner with the translation, vibration and rotation of molecules or other bodies because of interaction terms. When Cartesian coordinates are transformed into normal coordinates, as shown below, the interaction terms disappear and a motion can be expressed in terms of a single normal coordinate. 

It must be pointed out that another type of internal motion is the overall rotation of the molecule. The vibration and rotation of the molecule are shown schematically in figure Al.2.2. 

What quantity describes best the totahty of these solute-solvent interactions and how can the various contributions to them be estimated Let the pure solute B be vaporized in an imaginary process to a gas, and let this gas be very dilute, so that it obeys the ideal gas laws. In this condition each particle of the solute (molecule or ion) is very remote from any neighbor and has no environment with which to interact. If B is polyatomic, it does have its internal degrees of freedom, such as bond vibrations and rotation of the particle. 

If molecular gases are considered, infrared spectra richer than those seen in the rare gases occur. Besides the translational spectra shown above, various rotational and rotovibrational spectral components may be expected even if the molecules are non-polar. Besides overlap, other induction mechanisms become important, most notably multipole-induced dipoles. Dipole components may be thought of as being modulated by the vibration and rotation of the interacting molecules so that induced supermolecular bands appear at the rotovibrational frequencies. In other words, besides the translational induced spectra studied above, we may expect rotational induced bands in the infrared (and rotovibrational and electronic bands at higher frequencies as this was suggested above, Eq. 1.7 and Fig. 1.3). Lines at sums and differences of such frequencies also occur and are common in the fundamental and overtone bands. We will discuss the rotational pair and triplet spectra first. 

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