Molecular vibrations representations

SPACEEIL has been used to study polymer dynamics caused by Brownian motion (60). In another computer animation study, a modified ORTREPII program was used to model normal molecular vibrations (70). An energy optimization technique was coupled with graphic molecular representations to produce animations demonstrating the behavior of a system as it approaches configurational equiHbrium (71). In a similar animation study, the dynamic behavior of nonadiabatic transitions in the lithium—hydrogen system was modeled (72). 

In the following chapter this brief outline of representation theory will be applied to several problems in physical chemistry. It is first necessary, however, to show how functions can be adapted to conform to the natural symmetry of a given problem. It will be demonstrated that this concept isof particular importance in the analysis of molecular vibrations and in the th iy of molecular orbitals, among others. The reader is warned, however, that a serious development of this subject is above the level of this book. Hence, in the following section certain principles will be presented without proof. 

Fig. 3. Schematic representation of the molecular vibration of [2.2]paracyclophane, in which the benzene rings move to and fro in a concertina-like fashion with respect to each other...

This table shows how the representations of group Oh are changed or decomposed into those of its subgroups when the symmetry is altered or lowered. This table covers only representations of use in dealing with the more common symmetries of complexes. A rather complete collection of correlation tables will be found as Table X-14 in Molecular Vibrations by E. G. Wilson, Jr., J. C. Decius, and P. C. Cross, McGraw-Hill, New York, 1955. 

In applying the methods of group theory to problems related to molecular structure or dynamics, the procedure that is followed usually involves deriving a reducible representation for the phenomenon of interest, such as molecular vibration, and then decomposing it into its irreducible components. (A reducible representation will always be a sum of irreducible ones.) Although the decomposition can sometimes be accomplished by inspection, for the more general case, the following reduction... 

Molecular vibrations, as detected in infrared and Raman spectroscopy, provide useful information on the geometric and electronic structures of a molecule. As mentioned earlier, each vibrational wavefunction of a molecule must have the symmetry of an irreducible representation of that molecule s point group. Hence the vibrational motion of a molecule is another topic that may be fruitfully treated by group theory. 

Step 3. From 7 3/v subtract the symmetry species rtrans(= r -) for the translation of the molecule as a whole and TTot (based on Rx, Ry, and Rz) for the rotational motion to obtain the representation for molecular vibration, TTitv... 

For the planar ground state, the vibrational representation decompose as rv = 3Ai Bi 2B2, where the Ai modes transform as a translation along the z (C—S) axis and the mode as the single out-of-plane displacement along the x-direction. The last two modes lie in the molecular cr (y,z) plane and are antisymmetric to the a (x,z) plane. Figure 2.6 gives a schematic view of the... 

Let us take now a more complicated basis, and consider all the nuclear coordinates of HNNH shown in Figure 4-8a. These are the so-called Cartesian displacement vectors and will be discussed in Chapter 5 on molecular vibrations. Let us find the matrix representation of the crh operation (see Figure 4-8b). The horizontal mirror plane leaves all x and y coordinates unchanged while all z coordinates will go into their negative selves. In matrix notation this is expressed in the following way ... 

The first step in the symmetry determination of the dynamic properties is the selection of the appropriate basis. Appropriate here means the correct representation of the changes in the properties examined. In the investigation of molecular vibrations (Chapter 5), either Cartesian displacement vectors or internal coordinate vectors are used. In the description of the molecular electronic structure (Chapter 6), the angular components of the atomic orbitals are frequently used... 

For a molecule that has little or no symmetry, it is usually correct to assume that all its vibrational modes are both IR and Raman active. However, when the molecule has considerable symmetry, it is not always easy to picture whether the molecular dipole moment and polarizability will change during the vibration, especially for large and complex molecules. Fortunately, we can easily solve this problem by resorting to simple symmetry selection mles. The molecular vibration is active in IR absorption if it belongs to the same representation as at least one of the dipole moment components fjix, iiy, jj z) or, since the dipole moment is a vector, as one of the Cartesian coordinates (x, y, z). In contrast, the molecular vibration is active in Raman scattering if it belongs to the same representation as at least one of the polarizability components, etc.) or, since the polarizability is a tensor, as... 

The thermodynamic energy of a collection of N harmonic oscillators (approximate representations of molecular vibrations) is given by... 

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