The Number of Normal Vibrations for Each Species

A more general method of finding the number of normal vibrations in each species can be developed by using group theory. The principle of the method is that all the representations are irreducible if normal coordinates are used as the basis for the representations. For example, the representations for the symmetry operations based on three normal coordinates, , 2, and Qy, which correspond to the p, and Vy vibrations in the HzO molecule of.Bg. 1-7, are as follows  

Lei a representation be written with the 3N reciangular coordinates of an A/-atom molecule as its basis. If it is decompdsed into its irreducible components, the basis for these irreducible representations must be the normal coordinates, and the number of appearances of the same irreducible representation must be equal to the number of normal vibrations belonging to the species represented by this irreducible representation. As stated previously, however, the 3/V rectangular coordinates Involve six (or five) coordinates, which correspond to the translational and rotational motions of the molecule as a whole. Therefore the representations that have such coordinates as their basis must be subtracted from the result obtained above. Use of the character of the representation, rather than the representation itself, yields the same result. 

For example, consider a pyramidal XYy molecule that has six normal vibration.s. At first, the representations for the various symmetry operations must be written with the 12 rectangular coordinates in Fig. I-IO as their basis. Consider pure rotation Cp. If the clockwise rotation of the point (x,y, z) around the z axis by the angle 6 brings it to the point denoted by the coordinates (x. y j z ), the relations between these two sets of coordinates are given by 

The same result is obtained for y( Cg). If this symmetry operation is applied to all the coordinates of the XY3 molecule, the result is 

Pure rotation and identity are called proper rotation. 

From the definitions given in the footnotes of Appendix 111, it is obvious that m = 0, mo= 1, and m = 1 in this case. To check these numbers, we calculate the total number of atoms from the equation for N given above. Since the result is 4, these assigned numbers are correct. Then, the number of normal vibrations in each species can be calculated by inserting these numbers in the general equations given above 2, 0, and 2, respectively, for the Ai, A2, and E species. Since the E species is doubly degenerate, the total number of vibrations is counted as 6, which is expected from the 3N — 6 rule. 

A more general method of finding the number of normal vibrations in each species can be developed by using group theory. The principle of the method is that all the 

Since this equation is not applicable to the infinite point groups (C approaches have been proposed (see Refs. 75 and 76). 

For example, consider a pyramidal XY3 molecule that has six normal vibrations. At first, the representations for the various symmetry operations must be written with the 12 rectangular coordinates in Fig. 1.19 as their basis. Consider pure rotation C +. If the 

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The procedure for finding the number of normal vibrations in each species was described in Sec. 1.8. This procedure is, however, considerably simplified if internal coordinates are used. Again, consider a pyramidal XY3 molecule. Using the internal coordinates shown in Fig. 1.20c, we can write the representation for the operation as... 

APPENDIX III. GENERAL FORMULAS FOR CALCULATING THE NUMBER OF NORMAL VIBRATIONS IN EACH SPECIES... 

General Formulas for Calculating the Number of Normal Vibrations in Each Symmetry Species... 

Two structures have been proposed for (Gly) I an antiparallel-chain pleated sheet (APPS) and a similar rippled sheet (APRS) (see Section III,B,1). These structures have different symmetries the APPS, with D2 symmetry, has twofold screw axes parallel to the a axis [C (a)] and the b axis [C (b)], and a twofold rotation axis parallel to the c axis [62(0)] the APRS, with C2h symmetry, has a twofold screw axis parallel to the b axis ( 2(6)], an inversion center, i, and a glide plane parallel to the ac plane, o-Sj. Once these symmetry elements are known, together with the number of atoms in the repeat, it is possible to determine a number of characteristics of the normal modes the symmetry classes, or species, to which they belong, depending on their behavior (character) with respect to the symmetry operations the numbers of normal modes in each symmetry species, both internal and lattice vibrations their IR and Raman activity and their dichroism in the IR. These are given in Table VII for both structures. 

For each of the molecules of Problem 9.23, find the characters of r3Ar. Then find the symmetry species of the 3N-6 normal modes Of vibration. State the number of distinct vibrational frequencies for each molecule. 

In Section 4.3.f it was shown that there are 3N — 5 normal vibrations in a linear molecule and 3N — 6 in a non-linear molecule, where N is the number of atoms in the molecule. There is a set of fairly simple rules for determining the number of vibrations belonging to each of the symmetry species of the point group to which the molecule belongs. These rules involve the concept of sets of equivalent nuclei. Nuclei form a set if they can be transformed into one another by any of the symmetry operations of the point group. For example, in the C2 point group there can be, as illustrated in Figure 6.18, four kinds of set ... 

Since molecular vibrations in general are slightly anharmonic, both the infrared and Raman spectrum may contain weak overtone and combination bands. A combination energy level is one which involves two or more normal coordinates with different frequencies that have vibrational quantum numbers greater than zero. For example, a combination band which appears at the sum of the wavenumbers of two different fundamentals involves a transition from the ground vibrational level (belonging to the totally symmetric species) to an excited combination level where two different normal coordinates each have a quantum number of one and all the others have a quantum number zero. To obtain the spectral activity of the combination band transition it is necessary to determine the symmetry species of the excited wave-function. In quantum mechanics the total vibrational wavefunction is equal... 

A difference band appears at a wavenumber equal to the difference between the wavenumber of two different fundamentals. In this case the initial state is not the ground state but is one where for one normal coordinate the quantum number is one and the other quantum numbers are zero. The final state is one where some other normal coordinate with a higher frequency has a quantum number of one and the other quantum numbers are all zero. Each of the two levels involved has a wavefunction with the same symmetry as its normal coordinate whose quantum number is equal to one. From the selection rule given earlier, the spectral activity of the difference band transition is evaluated by determining the symmetry species of the direct product of the characters of the vibrational symmetry species of the two normal coordinates involved. Mechanically, this procedure is identical to that given... 

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