Degenerate symmetry coordinates

The degenerate symmetry coordinates for the same two sets are of species and E g (Table 10-2). Only one member of each degenerate pair will be constructed, using the simple method described in See. 6-4. Applying the general formula [Eq. (7), Sec. 6-4], namely,... 

Consider the doubly degenerate symmetry coordinates of species E in ammonia.. Vccording to Pablo 6-2 the three Nil stretches, r, r, and r. 

Then tlie theorem above states that there is one combination of r s in Ai of 7>d and one degenerate set of three in Fi, since Ai and F2 are connected with the special species Ai of 3C. If properly oriented, one combination of the F set will be in A, of 63,-, the other two in E of Gsv Proof of the Modified Rule for Degenerate Cases. Degenerate symmetry coordinates Sa, Si, will transform according to the formula... 

Flowever, when a symmetry operation is carried out on a degenerate normal coordinate it does not simply remain the same or change sign in general, it is transformed into a linear... 

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

S transforms o i 0 2 Q3 0 4 as r f into the symmetry coordinate basis X)r). Normalization factors are included in S but omitted from X),), as is usually the case in character tables. For the reason given in the text, in this group a4 0 5 are used to define similarity of orientation in the doubly degenerate E representation. 

When r is not [ (case (b) in Section 13.4), T, T form a degenerate pair which are generally labeled by 1H, 2E (sometimes with subscripts). Applying the projection operator 2 x(R) R to a tensor component will therefore yield symmetry coordinates from 1E, 2E... 

From these symmetry coordinates, Q is associated with the totally symmetric representation Aj, while the pair (Q2,Q3) is associated with the doubly degenerate representation , The geometric meaning of these coordinates is illustrated in Fig. 5. From the coordinates (Q2, Q3) we may define the polar coordinates... 

Since each of the three symmetry species, Xg, Xu, and Il appears only once, the symmetry coordinates will be good representations of the normal modes. There is no possibility for mixing. Figure 5-10 shows the forms of the normal vibrations of the C02 molecule. The two bending modes are degenerate they are of equal energy. 

In this sequence, the totally symmetric stretching vibrations are compared, since they are the only ones in this irreducible representation in which the symmetry coordinate is identical with the normal coordinate. However, the same trend is often observed for the degenerate stretching modes v p2), although in this case interaction with the deformation mode of the same symmetry va F2) can in principle lead to frequency changes ... 

Vibrational Spectroscopy. In spite of their complex molecular frameworks, the monomeric borohydrides display surprisingly simple vibrational spectra due to their high symmetry (T ), which requires that many fundamental vibrations be degenerate. Normal coordinate analyses have been carried out for ZrCBHtt) (14) and HfCBHtt) (15) and a similar study was completed for Np(BHit)if in order to compare vibrational energy level structures and elucidate the nature of the fundamental vibrations of Np(BHit)tf (10). ... 

Concerning trinuclear systems, the Jahn-Teller theorem predicts that the geometry of an isosceles triangle is unstable unless the ground electronic state is non-degenerate. In the geometry of the electronic wave functions are of either the A- or the F-type and there exist 37V - 6 = 3 symmetry coordinates ... 

The carbon atom is at the center of the molecule, and the carbon Ij and Is AOs are each sent into themselves by every symmetry operation. These AOs transform according to the totally symmetric species A]. The carbon 2p 2py, and 2p AOs are given by x, y, or z times a radial function. Their symmetry behavior is the same as that of the functions x, y, and z, respectively. From the formulas for rotation of coordinates [Eq. (15.52)], we see that any proper rotation sends each of the functions x, y, and z into some linear combination of x, y, and z. Any improper rotation is the product of some proper rotation and an inversion (Problem 12.15) the inversion simply converts each coordinate to its negative. Hence the three carbon 2p orbitals are sent into linear combinations of one another by each symmetry operation. They must therefore transform according to one of the triply degenerate symmetry species. Further investigation (which is omitted) shows the symmetry species of the 2p AOs to be T2. 

Schnepp (1967) has given the values of these sums for two infrared active symmetry coordinates of a crystal of structure n, appropriate for the solids a-Nj and COj. The symmetry coordinate designations are those of Walmsley and Pople (1964) and the coordinates and 5 are components of triply degenerate sets. The results are... 

As a second, slightly more complicated example, consider the four stretches and six bending and atj coordinates of methane (Fig. 6-4). Only the a set is involved in the doubly degenerate species, so that it is permissible to construct the symmetry coordinate according to the formula... 

To be sure, (22) gives only one member of a degenerate set of symmetry coordinates, but this is all that is needed for any ordinary calculations. 

If the species is degenerate, the problem of orientation can be solved by the same method as used in Sec. 6-4 for internal symmetry coordinates that is, select to be a coordinate, or simple combination of coordinates, symmetrical (or antisymmetrical) to some chosen set of group operations constituting a subgroup. 

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