Degenerate normal coordinates

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... 

The implication of this last equation is that the Dn(R) are in block form with each n,x , block corresponding to ny identical X values, i.e. each block corresponds to a set of degenerate normal coordinates. 

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) ... 

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). ... 

An example may clarify the method. Suppose that Qka and Qkb are two degenerate normal coordinates of methane (5d) of species E, while Qia, Qtb, and Qu are taken as of species The combination level with one (luantum of each vibration excited will have a symmetry V = E I o find the structure of F, use is made of the following character table (see. Appendix X). 

I he bawis of tho alternate method mentioned in Sec. 7-3 is the use of polar rather than cartesian degenerate normal coordinates. It is shown in treatises on quantum mechanics that the solution of the wave equation for the doubly degenerate harmonic oscillator expressed in terms of coordinates p and (j> defined by... 

Fig. 3.7. The schematic forms of the normal coordinates of H2O (colunm one) and their transformations by the symmetry operations of the group. A transformation either leaves the form of the normal coordinate unchanged, or else changes its sign which reverses every displacement vector. This distinguishes forms of non-degenerate normal coordinates which are either symmetrical or antisymmetrical with respect to specific symmetry elements possessed by the equilibrium configuration.

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