Degeneracy analysis

From the preceding analysis, it is seen that the coordinate space neai R can be usefully partitioned into the branching space described in tenns of intersection adapted coordinates (p, 9, ) or (x,y,z) and its orthogonal complement the seam space spanned by a set of mutually orthonormal set w, = 4 — M . From Eq. (27), spherical radius p is the parameter that lifts the degeneracy linearly in the branching space spanned by x, y, and z. 

The reorientation of the B—H complex at 100 K complicates the analysis of the stress splitting data. The ratios of the intensities of the stress split components were extrapolated to zero stress to determine the site degeneracies for each stress orientation and hence to deduce the symmetry of the complex (Herrero and Stutzmann, 1988b). A unique configuration could not be found to fit the data for all stress directions it was suggested that the configuration of the complex must depend upon the applied stress. For the [110] stress direction it was proposed that the H is displaced from the trigonal axis in the direction away from the C site, while for [100] stress the H is supposed to be displaced toward the C site. 

Luttinger-Tisza method is burdened by independent minimization variables, while analysis of the values of the Fourier components F k) makes it possible to immediately exclude no less than half of the variable set and to obtain a result much more quickly. Degeneracy of the ground state occurs either due to coincidence of minimal values of Vt (k) at two boundary points of the first Brillouin zone k = b]/2 and k = b2/2, or as a result of the equality Fj (k) = F2 (k) at the same point k = h/2. The natural consequence of the ground state degeneracy is the presence of a Goldstone mode in the spectrum of orientational vibrations.53... 

It remains to consider the isotopically heteronuclear systems to complete the symmetry analysis of this system. Because the experiments are performed under natural abundance conditions, only systems containing a single rare isotope ( O or 0) need be considered. However, because the spatial degeneracy of the electronic state of the ion and the neutral differ, the case where either the neutral or the ion is isotopically heteronuclear must be considered separately. The results in Table 4 show that when the neutral is made isotopically heteronuclear the /-based restriction is removed, while that based on is preserved. Conversely, when... 

When there are only two electrons the analysis is much simplified. Even quite elementary textbooks discuss two-electron systems. The simplicity is a consequence of the general nature of what is called the spin-degeneracy problem, which we describe in Chapters 4 and 5. For now we write the total solution for the ESE 4 (1, 2), where the labels 1 and 2 refer to the coordinates (space and spin) of the two electrons. Since the ESE has no reference at all to spin, 4 (1, 2) may be factored into separate spatial and spin functions. For two electrons one has the familiar result that the spin functions are of either the singlet or triplet type. 

For s equivalent oscillators, the degeneracy of vibrational level v is found from basic probability analysis to equal to the number of unique ways of putting v identical objects into s boxes ... 

As an example, consider a tetrahedral molecule in T symmetry, with two singly-occupied t2 symmetry orbitals, say tfy1. The direct product T2 (8) T2 reduces to A E Ti T2, so we obtain singlet states Mi, 1E, 1Ti, and 1T2, and triplet states Mi, 3E, 37), and 3T2. A handy check on the correctness of this sort of analysis is to add up the toted spin and spatial degeneracies of all the states and verify that it equals the spin and spatial degeneracy of the original orbital product (36 in this case). 

The simplest new phenomenon induced by this mechanism is a secondary bifurcation from the first primary branch, arising from the interaction between the latter and another nearby primary branch. It leads to the loss of stability of the first primary branch or to the stabilization of one of the subsequent primary branches, as illustrated in Fig. 1. The analysis of this branching follows similar lines as in Section I. A, except that one has now two control parameters X and p., which are both expanded [as in equation (5)] about the degeneracy point (X, p.) corresponding to a double eigenvalue of the linearized operator L. Because of this double degeneracy, the first equation (7) is replaced by... 

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