Orientational degeneracy

Each set of p orbitals has three distinct directions or three different angular momentum m-quantum numbers as discussed in Appendix G. Each set of d orbitals has five distinct directions or m-quantum numbers, etc s orbitals are unidirectional in that they are spherically symmetric, and have only m = 0. Note that the degeneracy of an orbital (21+1), which is the number of distinct spatial orientations or the number of m-values. 

So, the calculation of the shape of an IR spectrum in the case of anticorrelated jumps of the orienting field in a complete vibrational-rotational basis reduces to inversion of matrix (7.38). This may be done with routine numerical methods, but it is impossible to carry out this procedure analytically. To elucidate qualitatively the nature of this phenomenon, one should consider a simplified energy scheme, containing only the states with j = 0,1. In [18] this scheme had four levels, because the authors neglected degeneracy of states with j = 1. Solution (7.39) [275] is free of this drawback and allows one to get a complete notion of the spectrum of such a system. 

An applied stress lowers the symmetry of the crystal and can make defects with different orientations inequivalent. A review of stress techniques has been written by Davies (1988). The degeneracy of the ground state and also of the spectroscopic transition energies can be lifted. In this section we suppose that the defects cannot reorient and consider only the splitting of the transition energies. The stress-induced reorientation of defects is discussed in the next section. 

Most of the complexes to be described here have trigonal symmetry. For a trigonal center, the splitting of the spectral bands due to the lifting of orientational degeneracy is described by two parameters, A1 and A2. The parameter A is proportional to the hydrostatic component of the stress and gives rise to a shift in frequency that is independent of the stress direction, whereas A2 gives rise to a shift that depends on the orientation of the center. 

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. 

The orientation dependence of the stress alignment effect is consistent with the trigonal symmetry of the B—H complex. Stress along the [110] direction lifts the orientational degeneracy of the four BC sites about the boron while stress along the [100] direction does not. (A [111] stress also leads to a dichroism of the expected magnitude.) The sites perpendicular to... 

The most documented case is the Be-H complex in GaAs, which is characterized by a LVM at 2037 cm-1 (Nandhra et al., 1988). Stavola et al. (1989) have studied the effect of uniaxial stress on this LVM. The use of uniaxial stress allows the orientational degeneracies to lift and therefore gives the symmetry of the center evidenced by its LVM. 

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... 

The number of peaks actually observed in an infrared spectrum is often less than the maximum because some of the vibrations are energetically identical or degenerate. A real molecule will often have two or more vibrations that may differ only by their orientation in space. These will have exactly the same energy and result in one absorption peak. In addition to the degeneracy of vibrational modes, there is also the requirement that a vibration result in a change in the dipole moment of the molecule needs to be observed. 

In contrast to channel I, which remains non-degenerate in the field, channel II splits into three dissociation branches due to the Stark effect. Atomic calculations of the H atom in a cylindrical potential oriented along the z-axis show that the energy levels of 2p-orbitals split and H(2pj.) becomes more stable relative to H(2p c) and H(2py). Because of the lateral symmetry of the potential, the degeneracy of H(2pJ and H(2py) persists. For small values of w, H(2s) is slightly less stable than H(2p ) and H(2py) while the ordering reverses when w exceeds 0.15 a.u. As a result, there exist three dissociation limits for channel II, with a nonzero cylindrical potential, which correspond to H(2s), H(2pj,), and H(2p )/H(2py). 

Not considered in this review are the removal of energetic equivalence by an applied field or stress where, as in a spinel, the energetically equivalent sites have differently oriented crystal-field axes. This lifting of site degeneracy coupled with charge transfer between mixed-valence states leads to such phenomena as magnetic after effect and photoinduced anisotropy ... 

For the description of the order tensor of an isolated molecule or domain, this ambiguity has no practical significance. However, it becomes an important consideration when establishing the orientation of two or more domains, as will be discussed in Section 4. This degeneracy can be lifted by considering a small number of NOE derived constraints, stereo chemical considerations, or through acquisition of RDCs in an additional independent aligning medium.55,56... 

Interaction of the momenta is contained in a non-spherical part of the Coulomb interaction and in the spin-orbit interaction. The value of the energy of the interaction of two momenta depends on the angle between them, therefore, in such a case the definite inter-orientation of all one-electronic momenta is settled. Differently oriented states have different energy, i.e. the zero-order level splits into sublevels and its degeneracy disappears. 

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