States degeneracy

To start, we restrict our treatment to a tri-state degeneracy (the generalization is stiaightforward) and consider the following situation ... 

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... 

Fortunately C-terms are readily distinguished from A- and B-terms by their inverse temperature dependence. If present, they usually dominate the MCD spectrum at low temperatures, since they can be enhanced up to 70-fold on going from room temperature to liquid helium temperatures. For biological chromo-phores, which generally possess only low symmetry, ground state degeneracy can... 

Thus, the bound-state degeneracy factor, = 0/ 1 falls out in a very natural way. It arises simply because of the assumption that the state with na + 2 electrons is at too high an energy to be stable (see Look, 1981, for more detail). 

To determine the various state degeneracies, the gkim, we, of course, have to know something about the states themselves. Sometimes symmetry considerations are sufficient to determine the gklm, even when the various state energies are totally unknown. We will consider only two simple examples here. 

Fig. 8.9. The locus A of double-zero eigenvalue degeneracies of the Hopf bifurcation for cubic autocatalysis with decay. Also shown, as broken curves, are the loci of stationary-state degeneracies, corresponding to the boundaries for isola and mushroom patterns. The curve A lies completely within the parameter regions for multiple stationary states.

The final set of thermodynamic quantities to illustrate is the entropy, also listed in Table 8.1. The largest contribution by far is from translation, calculated from Eq. 8.106. The portion of the entropy attributable to rotational and vibrational degrees of freedom are calculated by Eqs. 8.108 and 8.109, respectively. The electronic contribution to S from Eq. 8.101 is large (certainly relative to the role it played for the other thermodynamic functions just considered), 5.763 (= R In 2, from the ground-state degeneracy contribution) J/mole-K. Thus the net value of S at 298.15 K is calculated to be 198.542 J/mole-K, com-... 

Point group Configuration representation representations Allowed states Degeneracy... 

In order to understand the ferromagnetic order that is sometimes found in perovskites containing 3d4 ions, it is necessary to investigate the effect of Jahn-Teller stabilizations when there is no static, cooperative distortion of the entire lattice. Lack of a static Jahn-Teller stabilization may indicate either (1) noncooperative, static distortions or (2) removal of the ground-state degeneracy by resonance between... 

The relative intensities of the two lines are expected to be proportional to the final state degeneracies, i.e.,... 

Spin-orbit splitting Removal of state degeneracy by spin-orbit coupling. 

Do the experimentally derived state degeneracies and band polarizations match those predicted by theoretical calculations ... 

MCD spectroscopy provides key orbital and spin state degeneracy information that cannot be derived from electronic absorption and NMR spectroscopy. 

Identification and Analysis of Faraday A Terms in the Spectra of Porphyrins with Excited State Degeneracy and Near Degeneracy... 

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