Degeneracy defined

Wlien = N/2, the value of g is decreased by a factor of e from its maximum atm = 0. Thus the fractional widtii of the distribution is AOr/A i M/jV)7 For A 10 the fractional width is of the order of 10 It is the sharply peaked behaviour of the degeneracy fiinctions that leads to the prediction that the thennodynamic properties of macroscopic systems are well defined. 

Figure 23. A Loiigiiet-Higgms loop around the Jahn-Teller degeneracy of CPDR at D h symmetry, preserve and Qinv n aje the phase-inverting and phase-preserving coordinates that define the loop.

In an octahedral crystal field, for example, these electron densities acquire different energies in exactly the same way as do those of the J-orbital densities. We find, therefore, that a free-ion D term splits into T2, and Eg terms in an octahedral environment. The symbols T2, and Eg have the same meanings as t2g and eg, discussed in Section 3.2, except that we use upper-case letters to indicate that, like their parent free-ion D term, they are generally many-electron wavefunctions. Of course we must remember that a term is properly described by both orbital- and spin-quantum numbers. So we more properly conclude that a free-ion term splits into -I- T 2gin octahedral symmetry. Notice that the crystal-field splitting has no effect upon the spin-degeneracy. This is because the crystal field is defined completely by its ordinary (x, y, z) spatial functionality the crystal field has no spin properties. 

Strictly, L is defined only as a quantum number for a spherical environment - the free ion. The use of L ff = 0 for A terms or Leff = 1 for L terms on the grounds that (2Leff + 1) equals the degeneracy of these terms is, however, legitimate as used here. There is a close parallel between the quantum mechanics of T terms in octahedral or tetrahedral symmetry on the one hand, and of P terms in spherical symmetry on the other. 

Here, AS° = AS i + AS n, but since identical vibrational levels have been assumed for the two electronic levels, AS jb = 0. In addition, AS j is related to the electronic degeneracies of the two levels by ASei = K In ( l/ h)- Finally, if the transformation takes place at constant pressure, AH° = NAE where AE is defined as above. Consideration of the Gibbs free energy for an assembly of N molecules thus produces the same relation for the HS fraction Uh as the potential energy for the isolated molecule, if only the interaction between the molecules is neglected. 

A) Measure the positions and amplitudes of all the lines in the spectrum and list them in order in a table (a spreadsheet program is convenient for this purpose). A well-defined measure of position in a complex spectrum is the x-axis point halfway between the maximum and minimum of the first-derivative line. The amplitude is the difference in height between the maximum and minimum. If convenient, measure the line positions in gauss if this is inconvenient, use arbitrary units such as inches, centimeters, or recorder chart boxes measured from an arbitrary zero. In your table, also provide headings for the quantum numbers (m1 m2, etc.) for each of the line positions, for the coupling constants (a, a2, etc.), and for the theoretical intensity (degeneracy) of each peak. 

The other degree of freedom in defining Vee related to the degeneracy of the 1-RDM can be avoided with a proper definition of the diagonal part (uee) (see Ref. [81]). As a consequence, the NOs with the same ON are determined only up to the unitary transformation among them. 

The degeneracy of the non-chiral complexes can be removed by incorporating chiral centers, usually as resolved amino acids, into the arms at close vicinity to the hydroxamate iron binding sites. Thus, only one of the energetically non-equivalent diastereomers predominates, leading to pure enantiomeric iron(III) complexes with defined hehcity that allows assessing stereospecific recognition by the ferrichrome receptor. 

---