Degenerate electron distribution

The ground term of the cP configuration is F. That of is also F. Those of and d are " F. We shall discuss these patterns in Section 3.10. For the moment, we only note the common occurrence of F terms and ask how they split in an octahedral crystal field. As for the case of the D term above, which splits like the d orbitals because the angular parts of their electron distributions are related, an F term splits up like a set of / orbital electron densities. A set of real / orbitals is shown in Fig. 3-13. Note how they comprise three subsets. One set of three orbitals has major lobes directed along the cartesian x or y or z axes. Another set comprises three orbitals, each formed by a pair of clover-leaf shapes, concentrated about two of the three cartesian planes. The third set comprises just one member, with lobes directed equally to all eight corners of an inscribing cube. In the free ion, of course, all seven / orbitals are degenerate. In an octahedral crystal field, however, the... 

Returning to the Case 2 systems, one may expect that the large twist of the double bond should have strong effects on the electron distribution. A considerable fraction of the ir-electron density originally in the double bond must be delocalized into the acceptor part of the molecule. The effect of this delocalization has been studied in some twisted 1,1-diacetylethylenes (84), in which the partial negative charge makes the acceptor part Ac—C —Ac similar to an acetylace-tonate anion (89). In this anion as well as in 84, four rotamers (one degenerate... 

The two molecules that have one- or three-electron n bonds show triplet ground states. This conforms to Hund s rule in atoms where one has unpaired electrons distributed among degenerate orbitals to produce the highest possible multiplicity. The other molecules all have electron pair bonds or unshared pairs and are in singlet states. 

One could easily extend these relations to crystals in which the electron distribution is degenerate by using Fermi statistics instead of Boltzmann statistics. 

A degenerate electron gas is an electron gas that is far below its Fermi temperature, thai is. which must be described by die Fermi distribution. The essential characteristic or this state is that a very large proportion of the electrons completely fill the lower energy levels, and are unable to lake pan in any physical processes until excited out of these levels. 

The evaluation of //, is readily accomplished under certain simplifying assumptions. The first is that — F kT, which is often the case since kF is approximately 0.025 eV at room temperature and — F is commonly greater than 0.2 eV. If this condition is met the term +1 can be omitted from Eq. (2.29) if it is not met then the electron distribution is said to be degenerate and the full Fermi function must be used. The second assumption is that the excited electrons... 

Methionine adds another degree of complexity because the radical cation, which is formed by the same pathway as in the cysteine case, namely, by exclusive electron transfer from sulphur, can stabilize by the formation of a cyclic structure with a two-centre three-electron bond between sulphur and nitrogen. CIDNP has provided unequivocal evidence for this species and allowed to probe its spin distribution, through the polarization pattern.Only an unprotonated amino group can function as a donor and effect the cyclic stabilization, so there is a pronounced dependence of the polarization pattern on the pPI. ° This pH-dependence was explained by a reversible pair substitution of the open-chain and cyclic radical cations, but later reinterpreted as arising from a fast (on the CIDNP timescale) equilibrium between the different forms in conjunction with degenerate electron exchange. ... 

The occupied valence bands consist of two low-lying a bands and a double degenerate tt band just below the Fermi level. Also the lowest unoccupied band is of tt symmetry. Thus, without the DC field the four energetically lowest valence bands are double occupied and all other bands are empty. We shall use this information below in quantifying the effects of the external DC field in different approximations, i.e., we shall analyze the occupation of the different bands as a function of band index. Moreover, in order to quantify the electronic distribution, we shall use the number of electrons inside the muffin-tin spheres (with radii of 1.1 a.u.) for the 24 atoms per Born von Karman zone. 

Devise a set of degenerate orbitals for cyclobutadiene that is different from those shown above and meets the other requirements as to energy and average electron distribution. 

Notice that the position of the positive charge of 1 is not the same along the four possible directions. Furthermore, notice that at the Cl where the excited- and ground-state energies are degenerate, the positive charge (i.e. the electronic distribution) is very different in the excited and ground state as shown in Scheme 1. These electronic features will now be discussed. 

Fig. 33a-c. d-valence electron distribution at the (111) surface, a. The out-of-plane lobes of the degenerate d,y, dy, and d atomic orbitals [37]. b. The linear combinations of the plane lobes of the djy, dy, and d, atomic orbitals symmetry adapted to the (111) surface, c. Scheme of surface d-electron density of states... 

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