Theory local-band

Generally, all band theoretical calculations of momentum densities are based on the local-density approximation (LDA) [1] of density functional theory (DFT) [2], The LDA-based band theory can explain qualitatively the characteristics of overall shape and fine structures of the observed Compton profiles (CPs). However, the LDA calculation yields CPs which are higher than the experimental CPs at small momenta and lower at large momenta. Furthermore, the LDA computation always produces more pronounced fine structures which originate in the Fermi surface geometry and higher momentum components than those found in the experiments [3-5]. 

Chemical bonds are defined by their frontier orbitals. That is, by the highest molecular orbital that is occupied by electrons (HOMO), and the lowest unoccupied molecular orbital (LUMO). These are analogous with the top of the valence band and the bottom of the conduction band in electron band theory. However, since kinks are localized and non-periodic, band theory is not appropriate for this discussion. 

If the localized electron tunnels out through the barrier (state 1 in Fig. 12 b) a certain amount of f-f overlapping is present. States like 1 in Fig. 12 b are called sometimes resonant states or "virtually bound" states. In contrast with case 2 in Fig. 12b, which we may call of full localization , the wave function of a resonant state does not die out rapidly, but keeps a finite amplitude in the crystal, even far away from the core. For this reason, overlapping may take place with adjacent atoms and a band may be built as in ii. (If the band formed is a very narrow band, sometimes the names of localized state or of resonance band are employed, too. Attention is drawn, however, that in this case one refers to a many-electron, many-atoms wave function of itinerant character in the sense of band theory whereas in the case of resonant states one refers to a one-electron state, bound to the central potential of the core (see Chap. F)). 

Further along the series, we saw that stoichiometric MnO, FeO, CoO, and NiO are insulators. This situation is not easily described by band theory because the d orbitals are now too contracted to overlap much, typical band widths are 1 eV, and the overlap is not sufficient to overcome the localizing influence of interelectronic repulsions. (It is this localization of the d electrons on the atoms that gives rise to the magnetic properties of these compounds that are discussed in Chapter 9.)... 

A simple alternative model, consistent with band theory, is the electron sea concept illustrated in Fig. 9-22 for sodium. The circles represent the sodium ions which occupy regular lattice positions (the second and fourth lines of atoms are in a plane below the first and third). The eleventh electron from each atom is broadly delocalized so that the space between sodium ions is filled with an electron sea of sufficient density to keep the crystal electrically neutral. The massive ions vibrate about the nominal positions in the electron sea, which holds them in place something like cherries in a bowl of gelatin. This model successfully accounts for the unusual properties of metals, such as the electrical conductivity and mechanical toughness. In many metals, particularly the transition elements, the picture is more complicated, with some electrons participating in local bonding in addition to the delocalized electrons. 

Most minerals fall into the class of insulator phosphors. The characteristics of the luminescence are usually defined by the electronic structure of an activator ion as modified by the crystal field of the host crystal structure. Although some energy transfer takes place between nearby ions, appearing as the phenomena of co-activation, luminescence poisons, and activator pair interactions, the overall luminescence process is localized in a "luminescent center" which is typically 2 to 3 nm in radius. From a perspective of band theory, luminescent centers behave as localized states within the forbidden energy gap. 

Using this local moment model, and using band theory or its variations, a number of workers have been able to formulate expressions which represent the measured magnetic data reasonably well, at least for the case where well-localized moments are developed on the solute atoms (II, 18). However, considerably more data has become available on other properties of dilute alloys, including data on resistivity and specific heat, neutron scattering, various magnetic resonance experiments, Moss-bauer measurements, Kondo effect, and the like. Measurements have been extended also to alloys of many other systems besides those involving the platinum metals. 

We note that the notions optical and electrical gap are here used in the context of the classical band theory of solids and can be confusing in application to molecular (van der Waals bonded) solids, where they have the opposite meaning the optical gap reflects the energy of excitonic (localized) states, while the electrical gap stands for the lowest energy between free carrier states. 

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