Orbits in a crystal

FIGURE 17.3 Splitting of the c/ orbitals in a crystal field of octahedral symmetry. 

Figure 13-33 (a) The band of orbitals resulting from interaction of the 3s orbitals in a crystal of sodium, (b) Overlapping of a half-filled 3s band (black) with an empty 3p band (red) of Najy crystal. 

Crystal field splitting energy The energy separation between sets of d orbitals in a crystal field. In an octahedral complex, this is denoted by Aq and is the energy separation between the and tjg sets of orbitals. 

For Iran sition metals th c splittin g of th c d orbitals in a ligand field is most readily done using HHT. In all other sem i-ctn pirical meth -ods, the orbital energies depend on the electron occupation. HyperCh em s m oiccii lar orbital calcii latiori s give orbital cri ergy spacings that differ from simple crystal field theory prediction s. The total molecular wavcfunction is an antisymmetrized product of the occupied molecular orbitals. The virtual set of orbitals arc the residue of SCT calculations, in that they are deemed least suitable to describe the molecular wavefunction, ... 

The situation in beryllium metal is more complex. We might expect all of the 2s molecular orbitals to be filled because beryllium has the electron configuration ls22s2. However, in a crystal of beryllium, the 2p MO band overlaps the 2s (Figure 5). This means that, once again, there are vacant MOs that differ only infinitesimally in energy from filled MOs below them. This is indeed the basic requirement for electron conductivity it is characteristic of all metals, including lithium and beryllium. 

Atomic orbital populations and resulting QS in a crystal field model. 

In a crystal-field picture, the electronic structure of iron in the five-coordinate compounds is usually best represented by a (d yf idyz, 4cz) ( zO configuration [66, 70], as convincingly borne out by spin-unrestricted DFT calculations on the Jager compound 20 [68]. The intermediate spin configuration with an empty d 2 yi orbital in the CF model, however, has a vanishing valence contribution to the... 

When a metal ion is surrounded by anions in a crystal, there is an electrostatic field produced by the anions that alters the energies of the d orbitals of the metal ion. The field generated in this way is known as a crystal field. Crystal field theory was developed in 1929 by Hans Bethe in an attempt to explain the spectral characteristics of metal ions in crystals. It soon became obvious that anions surrounding a metal in a crystal gave a situation that is very similar to the ligands (many of which are... 

A magnetic semiconductor thin him is made by doping ZnO with the 3d3 7 ion Co2+. The crystal hied splitting of the d orbitals in a tetrahedral site is opposite to that in an octahedral site, with the lower pair of levels labeled e and three upper orbitals labeled t2- (a) What is the spin state of the Co2+ ion in ZnO (b) What is the expected magnetic moment on the Co2+ ions (c) The spectrum has an absorption peak at 660 nm. What is the crystal held splitting of the Co2+ ion in the tetrahedral crystal held of ZnO ... 

In a crystal lattice where each atom contributes one atomic orbital, and where these orbitals are related to each other by the translations characteristic of the lattice, the molecular orbitals must belong to irreducible representations of the group of these translations and hence form so-called Bloch orbitals. 46)... 

If many atoms are bound together, for example in a crystal, their atomic orbitals overlap and form energy bands with a high density of states. Different bands may be separated by gaps of forbidden energy for electrons. The calculation of electron levels in the periodic potential of a crystal is a many-electron problem and requires several approximations for a successful solution. 

The sums are over all n/2 free-ion atomic orbitals in the crystal. As all Sav are zero if a and v are on the same center, it is clear that p r) increases in the regions near the nucleus. The terms involving Sav for near neighbors in the last term of Equation 10 are multiplied by the free-ion product, v r) (a and v are on different centers), which... 

The different shift mechanisms may be understood in more detail by considering the effect of the magnetic field on the populations and energies of the different crystal orbitals (Figure 7a). Transfer of electron density via the 90° interaction arises due to a direct delocalization of spin density due to overlap between the half-filled tzg. oxygen jt, and empty Li 2s atomic orbitals (the delocalization mechanism. Figure 7b).This overlap is responsible for the formation of the tzg (antibonding) molecular orbital in a molecule or the tzg crystal orbital (or band) in a solid. No shift occurs for the 180° interaction from this mechanism as the eg orbitals are empty. 

Interelectron interactions depend on the size, namely the greater the ion size, the more distant the electrons from each other, the less repulsion between them. Hence B and C decrease with a decreasing of oxidation state, from the first transition series to the second and third series and from the first to the last ions within each of the series. For an ion in a crystal the overlapping of transition metal and ligand orbitals leads to a decrease of B and C, namely... 

The formation of a Si crystal is shown in Fig. 1.10. Aside from the core, each Si atom has four valence electrons two 3s electrons and two 3p electrons. To form a Si crystal, one of the 3s electrons is excited to the 3p orbital. The four valence electrons form four sp hybrid orbitals, each points to a vertex of a tetrahedron, as shown in Fig. 1.10. Thpse four sp orbitals are unpaired, that is, each orbital is occupied by one electron. Since the electron has spin, each orbital can be occupied by two electrons with opposite spins. To satisfy this, each of the directional sp orbitals is bonded with an sp orbital of a neighboring Si atom to form electron pairs, or a valence bond. Such a valence bonding of all Si atoms in a crystal form a structure shown in (b) of Fig. 1.10, the so-called diamond structure. As seen, it is a cubic crystal. Because all those tetrahedral orbitals are fully occupied, there is no free electron. Thus, similar to diamond, silicon is not a metal. 

In the free ion, the five 3c/ orbitals all have the same energy. In a crystal, these levels are split for example, if the ion occupied an octahedral hole, the 3c/levels would be split into a lower, triply degenerate level and a higher, doubly degenerate (e level. This is depicted in Figure 8.2. 

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