Electronic Structure of Simple Tetrahedral Solids

Bond orbitals are constructed ft om s/r hybrids for the simple covalent tetrahedral structure energies are written in terms of a eovalent energy V2 and a polar energy K3. There are matrix elements between bond orbitals that broaden the electron levels into bands. In a preliminary study of the bands for perfect crystals, the energies for all bands at k = 0 arc written in terms of matrix elements from the Solid State Tabic. For calculation of other properties, a Bond Orbital Approximation eliminates the need to find the bands themselves and permits the description of bonds in imperfect and noncrystalline solids. Errors in the Bond Orbital Approximation can be corrected by using perturbation theory to construct extended bond orbitals. Two major trends in covalent bonds over the periodic table, polarity and metallicity, arc both defined in terms of parameters from the Solid State Table. This representation of the electronic structure extends to covalent planar and filamentary structures. 

In the zincblcnde structure there are an equal number of sulphur atoms, each in a position [Ill]a/4 relative to one of the zinc atoms (these displacements are represented by arrows in Fig. 3-1,a- the four shaded atoms represent sulphur atoms) other sulphur atoms are likewise displaced [lll]a/4 from the other zinc atoms in the figure, and lie outside of the cube, so are not shown. The zinc atom at the front lower-left corner of the cube and those at the center of the bottom face, left face, and front face form the corners of a regular tetrahedron that has a sulphur atom at its center. Every other sulphur atom in zincblcnde is also tetra-hedrally surrounded by zinc atoms in exactly the same way. Similarly, every 

A unit cube of the zinchlendc structure (a) rotated and (b) viewed in the [010] direction. 

The cube in Fig. 3-1,a is shown in perspective, or slightly rotated, to illustrate atomic positions. In Fig. 3-1,b the cube is rotated back to provide a more convenient diagram for purposes of calculations, since in terms of it the position of each atom can very easily be written down. 

A zincblcnde lattice view along a [111] direction. The facc-ccntcred cubic lattice looks exactly the same viewed in this way. 

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Although our discussion in the next seven chapters will center on simple tetrahedral structures, in which all electrons form simple two-electron bonds, it is desirable to introduce two other types of structures in these, some of the electrons form two-electron bonds (and are understandable in the same terms used for the tetrahedral solids) and other electrons are accommodated in pure p slates, similar to the 71 states discussed in Chapter 1 for diatomic molecules. For a discussion of the stability of these structures, see Friedel (1978). The two-electron bonds are simpler geometrically in these systems than in the tetrahedral solids and will provide very good problems. (Many of the problems at the ends of the following chapters will explore the theory of the bonds for these simpler systems.)... 

In all of these compounds, even the tetrahedral ones, a possible starting point for the calculation of properties is an ionic electronic structure with the effects of interatomic matrix elements treated in perturbation theory. As wc liave indiettted, and as will be seen in detail in the next section, it is even possible to treat tlic polar covalent nontransition-metal solids in this way. Thus we should be able to calculate properties of the transition-metal compounds just as we did for the simple ionic compounds. 

Of all solid state reactions, the formation of oxide spinels is at present the most thoroughly investigated [4, 5, 33]. The first reason for this is the relatively simple crystallographic structure of the spinel lattice. Essentially, this consists of a nearly close-packed face-centred-cubic sublattice of oxygen ions. The tetrahedral and octrahedral interstices of this sublattice are filled in a certain way by the cations. The second reason is that spinels are technically very interesting substances, and one would like to be able to find optimal methods for their preparation. For instance, ferrites are used as control or circuit elements in the electronics industry, and chromite brick is used as cladding in ovens which are used for the production of steel. Therefore, the formation of spinels will now be discussed in detail as a model of a classical solid state reaction. 

Let us look first for transition-metal compounds that arc truly covalent in the sense of tetrahedral structures and two-electron bonds, which we di.scu.sscd earlier. There are only a few examples. NbN and TaN both form in the wurtzite structure. We presume that bond orbitals of sp hybrids must be present to stabilize the structure this requires three electrons from each transition-metal ion. Both ions are found in column D5 of the Solid State Table, so we anticipate that the remaining two electrons would form a multiplet (as in the ground stale of Ti " ). Thus the effects of the d state are simply added onto an otherwise simple covalent system, just as they were added to a simple ionic system in the monoxides. MnS, MnSe, and MnTe also form a wurtzite structure and presumably may be understood in just the same way. This class of compounds is apparently too small to have been studied extensively. 

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