Empirical representations of covalent and metallic bonding

Covalent and metallic bonding have been conspicuously absent from the discussion so far. This is because the bonding is not easily rationalized in terms of simple pair functions. It is possible to represent interactions between atoms in metallic and covalent systems in terms of simple empirical functions, but these are not as easily justified as the Coulomb, Born–Mayer, and dispersive functions discussed above in the context of ionic crystals.

Metallic bonding

Traditionally much of solid state physics has been concerned with describing the distribution of electron energies in metals, and the response of this distribution to applied fields. For metals, unlike insulators, the distribution of electron energies is the primary factor in determining properties such as electrical and thermal conductivity, as well as magnetism. Clearly the distribution of electrons will also account for the cohesion of metals.

Metallic bonding arises from the delocalization of some of the electrons in the crystal. This gives rise to the simple picture of a uniform gas of electrons and a periodic array of positively charged atom cores (nuclei and bound electrons).

The Coulomb energy can be computed for the interactions between the atom cores and between the electrons and atom cores, using methods similar to those used in evaluating the Coulomb energy of ionic crystals. As for ionic solids, the result will be an energy that varies as−1/R, where R is the separation of nearest-neighbour atom cores.

The Coulomb interaction between the atom cores and electrons needs to be balanced by a force that prevents the atom cores from becoming too close. In a

uniform electron gas, the energy of each electron is related to its wave vector by E= ¯h²k²

2m (5.26)

The Pauli exclusion principle prevents two electrons from having the same quantum state. This means that only two electrons can have the same wave vector k, each with opposite spin. If there are N unit cells in the crystal, there will be N wave vectors in the reciprocal unit cell or Brillouin zone. Because the energy of each electron increases as k², the electrons occupy the set of smallest wave vectors possible, and occupy all the wave vectors within a sphere about the origin of reciprocal space until all electrons are accounted for. The surface of the sphere is called theFermi surface, and the surface will have radius kF, where the subscript F represents the Fermi surface. The energy of the electrons with wave vectors on the Fermi surface will be equal to F = ¯h²k_F²/2m. The physics of the Fermi surface unlocks an understanding of the physics of metals, and is covered in some detail in most standard textbooks on solid state physics.

In a crystal of volume V (assumed to be cube-shaped), the distribution of electron wave vectors will form a grid in reciprocal space. This is illustrated in Fig. 5.8. Each grid point has an associated volume of (2π )³/V. If the crystal contains n electrons, they will occupy a total volume of n(2π )³/2V in reciprocal space. Equating this with the volume of the Fermi sphere, (4π/3)k_F³, we have

k³_F= 3π²n

V (5.27)

The energy of the n electrons is given by the integral over the volume of the Fermi sphere. This yields (Problem 5.9) the mean energy per electron of

E

n = 3¯h²k_F² 10m = 3

5 F (5.28)

Since kF ∝ V^−1/3, and V^1/3 ∝ R, we have E/n ∝ R⁻². This energy is positive, and balances the negative Coulomb energy. Thus the binding energy of the metal is given as

U= UCoulomb+ Uelectrons= −a R + b

R² (5.29)

where a and b are constants. Clearly this function can be minimized to give an equilibrium value of the atomic separation R.

This description of metallic binding gives a rough idea of the basic principles, but is a gross oversimplication. It neglects interactions between the electrons, and it neglects any inhomogeneity of the electron gas. Atom cores are not points, and therefore the electron gas will only exist in the space between the atoms and not everywhere in the metal. However, the basic idea summarized by eqn 5.29 can accommodate a more accurate picture.

Of course, what this approach has not given is a picture of interactions between specific pairs of atoms. If two atom cores are displaced with respect to each other, there will be a local change in the electron distribution which will give a change in energy that is related to the size of the displacements. It is possible to write this change of energy in terms of interactions between atoms with some degree of justification.

Fig. 5.8 Two-dimensional representation of the distribution of allowed electron wave vectors in reciprocal space. The square shows the volume (2π )³/V associated with each grid point. The sphere represents the Fermi surface, and encompasses all grid points that are occupied by electrons in the ground state.

For clarity the number of grid points shown within the Fermi surface is many orders of magnitude smaller than in real materials.

Covalent bonding

Covalent bonds, like metallic binding, involve delocalized distributions of electrons. Unlike metals, the delocalization occurs across networks of bonds rather than throughout space. Another important difference between covalent and metallic systems lies in the energy distribution of the electrons. In a metal, there are energy states lying just above the Fermi energy, and electrons can easily be excited into theses states (and in this lies much of the physics of metals).

However, in covalent systems the energy band lying above the highest-energy filled state is at a significantly higher energy, and it is much harder for electrons to be excited into this band. This factor is the origin of many of these difference between metals and covalent systems.

The simplest examples of covalent structures are the elemental crystals of silicon and carbon (Section 2.2.4). The structures of silicon and carbon in its diamond phase give three-dimensional covalent networks. In the graphite form of carbon, this network is two-dimensional. Molecular crystals, such as I2, N2, and C60 (Sections 2.3.3 and 2.7.3), also have covalent bonds within the molecules, but without forming a network of covalent bonds. Some crystals containing different types of atoms can also be thought of in terms of networks of covalent bonds (Section 2.3.3). The primary examples are the phases of silica, SiO2, which form infinite networks of corner-linked SiO4 tetrahedra (Sections 2.3.3 and 2.7.3). However, we very quickly hit an ambiguity, because the distinction between covalent and ionic crystals is not clear-cut. In silica, the silicon and oxygen ions appear to be charged (calculations typically suggest that there is just over one electron unit of charge on the oxygen atoms, and a charge of just above+2e on the silicon) but with strong bonds between the Si and O atoms that involve electron delocalization. In fact, this is the most common situation, even in simple crystals such as MgO with the NaCl structure.

The electron distribution in a covalent crystal, whether close to that of an ionic crystal or fully shared by all atoms, is determined by the details of the atoms

and their coordination in the structure. Moreover, the electron distribution can be changed by changing pressure, or as a result of a phase transition through changing temperature. It is possible for a material to have metallic bonding in one phase and be an insulator in another.

It is often convenient to represent covalent bonds by an empirical function, such, as theMorse potential, which has the form

φ (r)= [exp(2α(r − r0))− 2 exp(α(r − r0))] (5.30) where r0is the equilibrium separation of the atoms, is the energy of the bond at r = r0, and α gives the curvature of the potential energy at r = r0 and determines the frequency of the bond-stretching vibration. In cases where the covalent bonding enforces a specific shape of the distribution of atoms, such as tetrahedral bonding of carbon and silicon atoms in their elemental crystal structures, or groups such as SiO4in silicates, the forces that determine the shape can be represented by bond-bending terms such as

φ (θ )= 1

2K(θ− θ0)² (5.31)

where θ is the bond angle, and θ0is the equilibrium angle with values such as 90^◦or 109.46^◦for octahedral and tetrahedral coordination respectively.

5.5 Quantum mechanical view of chemical