Bond Orbital Mode Approximation

The special points method depends upon retention of the translational periodicity of a lattice, which is lost if we consider defects, surfaces, or lattice vibrations. (Even the special vibrational mode with frequency listed in Table 8-1 entailed a halving of the translational symmetry.) It is therefore extremely desirable to seek an approximate description in terms of bond orbitals, so that the energy can be summed bond by bond as discussed in Chapter 3. We proceed to that now. 

The important accomplishment in this section has been the reduction of the problem to the point where the energy can be computed bond by bond, either approximately, by the Bond Orbital Approximation, or more accurately, by in-eluding the bonding antibonding matrix elements in perturbation theory. It is not so important for the understanding of the uniform shears, but will be important in other properties. Before we can examine these, however, we must consider another mode of shear in the lattice. 

It is possible in principle to calculate all of these modes from the theory of the electronic structure, which is equivalent to the calculation of all the force constants. Indeed we will see that this is possible in practice for the simple metals by using pseudopotential theory. In covalent solids, even within the Bond Orbital Approximation, this proves extremely difficult because of the need to rotate and to optimize the hybrids, and it has not been attempted. The other alternative is to make a model of the interactions, which reduces the number of parameters. The most direct approach of this kind is to reduce the force constants to as few as possible by symmetry, and then to include only interactions with as many sets of neighbors as one has data to fit- for example, interactions with nearest and next-nearest neighbors. This is the Born-von Karman expansion, and it has somewhat surprisingly proved to be very poorly convergent. This simply means that in all systems there arc rather long-ranged forces. 

Figure 2. Excited-state spectral features ofD -CuCl/-. A Energy level diagram showing the ligand-field (d - d) and charge-transfer (CT) optical transitions. The intensity of the transitions is approximated by the thickness of the arrow with the very weak ligand-field transitions represented as a dotted arrow. B Electronic absorption spectrum for D4h-CuCl42 (12). C Schematic of the a and tt bonding modes between the Cu 3dx2 y2 and Cl 3p orbitals.

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