For a pair-potential, removing an atom from the lattice involves breaking bonds. The cohesive energy of a lattice comes from adding the energies of those bonds. Hence, the cohesive energy is equal to the vacancy formation energy, aside from a small differ­ence from relaxation of the atoms around the vacancy.

In real metals, the vacancy formation energy is typi­cally one-third of the cohesive energy, the discrepancy coming yet again from the strengthening of bonds to undercoordinated atoms.

1.10.4.2.3 Cauchy pressure

Pairwise potentials constrain possible values of the elastic constants. Most notably, it is the ‘Cauchy’ relation which relates C12-C66. In a pairwise poten­tial, these are given by the second derivative of the energy with respect to strain, which are most easily treated by regarding the potential as a function of r2 rather than r; whence for a pair potential V(r2), it follows:

C12 = C66 = |X V"(r2j ij

where i, j run over all atoms and O is the volume of the system.

In metals, this relation is strongly violated (e. g., in gold, C12 = 157GPa, C44 = C66 = 42GPa).

1.10.4.2.4 High-pressure phases

Many materials change their coordination on pres­surization (e. g., iron from bcc (8) to hcp (12)) and some on heating (e. g., tin, from fourfold to sixfold). This suggests that the energy is relatively insensitive to coordination — for pair potentials, it is propor­tional. These problems suggest that a potential has to address the fact that electrons in solids are not uniquely associated with one particular atom, whether the bonding be covalent or metallic. Ultimately, bond­ing comes from lowering the energy of the electrons, and the number of electrons per atom does not change even if the coordination does.

1.10.4.2.5 Short ranged

It is worth noting that some properties that are claimed to be deficiencies of pair potentials are actu­ally associated with short range. So, for example, the diamond structure cannot be stabilized by near­neighbor potentials, but a longer ranged interaction can stabilize this, and the other complex crystal struc­tures observed in sp-bonded elements.4

1.10.3 From Quantum Theory to Potentials

To understand how best to write the functional form for an interatomic potential, we need to go back to quantum mechanics, extract the dominant features,
and simplify. Quantum mechanics can be expressed in any basis set, so there are several possible starting points for such a theory. Thus, a picture based on atomic orbitals (i. e., tight binding) or plane waves (i. e., free electrons) can be equally valid: for potential development, the important aspect is whether these methods allow for intuitive simplification.

When a potential form is deduced from quantum theory, approximations are made along the way. An aspect often overlooked is that the effects of terms neglected by those approximations are not absent in the final fitted potential. Rather they are incorporated in an averaged (and usually wrong) way, as a distor­tion of the remaining terms. Thus, it is not sensible to add the missing physics back in without reparameter­izing the whole potential.