A remarkable result by Ducastelle and Cyrot — Lackmann10 relates the tight-binding local density of states to the local topology. If we describe the density of states in terms of its moments where the pth moment is defined by

Ep„(E) dE

— 1

and recall that by definition

„(-E)=E d(E — Ei)

i

where i labels the eigenvalues, we get 1

Ep E d(E — E,) dE = E Ep = Tr[Hp] where H is the Hamiltonian matrix written on the basis of the eigenvectors. But, the trace of a matrix is invariant with respect to a unitary transformation, that is, change of basis vectors to atomic orbitals i. Therefore,

mt = Tr[Hf] = $>*]* ^mP0

ii

A sum of local moments of the density of states mp^. These diagonal terms of Hp are given by the sum of all chains of length p of the form HjjHjkHki. .. H„j. These in turn can be calculated from the local topol­ogy: a prerequisite for an empirical potential. They consist of all chains of hops along bonds between atoms which start and finish at і (e. g., see Figure 5). By counting the number of such chains, we can build up the local density of states.

Unfortunately, algorithms for rebuilding DOS and deducing the energy using higher moments tend to converge rather slowly, the best being the recursion method.1

The zeroth moment simply tells us how many states there are.

The first moment tells us where the band center is. Taking the band center as the zero of energy, the second moment is as follows:

m2i) = [H2]„ = 5>h ] = E h(rj )2 [3]

./ ./

where h is a two-center hopping integral, which can therefore be written as a pairwise potential.

This result, that the second moment of the tight- binding density of states can be written as a sum of pair potentials, provides the theoretical underpinning for the Finnis-Sinclair (FS) potentials. Referring back to the rectangular band model, we can take the second moment of the local density of states m2i) as a measure of the bandwidth.

This gives the relationship between cohesive energy, bandwidth, and number of neighbors (z,). In

the simplest form W, / that is, the band energy is proportional to the square root of the number of neighbors.

Note that this is only a part of the total energy due to valence bonding. There is also an electrosta­tic interaction between the ions and an exclusion — principle repulsion due to nonorthogonality of the atomic orbitals — it turns out that both of these can be written as a pairwise potential V(r).

The moments principle was laid out in the late 1960s.12 To make a potential, the squared hopping integral is replaced by an empirical pair potential f(rp), which also accounts for the prefactor in eqn [4] and the exact relation between bandwidth and second moment. Once the pairwise potential V(ry) is added, these potentials have come to be known as FS potentials.13

Ecoh = X V (П, )-X. Xfj) N

ijij where Vand f are fitting functions.

Further work14 showed that the square root law held for bands of any shape provided that there was no charge transfer between local DOS and that the Fermi energy in the system was fixed. For bcc, atoms in the second neighbor shell are fairly close, and are normally assumed to have a nonzero hopping integral.

Notice that the first three moments only contain information about the distances to the shells of atoms within the range of the hopping integral. Therefore, a third-moment model with near neighbor hopping could not differentiate between hcp and fcc (in fact, only the fifth moment differentiates these in a near­neighbor hopping model!). This led Pettifor to con­sider a bond energy rather than a band energy, and relate it to Coulson’s definition of chemical bond orders in molecules.15 Generalizing this concept leads to a systematic way of going beyond second moments and generating bond order potentials.

One can investigate the second-moment hypothesis by looking at the density of states of a typical transition metal, niobium, calculated by ab initio pseudopotential plane wave method, Figure 6, and comparing it with the density of states at extremely high pressure. The similarity is striking: as the material is compressed, the band broadens but the structure with five peaks remains unchanged. The s-band is displaced slightly to higher energies at high pressure, but still provides a

low, flat background, which extends from slightly

below the d-band to several electron volts above.

1.10.6.2 Key Points

• In a second-moment approximation, the cohesive (bond) energy is proportional to the square root of the coordination.

• Other contributions to the energy can be written as pairwise potentials.