Computational Physics

6.3 Approximations 129

neighbours. The numerical solution of the generalised eigenvalue problemHC=
ESCis treated inChapter 3.
Of course, we may relax the condition that we consider atomic orbitals as basis
functions, and extend the method to allow for arbitrary, but still localised, basis
functions. This even works for the valence orbitals in metals, although in that case
relatively many neighbours have to be coupled, so that the approach pays off only
for large unit cells. We may use the tight-binding approach for fixed interatomic
distances – for a tight-binding method in which the atoms are allowed to move,
thereby requiring varying distances, seeChapter 9.
The tight-binding method comes in two flavours. The first is thesemi-empirical
TB method, in which only a few valence orbital basis functions are used. Their
couplings are restricted to nearest neighbour atoms and the value of the couplings
are fitted to either experimental data such as band gaps and band widths, or to similar
data obtained using more sophisticated band structure calculations. Once satisfact-
ory values have been obtained for the TB couplings, more complicated structures
may be considered which are beyond reach of self-consistent DFT calculations.
We may also be more ambitious and use more TB parameters which are fitted
to DFT Hamiltonians. This is particulary useful when the TB Hamiltonians were
obtained using localised basis functions, such as Gaussian or Slater orbitals (see
Chapter 4for a dicussion of these basis sets). For DFT calculations, Slater type
orbitals are becoming increasingly popular, as the reason for choosing Gaussians
in Hartree–Fock calculations, i.e. the fact that integrals can be evaluated analytically,
ceases to be relevant in DFT with its highly nonlinear exchange correlation potential.
Therefore, the Hamiltonian naturally has a tight-binding form, and this means that
it issparse, that is, a small minority of the elements of the Hamiltonian are nonzero.
Such a Hamiltonian allows for iterative methods to be used.
In the next subsection we consider an appealing application of the tight-binding
method: graphene and carbon nanotubes.

6.3.3 Tight-binding calculation for graphene and carbon nanotubes

In this subsection, we calculate the band structure of a carbon nanotube within
the tight-binding approximation. It is a very instructive exercise which is strongly
recommended as an introduction to band structure calculations. Here we follow the
discussioninRefs.[ 3 , 8 ].
We assume that only elements of the Hamiltonian and overlap matrix coupling
two atomic orbitals are relevant, and that three-point terms do not occur. This is the
so-calledSlater–Kosterapproximation[9]. We shall first apply this to graphene,
which is a sheet consisting of carbon atoms ordered within a hexagonal lattice. This
is not a Bravais lattice, but it can be described as a triangular Bravais lattice with a