Set of supplementary notes.

70 CHAPTER 4. ELECTRONIC STRUCTURE
It is now fairly clear how to extend this method to multiple orbitals per unit cell. We simply
generalise the summation in (4.30):
|ψ〉 = ∑ e ikRn c (ν) ∣ 〉 k n
(ν)
, (4.32)
n,ν
where the index ν labels the different local orbitals ∣ ∣ n
(ν) 〉 , which exist in the n’th unit cell, and
is the associated coefficient, which determines to what level this local orbital is mixed into
the final state. This choice of Bloch state will give rise to an eigenvector/eigenvalue problem, in
which the number of energy eigenvalues at a particular wavevector k, and thereby the number
of bands, is equal to the number of local orbitals chosen per unit cell.
c (ν)
k
4.4.4 Tight binding vs. Nearly Free Electron approximation
Fundamentally, both the tight binding and the nearly free electron method do the same thing:
a Bloch state is constructed from a set of basis functions, and the associated coefficients are
determined by solving the eigenvector/eigenvalue equation which arises, when the Hamiltonian
is expressed in that basis. If the basis is complete, then the Bloch state is an exact eigenstate
of the Hamiltonian. However, a complete basis would give an infinite-dimensional eigenvector
problem, so the expansion of the Bloch state is done within a reduced basis set.
In the case of the nearly free electron approximation, the complete basis set required to form
|ψ k 〉 is the set of all plane wave states |k − G〉 (eigenstates to the kinetic energy operator), but
we can restrict this by concentrating on those states for which the matrix elements V G are large
and which are nearly degenerate with |k〉. If the periodic potential is comparatively smoothly
varying and weak (compared to the kinetic energy), then we can disregard Fourier components
V G with high wavevector G, and the set of basis functions is small. In this case, the nearly free
electron approximation is computationally efficient.
If, on the other hand, the periodic potential varies strongly compared to the kinetic energy,
then the plane wave expansion has to include a much larger set of states. In this case, the
free electron approximation is less useful, and the tight binding approximation may be more
appropriate. In the tight binding approximation (or linear combination of atomic orbitals), the
eigenstates of the local Hamiltonians Ĥ(0) n , i.e. the set of all atomic orbitals associated with
the different unit cells n, form a complete basis in which the Bloch states can be expanded.
Section 4.4.3 showed that the number of bands generated in this method is equal to the number
of atomic orbitals included per unit cell. If the hopping matrix elements are small compared
to the separation between the bands, then the bands generated from different sets of atomic
orbitals do not cross. In this case, the strength of the potential – which fixes the energies of
the atomic orbitals and also their spacing – is larger than the kinetic energy – which is related
to the bandwidth. To get a good description of a particular electronic band only a small set of
atomic orbitals needs to be included.
In short, in the nearly free electron scheme the kinetic energy appears on the diagonal of
energy matrix in the eigenvector equation, and the potential appears in the off-diagonal terms,
mixing basis states together. This is efficient, if the periodic potential is a weak perturbation.
In the tight binding scheme, conversely, the potential energy appears on the diagonal and the
hopping elements, which are the equivalent of the kinetic energy, form the off-diagonal terms.
This works best, if the potential is strong compared to the kinetic energy.