Set of supplementary notes.

4.5. PSEUDOPOTENTIALS 71
4.5 Pseudopotentials
The NFE method and the tight-binding method are not accurate methods of electronic structure determination;
nevertheless both of them exhibit the basic principles. They are commonly used to write down
simple models for bands, with their parameters fit to more sophisticated calculations, or to experiment.
It turns out that band gaps in semiconductors are usually fairly small, and the true dispersion can be
modelled by scattering from a few Fourier components of the lattice potential. The reason is that the
relevant scattering potential for valence band electrons is however MUCH smaller than the full atomic
potential ze 2 /r of an electron interacting with a nucleus of charge z. The effective potential for scattering
of the valence electrons by the atomic cores is a weak pseudopotential.
When we consider the band structure of a typical solid, we are concerned only with the valence
electrons, and not with those tightly bound in the core, which remain nearly atomic. If we solve the
full Schrödinger equation with the real Coulomb potential, we expect to calculate not just the valence
electronic states, but also the atomic like core states. A pseudopotential reproduces the valence states as
the lowest eigenstates of the problem and neglects the core states.
Figure 4.7: Pseudopotential: The true potential V (r) has a wavefunction for the valence electrons
that oscillates rapidly near the core. The pseudopotential V s (r) has a wavefunction Φ s (r)
that is smooth near the core, but approximates the true wavefunction far from the core region.
A weak pseudopotential acting on a smooth pseudo-wavefunction gives nearly the same energy eigenvalues
for the valence electrons as the full atomic potential does acting on real wavefunctions. Away
from the atomic cores, the pseudopotential matches the true potential, and the pseudo-wavefunction
approximates the true one.
A formal derivation of how this works can be given using the method of orthogonalised plane waves.
The atomic states are well described by the Bloch functions f nk of the LCAO or tight-binding scheme
(4.32). Higher states, which extend well beyond the atoms will not necessarily be of this kind, but they
must be orthogonal to the core levels. This suggests that we should use as a basis 3
|χ k >= |k > − ∑ n
β n |f nk > , (4.33)
where |k > is a plane wave, and the coefficients β n (k) are chosen to make the states χ orthogonal to the
core states |f nk >. The states in (4.33) are orthogonalised plane waves (OPW); away from the core, they
3 We use Dirac’s notation of bra and ket, where |k > represents the plane wave state exp(ik · r), and <
φ 1 |T |φ 2 > represents the matrix element ∫ dr φ ∗ 1(r)T (r)φ 2 (r) of the operator T .