Computational Physics

6.6 The linearised APW (LAPW) method 141
programming exercise
Write a program for calculating the determinant|H−E|.
CheckCheck that the determinant vanishes near the values which you can read
off from Figure 6.10 for a few points in the Brillouin zone.
The Fermi level for the potential supplied lies approximately at 0.29 a.u., so
one conclusion you can draw from the resulting band structure is that copper is a
conductor as the Fermi energy does not lie in the energy gap.
You will by now have appreciated why people have tried to avoid energy-
dependent Hamiltonians. In the next section we shall describe the linearised APW
(LAPW) method which is based on the APW method, but avoids the problems
associated with the latter.

6.6 The linearised APW (LAPW) method

A naive way of avoiding the energy-dependence problem in APW calculations
would be to use a fixed ‘pivot’ energy for which the basis functions are calculated
and to use these for a range of energies around the pivot energy. If the form of the
basis functions inside the muffin tin varies rapidly with energy (and this turns out
to be often the case) this will lead to unsatisfactory results.
The idea of the LAPW method [ 17 , 18 ] is to use a set of pivot energies for
which not only the solution to the radial Schrödinger equation is taken into account
in constructing the basis set, but also its energy derivative. This means that the
new basis set should be adequate for a range of energies around the pivot energy
in which the radial basis functions can be reasonably approximated by an energy
linearisation:
R(r,E)=R(r,Ep)+(E−Ep)R ̇(r,Ep). (6.39)

Here, and in the remainder of this section, the dot stands for the energy derivative, as
opposed to the prime, which is used for the radial derivative – for any differentiable
functionf(r,E):

̇f(r,E)= ∂

∂E

f(r,E) and (6.40a)

f′(r,E)=

∂

∂r

f(r,E). (6.40b)

The energy derivatives of the radial solution within the muffin tins are used alongside
the radial solutions themselves to match onto the plane wave outside the spheres.
Note that the APW Hamiltonian depends on energy only via the radial solutions
Rl, so if we take these solutions and their energy derivativesR ̇lat afixedenergy
into account, we have eliminated all energy dependence from the Hamiltonian.