Computational Physics

Exercises 163

by a pseudopotential. The new, weak charge density is calledpseudo-chargedens-
ity. That this replacement is possible can be seen by realising that the effect of a
muffin tin charge distribution can be formulated in terms of the multipole moments
of the charge density, and many different charge densities give the same multi-
pole moments. Note that this justification is also analogous to the pseudopotential
method, where the fact that many different potentials yield the same phase shifts
justifies the replacement of the full potential by a pseudopotential. For details we
refer to Weinert’s paper.

6.10 Other band methods

There are numerous band structure methods [ 1 , 10 , 36 ]), and we have considered
only two illustrative examples in this chapter. Another important approach is the
Korringa–Kohn–Rostocker (KKR) method, [ 37 – 39 ] based on a scattering approach
with a muffin tin form of potential. It leads to a matrix whose size is equal to the
number of different states used in the muffin tins.
Linearising the KKR method, one obtains the linear muffin tin orbital (LMTO)
method with localised, energy-independent wave functions which are centred at
eachatom–seeRefs.[40]and[41].

Exercises

6.1 In this exercise we want to establish the relation between the energy derivative and the
charge of a core wave function,Eq. (6.59). Our derivation will not rely on a spherical
shape for the core region. We use the normalisation convention that the value of the
wave function at the boundary of the core region is equal to some fixed number, so
that we have
∂ψ(rs)
∂E
=ψ( ̇ rs)= 0
whererslies at the core boundary.
(a) Starting from the Schrödinger equation, derive an equation satisfied byψ ̇. Note
that we use the full potential, which does not depend on energy.
(b) Green’s theorem applied to the core region for two arbitrary functions,ψ 1 andψ 2 ,
reads
∫

core

d^3 r[ψ 1 (r)∇^2 ψ 2 (r)−ψ 2 (r)∇^2 ψ 1 (r)]

=

∫

shell

d^2 a[ψ 1 (a)nˆ·∇ψ 2 (a)−ψ 2 (a)nˆ·∇ψ 1 (a)]

where the integral on the right hand side is a surface integral over the boundary of

the core region andnˆis a normal vector pointing out of the core boundary. Apply