Computational Physics

138 Solving the Schrödinger equation in periodic solids

given by

Hij=〈k+Ki|H|k+Kj〉=−EAij+Bij+

l∑max

l= 0

Cijl

R′l(R)

Rl(R)

. (6.34)

In this expression,R′l(R)isdRl(r)/dr, evaluated at the muffin tin boundaryr=R.
The coefficientsAij,BijandCijlare given by

Aij=

− 4 πR^2

j 1 (|Ki−Kj|R)

|Ki−Kj|

+δij,

Bij=

1

2

Aij(qi·qj)and

Cijl=( 2 l+ 1 )

2 πR^2

Pl

(

qi·qj

qiqj

)

jl(qiR)jl(qjR). (6.35)

Here, is the volume of the unit cell. Note that there is no divergence in the
expression for the matrix elementsAijif|Ki−Kj|vanishes, asj 1 (x)→x/3 for
smallx.
In addition to the inconvenient energy dependence of the Hamiltonian, another
problem arises from the occurrence ofRl(R)in the denominator in (6.34). For ener-
gies for which the radial solutionRlhappens to be zero or nearly zero on the border
of the muffin tin spheres, the matrix elements become very large or even diverge,
which may cause numerical instabilities. In the linearised APW (LAPW) method, an
energy-independent Hamiltonian is used in which the radial solution does not occur
in a denominator, and therefore both problems of the APW method are avoided.
The APW method is hardly used now because of the energy-dependence problem.
The reasons we have treated it here are that it is conceptually simple and that the
principle of this method lies at the basis of many other methods. Further details on
theAPWmethodcanbefoundinRefs.[ 14 – 16 ].

6.5.2 An APW program for the band structure of copper

Copper has atomic numberZ=29. We consider only the valence states. The core
states are two s- and two p-states, and theeleven valence electrons occupy the third
s-level and the first d-levels. Its crystal structure is fcc(Figure 6.1)with lattice
constanta=6.822 a.u. The unit cell volume is equal to 3a^3 /4. The reciprocal
lattice is a bcc lattice with basis vectors

b 1 =

2 π

a





− 1

1

1



, b 2 =^2 π

a





1

− 1

1



, b 3 =^2 π

a





1

1

− 1



. (6.36)

The Brillouin zone of this lattice is shown inFigure 6.9.