6.5 Augmented plane wave methods 139kzkykxXWK
LΓFigure 6.9. Brillouin zone of the fcc lattice.For a given vectorkin the Brillouin zone, we construct the APW basis vectors
asq=k+K. The norm of a reciprocal lattice vectorK=lb 1 +mb 2 +nb 3 is
given by
|K|=2 π
a√
3 l^2 + 3 m^2 + 3 n^2 − 2 lm− 2 nl− 2 nm. (6.37)We take a set of reciprocal lattice vectors with norm smaller than some cut-off
and it turns out that the sizes of these sets are 1, 9, 15, 27 etc. A good basis set
size to start with is 27, but you might eventually do calculations using 113 basis
vectors, for example. The set of such reciprocal lattice vectors is easy to generate
by considering all vectors withl,mandnbetween say−6 and 6 and neglecting all
those with norm beyond some cut-off.
The program must contain loops over sets ofk-points in the Brillouin zone
between for exampleandXinFigure 6.9. The locations of the various points
indicated inFigure 6.9, expressed in cartesian coordinates, are
=
2 π
a
0
0
0
, X=^2 π
a
1
0
0
, K=^2 π
a
3 / 4
3 / 4
0
,
W=
2 π
a
1
1 / 2
0
, L=^2 π
a
1 / 2
1 / 2
1 / 2
.
(6.38)
For eachk, the matrix elementsAij,BijandCijlin(6.35)are to be determined. Good
values for the cut-off angular momentum arelmax=3 or 4. Then, for any energy
E, the matrix elements ofHaccording to(6.34)can be found by first solving the
radial Schrödinger equation numerically fromr=0tor=Rand then using the