Ab initio investigations of magnetic properties of ultrathin transition ...

22 2 The FLAPW method
Then one can show also that the FLAPW basis functions (eq. 2.9) can be transformed,
for systems that possess inversion symmetry, into the following form
ϕG(k, −r) =ϕ ∗ G(k, r) = �
lm ′
e −iK(pμ ) (−i) l Ylm ′(Rμ ˆK) Y ∗
lm ′(ˆr){Aul(r)+B ˙ul(r)} (2.22)
The Hamiltonian and overlap matrix will have the same property as the FLAPW basis
functions in systems that possess inversion symmetry. These two matrices are real symmetric
rather than complex hermitian. The Hamiltonian depends explicitly on r via the
potential. The matrix elements are given by:
H G′ �
G
(k) = ϕ ∗ G ′(k, r)H(r)ϕG(k, rd 3 r (2.23)
Substituting r ′ = −r yields:
H G′ �
G
(k) = ϕG ′(k, r′ )H(r ′ )ϕ ∗ G(k, r ′ )d 3 r (2.24)
where (2.22) and H(r) =H(−r) have been used. In addition the Hamiltonian operator
is real, i.e. H(r) =H ∗ (r). Thus, we finally obtain:
H G′ G (k) =
=
�
ϕG ′(k, r′ )H ∗ (r ′ )ϕ ∗ G(k, r ′ d 3 r
�
H G′ �∗ G
(k) . (2.25)
The same relation holds for the overlap matrix. Because the two matrices are real, this
means a great simplification in actual calculation. Then, the diagonalization of a hermitian
matrix is no more difficult than in the real case. However, one complex multiplication
contains four real multiplications, and therefore the complex problem is more ”expensive”
than the real, and the diagonalization needs the biggest part of the computing time in each
iteration.
2.3.1 Film Calculations within FLAPW
Within the growing number of investigations in the area of thin films and surfaces, the
ability to treat two dimensional systems becomes very important nowadays. In this case,
either big spins allowed to be used or the basis set need to be reformed due to the broken
translation symmetry in the perpendicular direction of the surface. Only the 2-dimensional
symmetry parallel to the surface is left to be used to reduce the problem. For 10–15
atomic layers thickness, it is otherwise semi infinite to approximate surfaces by thin films.
This approximation, called the thin-slab approximation, can only yield good results if the
interaction between the two surfaces of the film is weak enough, so that each of them shows
the properties of the surfaces of an ideal semi-infinite crystal.