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

18 2 The FLAPW method
nonlinear problem, which is computationally much more demanding than a secular problem.
It is difficult to extend the APW method beyond the spherically averaged muffin-tin
potential approximation, because in the case of a general potential the optimal choice of
El is no longer the band energy. For a given choice of El, the radial functions ul vanish at
the muffin-tin radius, the boundary conditions on the spheres cannot be satisfied, i.e. the
planewaves and the radial functions become decoupled. It can cause numerical difficulties
if ul becomes very small at the sphere boundary.
An additional term was introduced in the basis within the muffin-tin spheres in a
method called the linearized augmented planewave method (LAPW) [76]. Extra variational
freedom is added to the basis, so that it is not necessary to set the El equal to
the band energy. This is done by including the derivative of the radial solution of the
Schrödinger equation with respect to the energy. The construction, then, can be regarded
as a linearization of the APW. In APW, the ul’s depend on the band energies and can
thus be understood as functions of r and ɛ. Hence, ul can be expanded into a Taylor-series
around El,
ul(ɛ, r) =ul(El,r)+ ˙ul(El,r)(ɛ − El)+O[(ɛ − El) 2 ]. (2.8)
Here ˙ul denotes the energy derivative of ul, ∂ul(ɛ, r)/∂ɛ, and O[(ɛ − El) 2 ] denotes errors
that are quadratic in the energy difference. Therefore, an error of order (ɛ − El) 2 in the
wavefunction is introduced in the LAPW method. According to the variational principle
the error in the calculated band energies is of the order (ɛ − El) 4 . Due to this high order,
the linearization works very well even over rather broad energy regions. A single set of
energy parameters is sufficient for the whole valence band in most cases. Sometimes the
energy region has to be split up in two (very rarely more) windows with separate sets of
energy parameters. The LAPW basis functions are of the form
⎧
⎨ e
ϕG(k, r) =
⎩
i(G+k)·r
interstitial region
�
[A μG
μG
L (k)ul(r)+BL (k)˙ul(r)]YL(ˆr) muffin-tin sphere μ
L
(2.9)
with the extra term B μ
L ˙ul(r)YL(ˆr) compared to the APW method. The additional coefficient
is determined by requiring that not only the value of the basis functions, but also
their derivatives with respect to r are continuous at the sphere boundaries. In any case is
useful to require normalization
� RMT
〈ul|ul〉 =
0
u 2 l (r)r 2 dr =1. (2.10)
Here RMT is the muffin-tin sphere radius. Taking the derivative of (2.10) with respect to
the energy it can easily be shown, that ul and ˙ul are orthogonal. ˙ul is calculated from a
Schrödinger-like equation, derived by taking the energy derivative of (2.7),