Diploma - Max Planck Institute for Solid State Research

42 4 EuRh 2 Si 2 – semi-localized electrons
–
–
WF FPLO orbital used energy contribution by the nextneighbour
contribution by
for projection window
[eV]
Rh sites the next-neighbour
Si sites
(a) 4f x(x 2 −3y 2 )−4f xz 2 [-0.2, 5.5] Rh x : 4d z 2, 4d xz , 4d x2 −y 2
∼ 4f x 3
Rh y : 4d xy , 4d xz
(b) 4f y(3x2 −y 2 )+4f yz 2 [-0.2, 5.5] Rh x : 4d xy , 4d xz
∼ 4f y 3
Rh y : 4d z 2, 4d xz , 4d x 2 −y 2
(c) 4f x(x 2 −3y 2 )+4f xz 2 [-0.2, 5.5] Rh x : 4d z 2, 4d xz , 4d x 2 −y 2 3p x , 3p y , 3p z
∼ 4f x(z 2 −y 2 )
Rh y : –
(d) 4f y(3x 2 −y 2 )−4f yz 2 [-0.2, 5.5] Rh x : –
3p x , 3p y , 3p z
∼ 4f y(z2 −x 2 )
Rh y : 4d z 2, 4d xz , 4d x2 −y 2
(e) 4f xyz [4.0, 5.0] – 3s, 3p x , 3p y , 3p z
(f) 4f z 3 [-1.7, -2.3] Rh x : 4d z 2, 4d x 2 −y 2, 4d xz –
Rh y : 4d z 2, 4d x 2 −y 2, 4d yz
(g) 4f z(x 2 −y 2 ) [-2.5, 3.5] Rh x : 4d z 2, 4d x2 −y 2, 4d xz
Rh y : 4d z 2, 4d x2 −y 2, 4d yz
–
Table 4.2: Symmetry information of the obtained WF localized at the Eu
site. The second and third column define the projection operator U k nµ for the
respective WF. The positions “off x” and “off y” are relative to the Eu site at
which the WF is localized. Contributions to a WF which were smaller than
10% of the maximum one have been neglected. The positions for Rh x and
Rh y are sketched on the right hand side.
scheme implemented in FPLO is used resulting in highly localized WFs [57, 61], whose
construction will be described shortly. The n’th Bloch band in momentum space
Ψ k n ≡ Ψ n (k) = ∑ Bsµ
c kn
sµ Φ Bsµ e ik(B+s) (4.1)
whereat Φ Bsµ denotes the local orbital basis set, gets transformed to the WF of type µ
localized at R:
W Rµ =
V
(2π)
∫BZ
3 d 3 k e ∑ −ikR Ψ k nUnµ k (4.2)
n
(V denotes the volume of unit cell in real space)
The related projection operator Unµ k has to be chosen in such a manner, that the resulting
WF is localized and has the symmetry we intended to get. As the atomic
basis orbitals are already localized, we can define a test function χ (an arbitrary linear
combination of the basis orbitals respecting the space group’s symmetry) and evaluate
the projection of the Kohn-Sham eigenfunctions onto χ. This measure defines to what
extent the Kohn-Sham function resembles χ which determines the composition of eigenfunctions
Ψ k n. Therefore Unµ k acts as a selector for the Kohn-Sham functions used to
create the WF. Furthermore, one can define an energy window to separate bonding and
anti-bonding orbital configurations. To demonstrate this procedure, one example will
be given: if we only choose a sole basis orbital to define the projector, and do not limit