Diploma - Max Planck Institute for Solid State Research

4.2 Hybridization: localized versus itinerant states 43
the energy window, we will get a WF which yields exactly this particular atomic orbital,
because for each k-point at least one Kohn-Sham function has a major contribution by
this orbital.
To create a (minimal) WF basis set, one should define molecular orbitals, in this
case mainly hybrids of Eu 4f and Rh 4d as it will be shown later, and check if the WF
Hamiltonian reproduces the band structure inside the chosen energy window. Because
this is a tough task, and we are only interested in the information of hybridizing orbitals,
a different approach has been applied. Defining a projector for each Eu 4f orbital and
restricting the energy to this particular band of the band structure should define a WF
which consists of the 4f orbital and its symmetry related counterpart at other sites (due
to “hybridization”). This rough procedure is solely suitable, because the dispersion of the
4f dominated bands is very small and all 4f levels can be separated in energy. Internally,
FPLO uses a real representation for the complex spherical harmonics of 4f orbitals (the
“general set”). Since the 4f levels are not decoupled (cf fig. 4.14), we use a different
superposition (similar to the “cubic set”) to create a representation so that all orbitals
are well separated by energy and / or symmetry.
In fig. 4.15 the 4f orbitals with respect to the applied projector are depicted, the
corresponding parameters are given in tab. 4.2. In that the energy windows are chosen
arbitrarily, the amount of hybridization cannot be compared between the WFs. Nevertheless,
comparing the WFs in fig. 4.15 the Eu 4f orbitals seem to hybridize primarily
with Rh 4d states despite of the ones in (c) and (e) although silicon is their nearestneighbour.
If the occupation of the 4f orbitals in LSDA+U [AL] reflects the ground
state occupancy, one could restrict the analysis to partially- and entirely-filled orbitals.
But since this scheme does not include spin-orbit coupling and respective excited final
states have mixed occupations of all 4f orbitals, this is not suitable. Regarding tab. 4.2,
one can at least conclude that the Rh 4d z 2 orbital possibly match some contribution to
a hybrid orbital because of its orientation towards the Eu layers and its occurance in
all WFs.
4.2.2 Estimation of the hybridization strength
Depending on the symmetry analysis, one has two possible options to get a first approximation
for the interaction strength. If one cannot relate the 4f orbitals with special
linear combinations of Rhodium 4d orbitals by e.g. a basis transformation or group theory,
then solely a quantative estimation within muffin-tin methods remains. Otherwise
the basis coefficients of the Rh 4d orbitals can be used directly as an estimate for the
coupling strength. The first option will be discussed below.
Remembering that in LMTO-ASA no interstitial regions exist, the redistribution of
charge density during the convergence process compared to the initial guess of a superposition
of atomic solutions causes non-vanishing contributions of initially unoccupied
orbitals in the considered sphere as well as in the surrounding ones. Since in our cal-