Diploma - Max Planck Institute for Solid State Research

2.1 Band Structure Theory 7
respecting the crystal symmetry (B is a Bravais vector and s a vector of the basis
pointing to a Wyckoff position). Thereby the problem is confined to a primitive unit
cell with periodic boundary conditions because the Hamiltonian commutes with the
translation operators of the lattice. Furthermore the amount of unique sites is reduced
by point symmetry. Whether the Bloch Ansatz is a result of the self-consistency cycle
or used for the definition of the basis set depends on the chosen scheme (cf. 2.1.3),
nevertheless its appearance in the self-consistent eigenstates arises from the translational
symmetry of the crystal.
2.1.2 Exchange-correlation Functionals
Since there is no general scheme to obtain a universal functional E xc which achieves
highly-accurate results for all input configurations it is necessary to choose a wellbalanced
approximation. In principle, the exchange-correlation functional can be expressed
as
E xc [ρ] = 1 2
∫
∫
d 3 r 1 ρ (r 1 )
d 3 r 2
1
|r 1 − r 2 | ɛ xc [ρ, ∇ρ, . . .] (r 1 , r 2 )
where the exchange-correlation density ɛ xc depends on both spatial coordinates. Most
functionals used correspond to one of these three classes [17, p. 479–481]:
(a) Local Density Approximation (LDA) type functionals are most widely-used
since they are simple, fast and yield good results for systems whose electrons are
itinerant.
The exchange correlation density (whose expectation value is the exchange
correlation energy) depends only on the density at the same spatial position
as the density evaluated for the expectation value. Thus it is a local correction.
(b) Generalized Gradient Approximation (GGA) are based on the LDA with
higher expansion terms (∇ρ . . .), therefore in general the lattice constants and total
energy are typically better than obtained by LDA [? ]. But in general it is not possible
to decide whether LDA or GGA is more sufficient because by the construction
principle the absolute value of the remainder depends on the compound [18, 19].
(c) Hybrid Functionals are constructed by empirical fits between HF (exact exchange),
exchange as well as correlation energies of LDA and GGA [20]. Due to the
portion of exact exchange the description of band gaps in semiconductors is better
than in LDA/GGA but the computational effort exceeds that of the others.
For all calculations presented in this diploma thesis an LDA-type (spin-dependent: LSDA)
functional [21] was used although it has been proven to be not accurate for rather localized
electrons (d or f electrons). The main drawback is, that the assumption of a
slowly-varying charge density is not fulfilled anymore.
If one would try to calculate