Diploma - Max Planck Institute for Solid State Research

2.1 Band Structure Theory 5
Figure 2.1: DFT scheme – the GS density can be calculated iteratively with eq. 2.9 and 2.10
because the Hamiltonian, especially the effective potential, is solely determined by the density
of the previous iteration step (HK1)
The approach of density functional theory to solve eq 2.2 will be discussed in detail
below using explicitly the symmetry of crystals. Furthermore, there are also schemes
for disordered materials[13].
2.1.1 Density Functional Theory
Hohenberg and Kohn [14] demonstrated that the external potential Ṽext ≡ 〈 ψ | V ne | ψ 〉
is solely defined by the ground state (GS) density ρ (HK1) as well as that Ṽext is determined
by an universial functional F HK [ρ] which does not depend on the external
potential V ext (HK2). Considering the expectation value of H e with a product wavefunction
ansatz for non-interacting electrons eq. 2.2 transforms into
E e [ρ] =
(
)
T e + V ee + Ṽext [ρ] (2.4)
Thereby one loses the exact solution of the electronic manybody problem due to the
approximation of a product wavefunction.
The variation of 2.4 under the constraint of charge conservation ∫ ∞
−∞ ρ d3 r = N e yields
µ = δE e [ρ]
δρ (r) = ṽ ext (r) + δF HK [ρ]
δρ (r)
with
F HK [ρ] = T e [ρ] + V ee [ρ]
(2.5)
which is regarded as the basic equation in DFT. Since the exact solution of H e is
approximated in eq. 2.5, one tried to incorporate the manybody phenomena into the
Hamiltonian H e . In particular, Kohn and Sham have shown that there always exists
a system of non-interacting electrons which has the same density as the system of
interacting electrons [15]. They constructed F HK as a sum of a non-interacting electron