Kohn-Sham Theory reduces the many body problem of the N-particle wave function to N single particle problems. Nevertheless, for every bulk material there does still remain an innite number of non-interacting electrons moving in a potential caused by an innite number of nuclei (always bearing in mind that an innite crystal is always an approximation of a nite real crystal).
For crystalline materials the spatial periodicity of the electrostatic potential of the nuclei imposes a periodic boundary condition on the wave functions which are solu- tions of the Schrödinger equation. This concept was rst expressed by Bloch, known as Bloch's Theorem [54].
2.5.1 Bloch's Theorem
In an ideal crystal with the translation vector R the electrostatic potential V(r)
obeys translational invariance, i.e. it is a periodic function with the periodicity of the crystal lattice:
V(r) =V(r+R). (2.33)
2.5 The Plane Wave Method
This is also valid for the eective potential Vef f(r) of the Kohn-Sham equations:
Vef f(r) = Vef f(r+R). (2.34)
Every eigenfunctionψk(r), that is a solution to the single particle Schrödinger equa-
tion as given by the Kohn-Sham equations −1 25 2+V ef f(r) ψk(r) = Ekψk(r) (2.35)
can be expressed as a product of a plane wave exp[ikr] (k denoting a reciprocal
space vector) and a functionuk(r)with the same periodicity as the real space lattice
(uk(r) = uk(r+R)):
ψk(r) =uk(r)eikr. (2.36)
Both, the periodic electrostatic potentialVef f(r)and the Bloch waves ψk(r), can be
expanded as a Fourier series in terms of reciprocal lattice vectors G Vef f(r) = 1 Ω X G Vef f(G)eiGr (2.37) ψk(r) = 1 √ Ω X G uk(G)ei(k+G)r (2.38)
with Ωbeing the volume of the unit cell of the system (Ω =a1·(a2×a3)).
The reciprocal lattice vectors G are dened as
G=ub1+vb2+wb3. (2.39)
Here, bi is the reciprocal lattice basis and u, v, w ∈Z, with the primitive reciprocal
lattice basis bi satisfying the condition
biaj = 2πδij (2.40)
2 Theory and Methods
2.5.2 Expansion into Plane Waves
Since the functions uk(r) of the bloch waves ψk(r) (2.36) are periodic, they can be
expanded into a set of plane waves: uk(r) =
X G
ckGeiGr. (2.41)
This gives for ψk(r):
ψk(r) = uk(r)eikr=
X G
ckGei(k+G)r. (2.42) Using this form for ψk(r), inserting it into the Schrödinger equation (2.35), multi-
plying from the left with exp[−i(k+G0)r]and integrating over r, gives:
X G 1 2|k+G| 2 δG,G0 +Vef f(G0−G) ckG =kckG. (2.43) The Kronecker delta δG,G0 denotes the orthonormality of the plane waves with re-
spect to the reciprocal lattice vectors G:
δG,G0 = Z ψG(r)∗ψG0(r)dr = 1 if G=G0 0 if G6=G0. (2.44) For practical use, only such plane wave vectors (k+G) are kept, that give kinetic
energies kin, that are lower than a chosen cut-o energyEcut:
1
2|k+G|
2 =
kin ≤Ecut. (2.45)
The cut-o energy has to be carefully chosen to be large enough to achieve conver- gence for the intended calculations [56, 57].
2.5.3 K-point Sampling
By employing Bloch's Theorem, the problem of calculating the eigenstates of an innite number of electrons extended innitely in space is reduced to calculating
2.5 The Plane Wave Method
eigenstates of a nite number of electrons at an innite number of k-points in a single unit cell in k-space (=ˆ reciprocal space).
This problem is solved by assuming, that the electronic wave functions are almost identical at k-points, which are very close to each other. Therefore, the wave function of a region in k-space can be represented by the wave function of one single k-point of this region. The1st Brillouin zone (unit cell in k-space) now can be sampled by
a regular mesh of k-points.
This simplies the calculation of the electronic density ρ(r) by approximating the
integration over the 1st Brillouin zone by a sum over the N
kpt k-points: ρ(r) = 2 Ω (2π)3 X j Z BZ |ψkj(r)|2Θ(F −kj)d3k= = 2 1 Nkpt X j X k fkj|ψkj(r)|2. (2.46)
Here Θ(F −kj) is a step-function which is either 1 for kj ≤ F and zero for
kj > F and fkj are occupation numbers (which are either 1 ore zero for insulators
or semiconductors). F is the Fermi energy, which gives the energy of the highest
occupied state in a quantum mechanical system at absolute zero point temperature [57].
In the present work, the Monkhorst-Pack Scheme [58] as implemented in the Vienna ab-initio Simulation Package (VASP) [5963], was used to create equally spaced k- point meshes in the Brillouin zone:
k=b1 n1+ 0.5 N1 +b2 n2+ 0.5 N2 +b3 n3+ 0.5 N3 , (2.47) with n1 = 0, ...,(N1−1) n2 = 0, ...,(N2−1) n3 = 0, ...,(N3−1)
and theNi being the numbers of subdivisions along each reciprocal lattice vector bi.
2 Theory and Methods
total number of k-points. k-point grids with an even number of subdivisions Nn are
shifted of the Γ-point (center of the Brillouin zone with the reciprocal coordinates
(000)).
For hexagonal lattices Gamma centered Monkhorst-Pack grids were preferred, as standard Monkhorst-Pack grids do not represent the full hexagonal symmetry:
k=b1 n1 N1 +b2 n2 N2 +b3 n3 N3 . (2.48)