6.8 Extracting information from band structures 161betweenEandE+dE, divided by dE. Another quantity of interest is the charge
density, which is needed for calculating the Hartree and exchange and correlation
potentials in the DFT self-consistency loop. The charge density is given by
n(r)=∑
k,n′
|ψk,n(r)|^2 =∫EF
−∞dEn(r,E), (6.107)where the sum in the second expression is over the occupied levels, i.e. those with
energy below the Fermi energyEF.
The charge density can also be found from an integration over the energy of the
local density of states, which is defined as the charge density resulting exclusively
from states at energyE. An elegant way of finding this quantity using Green’s
functions is described in Problem 6.3. Such an approach is necessary when the
total charge of the system is not known, as is the case for a small system coupled
to a large reservoir which determines the chemical potential: for a metal, this is (to
very good approximation) the Fermi energy of the large system. This Fermi energy
is defined with respect to the vacuum energy as thework function: the energy needed
to remove an electron from the large system. There exist Green’s function methods
in which the small system (e.g. an atom or a molecule) is coupled to the surface
Green’s function of a metal. The electronic structure can then still be determined in
a self-consistency loop, in which the charge density is determined from the Green’s
function of the combined system plus reservoir rather than from the eigenstates of
the Hamiltonian [28].
To find physical properties or quantities, we often must perform an integration
over the Brillouin zone, as the vectors (together with the band labels) in this zone
are quantum numbers of the stationary states. Taking the crystal symmetry into
account, these integrations only need to be carried out in the ‘irreducible wedge’
of the Brillouin zone: this can be used to fill the whole Brillouin zone by crystal
symmetry transformations. For example, in a two-dimensional lattice having the
symmetry of the square, the Brillouin zone is also a square, but to integrate quantity
over the Brillouin zone, an integration over a wedge of area 1/8 of the whole square
needs to be carried out. For the Brillouin zone of the fcc lattice in Figure 6.9, this
irreducible wedge is the volume bounded by the labelled points.
There exist many different methods for performing Brillouin zone integration
[21]. The most popular methods are those usingspecial points[ 29 , 30 ] and tetrahed-
ron methods. In the latter, (part of) the Brillouin zone is divided up into tetrahedra,
in each of which either a linear or a quadratic approximation of the function to be
integrated is made. For calculating the density of states, the quadratic works very
well since it is capable of reproducing all known Van Hove singularities [ 31 – 34 ].