Computational Physics

6.7 The pseudopotential method 157

Table 6.2. Parameters for the GTH parametrisation

of the exchange-correlation energy.

a 1 0.4581652932831429 b 1 1.0

a 2 2.217058676663745 b 2 4.504130959426697

a 3 0.7405551735357053 b 3 1.110667363742916

a 4 0.01968227878617998 b 4 0.02359291751427506

Here,rs=[ 3 /( 4 πn)]^1 /^3 is the radius of the spherical volume per atom; the numbers
aiandbiare given inTable 6.2. The exchange-correlation potential is given as the
derivative of the energy with respect ton. It must be calculated in real space, where
it is periodic (as it depends on the density, which is periodic), and then Fourier-
transformed so that it can be added to the (K-space) Hamiltonian. The procedure is
therefore to first fill a grid with the values ofVxc(r). This is then Fourier-transformed
toVxc(K). Then, the contributionVxc(K,K′), whereKandK′lie inside the circle
C1 of Figure 6.14, is found by first translatingK−K′to a point inside the unit cell
UCof the reciprocal grid, and then taking forVxc(K,K′)the Fourier-transformed
exchange-correlation potential at that reciprocal grid point.
The Hartree potential

VH(r)=

∫

n(r′)

|r−r′|

d^3 r′ (6.88)

can be Fourier-transformed to give

VH(K,K′)=VH(K−K′)=

4 π

|K−K′|^2

n(K−K′). (6.89)

For the density, the differenceK−K′has to be translated to lie inside the unit cell
UC(see Figure 6.13), just as in the case of the exchange correlation potential. For
the denominator, we simply take the norm of the smallest periodic image of the
differenceK−K′.

CheckIf we incorporate both the exchange-correlation and the Hartree poten-

tials, we have a complete self-consistent pseudopotential Kohn–Sham program.

For the calculation with one hydrogen atom in a cubic cell of size 5 and a cut-off

of 1.3 atomic units, we obtain the energy spectrum:

−0.468131; 0.249348; 0.373144; 0.373144;

0.373144; 0.404949; 0.404949.