Computational Physics

156 Solving the Schrödinger equation in periodic solids

The projector functions have the formpli:

p^01 =

1

√

4 rs

√

2 rsπ^5 /^4 e−(Krs)

(^2) / 2
, (6.85a)
p^02 =

1

√

8 rs

√

2 rs

15

π^5 /^4 e−(Krs)

(^2) / 2
[ 3 −(Krs)^2 ], and (6.85b)
p^11 (K)=

1

8 r 12

√

r 1

3

π^5 /^4 e−(Krs)

(^2) / 2
K. (6.85c)
This pseudopotential can be directly incorporated into the Kohn–Sham
Hamiltonian. It does not depend on the density, so if we simply want to calu-
late the energies and eigenfunctions of a particle moving in the pseudopotential,
we just have to diagonalise the Hamiltonian which consists of the kinetic energy
plus pseudopotential. A self-consistency cycle is not necessary.
The Fourier transform of the local part of the pseudopotential for a core located
atRnmust be multiplied by exp(−iK·Rn). The nonlocal part must be multiplied
by exp[i(K−K′)·Rn].
CheckDoing this for a cubic cell with an edge length of 5 a.u. containing one
hydrogen-core and an energy cut-off of( 1 / 2 )Kmax^2 =1.3, we obtain the eigen-
values−0.03572203 (once), 0.68175686 (once), 0.80555307 (three times) and
0.83735807 (twice). If we fill all seven levels, the density should be 0.05600 on
any real-space grid point. Try this first for a hydrogen at the origin, and then
some arbitrary position within the cell.
6.7.6 Exchange-correlation and Hartree potentials
The exchange-correlation and Hartree potentials are density-dependent; therefore,
including them makes a self-consistency cycle necessary. We shall first consider the
general problem of including density-dependent potentials into the problem. First,
we must have the density at our disposal. After diagonalising the Hamiltonian, we
calculate the Fourier transformsφ(j)(r)of the eigenfunctionsc(Kj)as in(6.63). Then
we calculate the density on all real-space grid points inside the cell according to
(6.66). Finally, we calculate the density in reciprocal space according to
n(K)=

1

N^3

∑

r

n(r)e−iK·r. (6.86)

For the exchange-correlation potential, we use the GTH parametrisation of the
pseudopotential of Perdew and Wang[27]. This is a form of Padé approximant:

εxc=−

∑ 4

i= 1 air

i− 1

∑ s

4

i= 1 birsi

. (6.87)