Computational Physics

154 Solving the Schrödinger equation in periodic solids

Note thatKandK′are indices of the Hamiltonian – therefore they lie inside the
C1 in Figure 6.14. Their difference will be insideC2.
For thej-th component (j=x,y,z)ofK,Kj= 2 πnj/L, the periodic image lying
on the grid inFig. 6.14is found as follows:

Kj= 2 π(nj mod GridSize), (6.75)

where[...]denotes the integer part.
Forr 0 on the real-space grid, the structure factor in front of the integral is periodic
inK-space. However, an atom does not know how we define our grid and may be
located in between grid points (of course, with only one atom in the cell, we could
always move the atom toR=0). This nonperiodicity of the structure factor is
responsible for the difference between potential at the points(2, 0, 0)and(−2, 0, 0)
(in units of the reciprocal grid constant) – hence we cannot take a grid size of 4
units, but instead need a 5× 5 ×5 grid, as mentioned above.
We now give a specific form of the pseudopotential. We shall use the Goedecker–
Teter–Hutter(GTH)potentialdescribedinrefs.[24]and[26]. This potential for a
core, located at the origin, with s and p electrons has the form:^3

V(r,r′)=Vcore(r)+Vloc(r)δ(r−r′)+Vnonloc(r,r′) (6.76)

with

Vcore=−

Zeff

r

erf

(

r

√

2 ξ

)

; (6.77a)

Vloc(r)=exp[−(r/ξ )^2 / 2 ]×[C 1 +C 2 (r/ξ )^2 ], (6.77b)

and

Vnonloc(r,r′)=

∑^2

i= 1

Y 00 (rˆ)pi^0 (r)hi^0 p^0 i(r′)Y 00

∗

(rˆ′)

+

∑

m=1,0,− 1

Ym^1 (rˆ)p 11 (r)h^11 p^11 (r′)Ym^1

∗

(rˆ′) (6.78)

In these expressions,ξ,Ci,hliare parameters, and thepliare the functions

pl 1 (r)=

√

2

rle−(^1 /^2 )(r/rl)

2

rl+^3 /^2

√

(l+ 3 / 2 )

and (6.79)

pl 2 (r)=

√

2

rl+^2 e−(^1 /^2 )(r/rl)

2

rl+^7 /^2

√

(l+ 7 / 2 )

(6.80)

(^3) The form given here is somewhat simpler than the full GTH potential. For the atoms considered here,
however, the present form is sufficient.