Computational Physics

6.7 The pseudopotential method 155

Table 6.1.Parameters for the GTH pseudopotential

Hydrogen Silicon

ξ 0.2 0.44

C 1 −4.0663326 −6.9136286

C 2 0.6778322D0 0.0

rl= 0 – 0.4243338

hl 1 =^0 – 3.2081318

hl 2 =^0 – 2.5888808

rl= 1 – 0.4853587

hl 2 =^0 – 2.6562230

Source:[24, 26]

Only values for hydrogen and silicon are listed.

The Gamma-function in the denominators ensures proper normalisation:
∫
pli(r)pli(r)r^2 dr=1. (6.81)

Let us spend a few moments studying this potential. The very first term,
−Zeff/rerf(r/ 2 ξ), is the Coulomb potential of a Gaussian charge distribution with
total chargeZeffand widthξ: for large arguments, that is, far from the ion core,
the error function erf tends to 1. The remaining terms are short-ranged and allow
therefore for refinement of the shape of the radial charge distribution. The nonlocal
term is, as usual, a projection onto the differentlsubspaces. For a complete list of
pseudopotentialparameters,werefertoRefs.[24]and[26]; here we give those for
hydrogen and silicon – seeTable 6.1.
The Fourier transform of the GTH potential can be calculated analytically,
yielding the following closed forms.

Vcore(K)=− 4 π

Zeff

e−(Kξ)

(^2) / 2
K^2

; (6.82)

Vloc(K)=

√

( 2 π)^3

ξ^3

e−(Kξ)

(^2) / 2
{C 1 +C 2 [ 3 −(Kξ)^2 ]} (6.83)
and
Vnonloc(K,K′)=

∑^2

i= 1

Y 00 (Kˆ)pi^0 (K)hi^0 p^0 i(K′)Y 00

∗

(Kˆ′)

−

∑

m=1,0,− 1

Ym^1 (Kˆ)p 11 (K)h^11 p^11 (K′)Ym^1

∗

(Kˆ′). (6.84)