Notes on pseudopotential generation

A.2 Fully relativistic case
The relativistic KS equations are Dirac-like equations for a spinor with a “large” R nlj (r)
and a “small” S nlj (r) component:
( d
dr + κ )
r
c
c
( d
dr − κ r
)
R nlj (r) = ( 2mc 2 − V (r) + ɛ ) S nlj (r) (9)
S nlj (r) = (V (r) + ɛ) R nlj (r) (10)
where j is the total angular momentum (j = 1/2 if l = 0, j = l+1/2, l−1/2 otherwise);
κ = −2(j − l)(j + 1/2) is the Dirac quantum number (κ = −1 is l = 0, κ = −l − 1, l
otherwise); and the charge density is given by
n(r) = ∑ nlj
A.3 Scalar-relativistic case
Θ nlj
R 2 nlj(r) + S 2 nlj(r)
4πr 2 . (11)
The full relativistic KS equations is be transformed into an equation for the large
component only and averaged over spin-orbit components. In atomic units (Rydberg:
¯h = 1, m = 1/2, e 2 = 2):
( )
− d2 R nl (r) l(l + 1)
+ + M(r) (V (r) − ɛ) R
dr 2 r 2 nl (r)
(
−
α2 dV (r) dRnl (r)
+ 〈κ〉 R )
nl(r)
= 0, (12)
4M(r) dr dr r
where α = 1/137.036 is the fine-structure constant, 〈κ〉 = −1 is the degeneracyweighted
average value of the Dirac’s κ for the two spin-orbit-split levels, M(r) is
defined as
M(r) = 1 − α2
(V (r) − ɛ) . (13)
4
The charge density is defined as in the nonrelativistic case:
A.4 Numerical solution
n(r) = ∑ nl
Θ nl
R 2 nl(r)
4πr 2 . (14)
The radial (scalar-relativistic) KS equation is integrated on a radial grid. It is convenient
to have a denser grid close to the nucleus and a coarser one far away. Traditionally
a logarithmic grid is used: r i = r 0 exp(i∆x). With this grid, one has
and
df(r)
dr
= 1 r
∫ ∞
0
df(x)
dx ,
f(r)dr =
∫ ∞
0
d 2 f(r)
dr 2
f(x)r(x)dx (15)
= − 1 r 2 df(x)
dx
+ 1 r 2 d 2 f(x)
dx 2 . (16)