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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: 
ri = r0exp(iΔx). With this grid, one has
|  f (r)dr =  f (x)r(x)dx | (15) | 
 
and
We start with a given self-consistent potential V and
a trial eigenvalue ε. The equation is integrated
from r = 0 outwards to rt, the outermost classical 
(nonrelativistic for simplicity) turning point, defined
by 
l (l+1)/rt2 +  V(rt) - ε
V(rt) - ε = 0.
In a logarithmic grid (see above) the equation to solve becomes:
 = 0.
In a logarithmic grid (see above) the equation to solve becomes:
|   | = |   +   + M(r)  V(r) - ε   Rnl(r) |  | 
|  |  | -      + 〈κ〉   . | (17) | 
 
This determines 
d2Rnl(x)/dx2 which is used to
determine 
dRnl(x)/dx which in turn is used to
determine Rnl(r), using predictor-corrector or whatever
classical integration method. 
dV(r)/dr is evaluated
numerically from any finite difference method. The series
is started using the known (?) asymptotic behavior of Rnl(r) 
close to the nucleus (with ionic charge Z)
| Rnl(r)  rγ,        γ =  . | (18) | 
 
The number of nodes is counted. If there are too few (many)
nodes, the trial eigenvalue is increased (decreased) and
the procedure is restarted until the correct number n - l - 1
of nodes is reached. Then a second integration is started 
inward, starting from a suitably large 
r∼10rt down 
to rt, using as a starting point the asymptotic behavior 
of Rnl(r) at large r: 
| Rnl(r)  e-k(r)r,        k(r) =  . | (19) | 
 
The two pieces are continuously joined at rt and a correction to the trial 
eigenvalue is estimated using perturbation theory (see below). The procedure 
is iterated to self-consistency.
The perturbative estimate of correction to trial eigenvalues is described in
the following for the nonrelativistic case (it is not worth to make relativistic
corrections on top of a correction). The trial eigenvector Rnl(r) will have 
a cusp at rt if the trial eigenvalue is not a true eigenvalue:
| A =  -  ≠0. | (20) | 
 
Such discontinuity in the first derivative translates into a
δ(rt) in the second derivative:
|  =  + Aδ(r - rt) | (21) | 
 
where the tilde denotes the function obtained by matching the
second derivatives in the r < rt and r > rt regions.
This means that we are actually solving a different problem in which 
V(r) is replaced by 
V(r) + ΔV(r), 
given by
| ΔV(r) = -   δ(r - rt). | (22) | 
 
The energy difference between the solution to such fictitious potential 
and the solution to the real potential can be estimated from
perturbation theory:
| Δεnl = - 〈ψ| ΔV| ψ〉 =  Rnl(rt)A. | (23) | 
 
 
 
 
 
 
 
 
  
 Next: B. Equations for the
 Up: A. Atomic Calculations
 Previous: A..3 Scalar-relativistic case
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