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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(i
x). 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
. 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) - 
= 0.
In a logarithmic grid (see above) the equation to solve becomes:
  |
= |
 +  + M(r) V(r) -   Rnl(r) |
|
|
|
-     +     . |
(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)
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:
Such discontinuity in the first derivative translates into a
(rt) in the second derivative:
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) +
V(r),
given by
The energy difference between the solution to such fictitious potential
and the solution to the real potential can be estimated from
perturbation theory:
Next: B. Equations for the
Up: A. Atomic Calculations
Previous: A..3 Scalar-relativistic case
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Filippo Spiga
2016-10-04