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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
Contents
Layla Martin-Samos Colomer
2012-05-10