Keywords: pseudo-gen

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## A..4 Numerical solution

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(ix). With this grid, one has

f (r)dr = f (x)r(x)dx | (15)

and

= , = - + . | (16)

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)

Rnl(r) r, = . | (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 (rt) in the second derivative:

= + A(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) + V(r), given by

V(r) = - (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:

= - |V| = Rnl(rt)A. | (23)

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