Atomic Structure Theory

64 2 Central-Field Schrödinger Equation
Pκ(r[i]) = r[i] γ u[i], (2.168)
Qκ(r[i]) = r[i] γ v[i]. (2.169)
These equations are used in the routine outdir to give the k values required
to start the outward integration using a k + 1-point Adams-Moulton scheme.
indir:
The inward integration is started using an asymptotic expansion of the radial
Dirac functions. The expansion is carried out for r so large that the potential
V (r) takes on its asymptotic form
V (r) =− ζ
r ,
where ζ = Z − N + 1 is the ionic charge of the atom. We assume that the
asymptotic expansion of the radial Dirac functions takes the form
Pκ(r) =r σ e −λr
Qκ(r) =r σ e −λr
c 2 + E
2c 2
c 2 + E
2c 2
1+ a1
r
c2 − E
+
−
2c 2
1+ a1
r
c2 − E
2c 2
a2
+
r
b1
r
a2
+
r
b1
r
+ ···
+ b2
r
+ ···
+ b2
r
+ ···
, (2.170)
+ ···
, (2.171)
where λ = c2 − E2 /c2 . The radial Dirac equations admit such a solution
only if σ = Eζ/c2λ. The expansion coefficients can be shown to satisfy the
following recursion relations:
b1 = 1
κ +
2c
ζ
, (2.172)
λ
bn+1 = 1
κ
2nλ
2 − (n − σ) 2 − ζ2
c2
bn , n =1, 2, ··· , (2.173)
an = c
κ +(n− σ)
nλ
E ζλ
−
c2 c2
bn , n =1, 2, ··· . (2.174)
In the routine indir, (2.170) and (2.171) are used to generate the k values of
Pκ(r) andQκ(r) needed to start the inward integration.
2.8.2 Eigenvalue Problem for Dirac Equation (master)
The method that we use to determine the eigenfunctions and eigenvalues of
the radial Dirac equation is a modification of that used in the nonrelativistic