Atomic Structure Theory

P n (r) and Q n (r)/(Z)
0.4
0.0
-0.4
0.0
-0.4
2.8 Numerical Solution to Dirac Equation 61
2s 1/2
-0.8
0.0 4.0 8.0 12.0
0.4
2p 1/2
-0.8
0.0
0.8
4.0 8.0 12.0
0.4
0.0
2p3/2 -0.4
0 4 8 12
r (a.u.)
Fig. 2.5. Radial Dirac Coulomb wave functions for the n = 2 states of hydrogenlike
helium, Z = 2. The solid lines represent the large components P2κ(r) and the dashed
lines represent the scaled small components, Q2κ(r)/αZ.
dy
= f(y, t) ,
dt
where y(t) andf(y, t) are the two-component arrays:
(2.153)
Pκ
y = , (2.154)
f(y, t) =r ′
where, r ′ (t) = dr
dt .
Qκ
−(κ/r) Pκ(r) − α[Wκ − V (r)+2α−2
]Qκ(r)
, (2.155)
(κ/r) Qκ(r)+α[Wκ − V (r)]Pκ(r)
2.8.1 Outward and Inward Integrations (adams, outdir, indir)
adams:
We integrate (2.154) and(2.155) forward using the Adams-Moulton algorithm
given in (2.52):