Atomic Structure Theory

2.6 Separation of Variables for Dirac Equation 55
φ(r) in the interval 0 - R. In the second panel, we show the corresponding
value of N(r), the number of electrons inside a sphere of radius r. Inthe
bottom panel, we give the electron contribution to the potential. Comparing
with Fig. 2.3, we see that the electron-electron potential U(r) obtained
from the Thomas-Fermi theory has the same general shape as the electroninteraction
contribution to the parametric potential Vb(r). This is consistent
with the previous observation that Vb(r) led to an accurate inner-shell energy
for sodium.
2.6 Separation of Variables for Dirac Equation
To describe the fine structure of atomic states from first principles, it is necessary
to treat the bound electrons relativistically. In the independent particle
picture, this is done by replacing the one-electron Schrödinger orbital ψ(r)
by the corresponding Dirac orbital ϕ(r). The orbital ϕ(r) satisfies the singleparticle
Dirac equation
hDϕ = Eϕ, (2.117)
where hD is the Dirac Hamiltonian. In atomic units, hD is given by
hD = c α · p + βc 2 + V (r) . (2.118)
The constant c is the speed of light; in atomic units c =1/α = 137.035999 ...,
the inverse of Sommerfeld’s fine-stricture constant. The quantities α and β in
(2.118) are 4 × 4 Dirac matrices:
α =
0 σ
σ 0
1 0
,β= . (2.119)
0 −1
The 2 × 2 matrix σ is the Pauli spin matrix, discussed in Sec. 1.2.1.
The total angular momentum is given by J = L+S, whereLis the orbital
angular momentum and S is the 4 × 4 spin angular momentum matrix,
S = 1
2
σ 0
0 σ
. (2.120)
It is not difficult to show that J commutes with the Dirac Hamiltonian. We
may, therefore, classify the eigenstates of hD according to the eigenvalues
of energy, J 2 and Jz. The eigenstates of J 2 and Jz are easily constructed
using the two-component representation of S. They are the spherical spinors
Ωκm(ˆr).
If we seek a solution to the Dirac equation (2.118) having the form
ϕκ(r) = 1
r
iPκ(r) Ωκm(ˆr)
Qκ(r) Ω−κm(ˆr)
, (2.121)