Atomic Structure Theory

7.7 Second-Order Perturbation Theory 231
Table 7.11. Dirac HF E HF and second-order E (2) Coulomb energies are given for
one-electron states of P +4 along with first-order B (1) and second-order B (2) Breit
energies.
nlj E HF
E (2)
B (1)
B (2)
3s1/2 -2.37233 -0.01744 0.00044 -0.00022
3p1/2 -1.97077 -0.01490 0.00053 -0.00020
3p3/2 -1.96707 -0.01478 0.00037 -0.00022
3d3/2 -1.44699 -0.01169 0.00009 -0.00009
3d5/2 -1.44694 -0.01167 0.00005 -0.00008
of valence energies of the one-electron ion that shares the same neonlike core;
in this case P +4 . These one-electron energies are listed in Table 7.11; both
E HF and B (1) are included in the first-order Hamiltonian. Let us examine,
for a specific example, the odd-parity J = 3 states. There are three configurations
in the model space (3p 1/2 3d 5/2), (3p 3/2 3d 3/2), and (3p 3/2 3d 5/2) that
contribute to these states. The first-order Hamiltonian matrix between these
three configurations is
H (1) ⎡
⎤
−2.93578 −0.10585 0.03247
= ⎣ −0.10585 −2.93268 −0.05918 ⎦ .
0.03247 −0.05918 −2.89656
The three eigenvalues of the first-order Hamiltonian matrix are
Eλ = −3.04277 −2.93349 −2.78876 .
The lowest eigenvalue of the even-parity J = 0 matrix, which represents the
(3s) 21 S0 ground state of P +3 , is found to be −4.23631. It follows that the firstorder
excitation energies of the three J=3 states are [1.1935 1.30282 1.44755],
corresponding to the (3p3d) 3 F3, 3 D3, and 1 F3 states. Similar calculations are
made for even and odd parity states with J =0··· 4, leading to the first-order
energies listed in Table 7.12; where we compare first- and second-order energies
of low-lying states of P +3 with measured energies from the [39]. For the
states considered, the differences between first-order theoretical energies and
observed energies range from 1% to 6%. Including second-order corrections,
discussed in the next section, substantially reduces these differences.
7.7 Second-Order Perturbation Theory
From the effective Hamiltonian, one immediately sees that the second-order
energy matrix between uncoupled states designated by k and l is
E (2)
kl
= 〈Ψ (0)
k |Vχ(1) |Ψ (0)
l 〉, (7.117)