Atomic Structure Theory

T (2)
deriv (−1)J [JI][JF ]
v≤w
x≤y
ηvwηxyC I vwC F xy ∆ FI
xy,vw
(−1) jx+jy+JI
JI JF J dTxv
jx jv jy dω δyw
+(−1) jw+jx
JI JF J dTxw
jx jw jy dω δyv
+(−1) JI
+JF +1 JI JF J dTyv
jy jv jx dω δxw
+(−1) jw+jx+JF
8.7 Summary Remarks 257
JI JF J
jy jw jx
dTyw
dω δxv
, (8.39)
along with the RPA and correlation corrections. The term ∆ FI
xy,vw in the above
equation,
∆ FI
xy,vw = E F − E I − ɛx − ɛy + ɛv + ɛw ,
is the first-order change in the transition energy.
As an example of a second-order calculation of transition matrix elements
for systems with two valence electrons, we turn once again to the magnesiumlike
ion P IV. In Table 8.11, second-order transition energies, electric-dipole
matrix elements, Einstein A coefficients, and lifetimes are listed. These parameters
are found to be in good agreement with values from CI calculations
of the type described earlier in the chapter.
Table 8.11. Second-order transition energies ω (a.u.), reduced electric-dipole matrix
elements D, line strengths S, and transition rates A (1/ns) are listed for singletsinglet
transitions in P IV. Lifetimes τ (ns) of the initial states obtained from secondorder
calculations are also listed.
Initial Final ω (a.u.) D (a0) S (a 2 0) A (1/ns) τ (ns)
(3s3p) 1 P1 (3s3s) 1 S0 0.4922 1.998 3.99 3.40 0.294
(3p3d) 1 D2 (3s3d) 1 D2 0.4965 2.648 7.01 3.68 0.272
(3p3d) 1 D2 (3p3p) 1 D2 0.2048 0.410 0.168 0.0062 —
(3s3d) 1 D2 (3s3p) 1 P1 0.2271 0.902 0.814 0.0490 24.5
(3s3p) 1 P1 (3p3p) 1 S0 0.3831 1.554 2.41 2.91 0.344
(3p3p) 1 D2 (3s3p) 1 P1 0.5188 3.551 12.6 7.54 0.133
8.7 Summary Remarks
In this chapter, we have discussed correlation corrections to matrix elements
of one- and two-particle operators. Calculations of one-particle transition matrix
elements, starting from a local central (model) potential, are gauge independent
order-by-order, provided derivative terms, accounting for possible