Atomic Structure Theory

Energy (a.u.)
10 1
10 0
10 -1
10 -2
8.4 Matrix Elements of Two-Particle Operators 249
B (1)
-E (2)
-B (2) Ne-like Ions
10
0 20 40 60 80 100
Nuclear Charge Z
-3
Fig. 8.1. First- and second-order Breit corrections to the ground-state energies of
neonlike ions shown along with the second-order correlation energy. The first-order
Breit energy B (1) grows roughly as Z 3 , B (2) grows roughly as Z 2 , and the secondorder
Coulomb energy E (2) is nearly constant.
Terms on the second line of (8.32) are second-order corrections to the “effective”
one-particle operator
ij tij : a †
i aj :. This term often dominates the
second-order correlation corrections. In such cases, one replaces the term by
its RPA counterpart.
Breit Interaction in Cu
As an example, we present a breakdown of the Breit corrections to energies
of 4s and 4p states of copper in Table 8.6. The terms B (2)
s and B (2)
d refer
to the sums over single and double excited states, respectively, on the first
line of (8.32). These terms are seen to be relatively small corrections to the
lowest-order Breit interaction B (1) .ThetermB (2)
e is the contribution from
the effective single-particle operator on the second line of (8.32). This term is
the dominant second-order correction; indeed, it is larger than the first-order
correction and has opposite sign. Iterating this term leads to the term BRPA.
The RPA correction is seen to be substantially different from B (2)
e for each
of the three states listed in the table. We replace B (2)
e by BRPA in the sum
Btot. The relatively small term B (3) is the Brueckner-orbital correction associated
with the effective single-particle operator and is expected to dominate
the residual third-order corrections. The extreme example given in Table 8.6
illustrates the importance of correlation corrections to two-particle operators;
the correlated matrix elements are larger in magnitude and differ in sign from
the lowest-order values!