Description

C (b) hΦabij |VcT2T3|ΦiC C C
2. Interpret the following graph and fully simplify your answer.

Extra Credit. Prove Rule 3 for a closed graph with a single bare excitation operator of the following form.

Appendix.
Axiom 1. The algebraic of a graph G is obtained from a corresponding summand graph Σ(G) as follows.
1
Σ(G)
Rule 1. Each set of k equivalent lines or equivalent subgraphs contributes a factor of k! to the degeneracy.
Rule 2. The overall sign of a closed graph is (−)h+l, where h and l denote the total number of hole lines and loops. Rule 3. For bare excitation operators, cancel the degeneracy factors from their equivalent coefficient lines by replacing the full antisymmetrizer, , with a reduced antisymmetrizer over inequivalent coefficient lines,

1
Here {p1,…,pm} = P1 ∪ ··· ∪ Ph and {q1,…,qm} = Q1 ∪ ··· ∪ Qk are the upper and lower indices on the bare excitation operator ˜ , and the Pi’s and Qi’s label subsets of equivalent coefficient lines.
2 For equivalent lines connecting two bare excitation operators, this cancellation can only be performed once.