Diagrams, Goldstone

Fig. 3. Goldstone diagrams for the second order contribution to the correlation energy in the ground state...

Draw all possible distinct Hugenholtz diagrams for a pertinent order of the perturbation expansion ) and generate from them Goldstone diagrams by (80). 

The sign of the total expression corresponding to the pertinent Goldstone diagram is given by (—l),+ i where l is the number of closed loops and h is the number of hole lines. 

Sum over all indices appearing in the pertinent Goldstone diagram. 

Sum the individual contributions of all Goldstone diagrams which are generated from the parent Hugenholtz diagram. 

Multiply the whole expression (the sum of the Goldstone diagram contribu-... 

Second-order Hugenholtz (A) and Goldstone (8, C) diagrams. The second-order contribution, e( ), contains also two other Goldstone diagrams that are equivalent to diagrams B and C. 

There are twelve third-order perturbation diagrams, consequently twelve Fock operators containing third-order correction, F[P, in the Goldstone diagram, we write only the first one here. 

For the partition 2, 1, 1,0 (Fig. 24c), we have 6 skeleton Hugenholtz diagrams that, when all 12 permutations are taken, results in 72 graphs. This, in turn, corresponds to 12 x 40 = 480 and 12 x 720 = 8640 antisymmetrized and Goldstone diagrams, respectively. 

The total number of antisymmetrized and Goldstone diagrams amounts to 840 and 13,680, respectively. 

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