Diagrams, Hugenholtz

Using canonical HF orbitals the diagrams representing the second and third order energy corrections are shown in Figure 1. In the fourth order we have 39 Hugenholtz diagrams. 

Fig. 4. Hugenholtz diagram 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). 

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

At this point, we introduce a graphical representation of the CIS equation with Hugenholtz diagrams. The three matrix elements in Eq. (2-18) are depicted by... 

Figure 3 Correspondence between Brandow and Hugenholtz diagrams 54 N. H. Hugenholtz, Physica, 1957, 23, 481.

Fig. 23. Example of fifth-order skeleton Hugenholtz diagram (a) and (b) skeleton diagrams corresponding to it. Arrows can be added to (a) in 24 ways, to each diagram in (b) in 8 ways.

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 fifth order Hugenholtz diagrams were listed by Lyons, Moncrieff and Wilson.194 These authors generated the (symmetric) connectivity matrices for undirected Hugenholtz diagrams. Putting... 

Figure 28 Undirectedfifth order Hugenholtz diagram corresponding to 1 ...

The corresponding undirected Hugenholtz diagram is shown in Figure 28. 

The connectivity matrices for the fifth-order directed Hugenholtz diagrams take the form... 

The directed Hugenholtz diagrams corresponding to the undirected diagram (172) shown above are... 

An analysis of the fifth order diagrams and terms is made in Tables 10 and 11. Table 10 shows the number of undirected Hugenholtz diagrams for different levels of excitation in 1234 and 1234. Table 11 shows the number of directed Hugenholtz diagrams for different levels of excitation in 12 4 and 1234. 

Rules for Consiriicling All Hugenholtz Diagrams for a Ciiven Order n... 

A Hugenholtz diagram is translated into one of its equivalent Brandow diagrams when the dots (which represent the two-electron interaction) are extended into dashed lines, where one arrow is entering and one leaving at both end points of the dashed line, e.g., The Hugenholtz... 

FIG. 3.2. Translation of a third-order Hugenholtz diagram into corresponding Brandow diagrams. 

Write all Hugenholtz diagrams that contribute in third order when the perturbation U = — K -f consists of the electronic repulsion W and a one-electron perturbation V. 

In Fig. 6.3 we have displayed the diagrammatic representation of Eq. (6.180) in terms of Hugenholtz diagrams. Using the rules in Table II of Chapter 3 for interpreting diagrams with the modifications to rule 3 similar to those discussed in Section 6.D.2.d, we may interpret the A matrix as giving rise... 

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