Representation of the CCSD Energy Equation

As discussed in detail earlier, products of normal-ordered operators can be simplified algebraically using Wick s theorem by evaluating pairs of contractions between the component annihilation and creation operators. Many of these contractions produce mathematically redundant terms that can be combined after complicated manipulation to eventually produce a much simpler expression. Diagrams provide a straightforward scheme by which these redundancies may be eliminated. 

As an example, consider the CCSD energy equation derived earlier in Eq. [134] using Wick s theorem. Each term of the general expression 

Next we consider the term from Eq. [159] that is linear in 

Note that to avoid external lines, both lines from the T diagram must connect to each line from the fragment. The diagram may be interpreted algebraically using the following rules  

According to these rules, the final algebraic interpretation of the diagram above is therefore 

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