Simplification of the Coupled Cluster Hamiltonian

The concepts of normal ordering and Wick s theorem provide the mathematical tools needed to derive programmable coupled cluster equations from the more formal expressions given in Eqs. [50] and [51]. If we truncate the cluster operator such that T = Tj + T2 insert it into the similarity-transformed normal-ordered Hamiltonian, H = e lij e, we obtain 

Therefore, a factor of Vi does not appear in front of this term in the expansion above. 

The generalized form of Wick s theorem (see Eq. [91]) says that this product of normal-ordered operator strings may be written using only contractions between the two strings. That is, 

For the second term of the expanded commutator, Ti n where the operator strings from and T are simply reversed in order, Wick s theorem gives only one term, namely. 

All other contractions, which involve a and a,- on the left, are zero by Eq. [94]. The final equality in this expression arises from the fact that, by construction, all operators within the braces anticommute. Therefore, using Eqs. [110] and [111], we may write 

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