Factorization of the Coupled Cluster Equations

Every term in the coupled cluster amplitude equations that is nonlinear in T may be factored into linear components. As a result, each step of the iterative solution of the CCSD equations scales at worst as ca. 0(X ) (where X is the number of molecular orbitals). The full CCSDT method in which all Tycon-taining terms are included requires an iterative 0(X ) algorithm, whereas the CCSD(T) method, which is designed to approximate CCSDT, requires a noniterative O(X ) algorithm. The inclusion of all T4 clusters in the CCSDTQ method scales as 

The most efficient scheme for factorization of the amplitude equations as described above is not obvious, however, and over the past 20 years numerous researchers have developed sets of intermediates to streamline their own coupled cluster programs.21,22,146,147 ny of these factorizations have been based on careful inspection of the amplitude equations. 

Scuseria, Janssen, and Schaefer, for example, developed a set of intermediates based on their reformulation of the CCSD amplitude and energy equations in a unitary group formalism designed to offer special efficiency when the refer- 

Diagrammatic techniques also provide a route to the construction of efficient coupled cluster intermediates. Kucharski and Bartlett, for example, described a particularly clever approach by which one uses matrix elements of the similarity transformed Hamiltonian as the desired intermediates. Consider the matrix element of H between the reference (on the left) and a singly excited determinant (on the right). Diagrammatically, this matrix element is resolved into two terms as 

We have chosen the double bar with the sign in the final diagram to simply denote the sum of the two diagrams corresponding to the matrix element. If we contract this diagram with a T2 operator fragment from below, we obtain two contributions to the Tj amplitude equations, namely. 

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