Coupled-cluster theory similarity-transformed Hamiltonian

Note that in contrast to a general similarity transformation (e.g., as found in the usual coupled-cluster theory) the canonical transformation produces a Hermitian effective Hamiltonian, which is computationally very convenient. When U is expressed in exponential form, the effective Hamiltonian can be constructed termwise via the formally infinite Baker-Campbell-Hausdorff (BCH) expansion,... 

The first term in Equation 6.2 is a direct term similar to the matrix elements of similarity-transformed Hamiltonians in single reference coupled cluster theory, while the second term is a coupling term that describes the coupling between model functions. It has been shown that the structme of the coupling term is crucial for ensuring that the theory is both size-extensive and intruder-free. 

The use of a similarity-transformed Hamiltonian in linked coupled-cluster theory means that the energy and amplitude equations contain terms that consist either of the Hamiltonian itself or of nested commutators of the Hamiltonian with cluster operators. For a system containing two noninteracting subsystems A and B, these nested commutators separate additively into nested commutators, each involving a single subsystem, for example. 

The arguments for size-extensivity in CCPT follow closely those for linked coupled-cluster theory in Section 13.3. Thus, whereas the order-by-order size-extensivity of the energy follows ftom the exponential ansatz for the wave function, the termwise size-extensivity follows from the use of a similarity-transformed Hamiltonian, assuming that both the zero-order Fock operator and the first-order fluctuation operator separate for noninteracting systems. 

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