  There have been assertions in the literature that the variational and unitary
forms of coupled cluster theory lead to the same energy functional. Numerical
evidence from previous authors was inconsistent with this claim, yet the small
energy differences found between the two methods and the relatively large
number of variational parameters precluded an unequivocal conclusion. Using the
Lipkin Hamiltonian, we here present conclusive numerical evidence that the two
theories yield different energies. The ambiguities arising from the size of the
cluster parameter space are absent in the Lipkin model, particularly when
truncating to double excitations. We show that in the symmetry adapted basis
under strong correlation the differences between the variational and unitary
models are large, whereas they yield quite similar energies in the weakly
correlated regime previously explored. We also provide a qualitative argument
rationalizing why these two models cannot be the same. Additionally, we study a
generalized non-unitary and non-hermitian variant that contains excitation,
de-excitation and mixed operators with different amplitudes and show that it
works best when compared to the traditional, variational, unitary, and extended
forms of coupled cluster doubles theories.
