On the difference between variational and unitary coupled cluster theories

TL;DR

This paper conclusively demonstrates that variational and unitary coupled cluster theories produce different energies, especially under strong correlation, using the Lipkin Hamiltonian, and introduces a generalized non-unitary variant with improved performance.

Contribution

It provides conclusive numerical evidence of differences between variational and unitary coupled cluster theories and introduces a generalized non-unitary variant that outperforms traditional methods.

Findings

Variational and unitary coupled cluster theories yield different energies.

Differences are significant under strong correlation in the Lipkin model.

The generalized non-unitary variant performs best among tested methods.

Abstract

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.