Algebraic connectivity analysis in molecular electronic structure theory II: total exponential formulation of second-quantised correlated methods

Abstract

The fundamentality of the exponential representation of a second-quantised correlated wave function is emphasised with an accent on the physical sense of cluster amplitudes as cumulants of the correlated ansatz. Three main wave function formalisms, namely, the configuration-interaction theory, the coupled-cluster approach, and the many-body perturbation theory (as well as their extensions, e.g. the equation-of-motion coupled-cluster method, multireference schemes, etc.), are represented in an exponential form, leading to a formulation of the working equations in terms of cluster amplitudes. By expressing the corresponding many-body tensor equations in terms of cluster amplitudes, we could unambiguously check connectivity types and the asymptotic behaviour of all tensors/scalars involved (in the formal limit of an infinite number of correlated particles). In particular, the appearance of disconnected cluster amplitudes corresponds to unphysical correlations. Besides, we demonstrate that the equation-of-motion coupled-cluster approach, as well as certain excited-state configuration-interaction methods, can be recast in a fully connected (exponential) form, thus breaking the common belief that all truncated configuration-interaction methods violate connectivity. Our work is based on the recently developed algebraic framework which can be viewed as a complement to the classical diagrammatic analysis.

Keywords:

• electron correlation
• size-extensivity
• size-intensivity
• size-consistency
• generalised extensivity
• tensor connectivity
• cumulant
• exponential
• coupled cluster
• perturbation theory
• configuration interaction
• equation of motion
• multireference

Acknowledgements

The authors greatly acknowledge the US Air Force Office of Scientific Research (AFOSR) for the financial support. Valuable comments from unknown referees are appreciated as well. A careful examination of the paper by Prof. Marcel Nooijen, who also presented alternative proofs to some of our propositions, is deeply appreciated..