The coupled-cluster formalism – a mathematical perspective

ABSTRACT

The Coupled-Cluster (CC) theory is one of the most successful high precision methods used to solve the stationary Schrödinger equation. In this article, we address the mathematical foundation of this theory with focus on the advances made in the past decade. Rather than solely relying on spectral gap assumptions (non-degeneracy of the ground state), we highlight the importance of coercivity assumptions – Gårding type inequalities – for the local uniqueness of the CC solution. Based on local strong monotonicity, different sufficient conditions for a local unique solution are suggested. One of the criteria assumes the relative smallness of the total cluster amplitudes (after possibly removing the single amplitudes) compared to the Gårding constants. In the extended CC theory the Lagrange multipliers are wave function parameters and, by means of the bivariational principle, we here derive a connection between the exact cluster amplitudes and the Lagrange multipliers. This relation might prove useful when determining the quality of a CC solution. Furthermore, the use of an Aubin–Nitsche duality type method in different CC approaches is discussed and contrasted with the bivariational principle.

GRAPHICAL ABSTRACT

KEYWORDS:

• Coupled-cluster theory
• extended coupled-cluster theory
• bivariational principle
• Gårding inequalities
• local strong monotonicity

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

A. Laestadius http://orcid.org/0000-0001-7391-0396

Additional information

Funding

This work was supported by H2020 European Research Council [639508] and Norges Forskningsråd [262695].