The Coupled-Cluster Formalism - a Mathematical Perspective

TL;DR

This paper explores the mathematical foundations of Coupled-Cluster theory, emphasizing coercivity assumptions for solution uniqueness, and introduces new criteria and connections to improve understanding and assessment of the method.

Contribution

It provides a rigorous mathematical analysis of Coupled-Cluster theory, highlighting coercivity conditions, and establishes a link between cluster amplitudes and Lagrange multipliers using the bivariational principle.

Findings

Coercivity assumptions are crucial for local uniqueness of solutions.

A new criterion based on the smallness of cluster amplitudes relative to Gårding constants.

Derived a relation between exact cluster amplitudes and Lagrange multipliers.

Abstract

The Coupled-Cluster theory is one of the most successful high precision methods used to solve the stationary Schr\"odinger 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 - G\aa rding type inequalities - for the local uniqueness of the Coupled-Cluster 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 G\aa rding constants. In the extended Coupled-Cluster 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 Coupled-Cluster solution. Furthermore, the use of an Aubin-Nitsche duality type method in different Coupled-Cluster approaches is discussed and contrasted with the bivariational principle.