Coupled-Cluster theory revisited

Abstract

In a series of two articles, we propose a comprehensive mathematical framework for Coupled-Cluster-type methods. These methods aim at accurately solving the many-body Schrodinger equation. In this first part, we rigorously describe the discretization schemes involved in Coupled-Cluster methods using graph-based concepts. This allows us to discuss different methods in a unified and more transparent manner, including multireference methods. Moreover, we derive the singlereference and the Jeziorski–Monkhorst multireference Coupled-Cluster equations in a unified and rigorous manner.

Recent mathematical advances in coupled cluster theory

“…In a series of two articles [41, 42] Csirik and Laestadius propose a novel and comprehensive mathematical framework for Coupled‐Cluster‐type methods.…”

“…Let $$$ \left({\mathbf{t}}_{\ast, 0},{\mathbf{t}}_{\ast, \perp}\right) $$$ be an amplitude vector, where $$$ {\mathbf{t}}_{\ast, 0} $$$ is a root to the truncated CC equations and $$$ {\mathbf{t}}_{\ast, \perp } $$$ is a root of the remainder term. Under some assumptions, the authors then prove the existence of a bounded solution path tracking from $$$ \left({\mathbf{t}}_{\ast, 0},{\mathbf{t}}_{\ast, \perp}\right) $$$ to a solution to the untruncated CC equations, see theorem 4.34 in [42]. In essence, the imposed assumptions concern the fluctuation potential (and how it couples the truncated space to the “rest”, which can be controlled by the CC truncation level) and the size of the $$$ {\mathbf{t}}_{\ast, \mathbf{\perp}} $$$ (which must not be too large).…”

Measuring Electron Correlation: The Impact of Symmetry and Orbital Transformations

“…17,18,80 The most popular of them is CC theory, 81 which was introduced into chemistry by C ̌izěk, 82 Paldus and Shavitt, 83 was reformulated in terms of a variational principle by Arponen 84 and whose mathematical properties are still an active area of research. 85,86 Even more actively studied are its local variants 87,88 and its explicitly correlated, 89 excited state 90 and multireference 91 extensions. Beyond these methods, there is also room for a "top-down" approach in which one aims directly at the exact solution (full CI, full CC) and applies some strategy to neglect contributions that are less important while trying to save as much of the accuracy as possible.…”

State-Specific Coupled-Cluster Methods for Excited States

“…Živković and Monkhorst were the first to study the conditions under which higher roots of the CC equations exist, , while Adamowicz and Bartlett explored the feasibility of reaching certain excited states of the LiH molecule . Subsequently, Jankowski et al clearly evidenced that some of these nonstandard CC solutions are unphysical. − A pivotal development in the study of nonstandard CC solutions was the introduction of the homotopy method by Kowalski, Jankowski, and others. − We refer the interested reader to the review of Piecuch and Kowalski for additional information on this topic (see also refs – . for a more mathematical perspective).…”

“… 132 − 134 A pivotal development in the study of nonstandard CC solutions was the introduction of the homotopy method 135 by Kowalski, Jankowski, and others. 136 − 143 We refer the interested reader to the review of Piecuch and Kowalski for additional information on this topic 144 (see also refs ( 145 – 148 ). for a more mathematical perspective).…”