Recent mathematical advances in coupled cluster theory

TL;DR

The paper surveys mathematical advances in coupled cluster (CC) theory after 2009, emphasizing rigorous foundations for solvability and root structure. It covers local invertibility analyses based on local strong monotonicity, a graph-based excitation framework, an inf-sup condition approach for CC Jacobians, and algebraic-geometry and homotopy techniques to analyze CC roots. It reports universal positive invertibility constants in the inf-sup framework, tighter root bounds via truncation varieties and the BKK theorem, and numerical demonstrations on model systems, illustrating improved reliability and scalability prospects. By linking CC theory to Lie-algebraic structure, topological degree theory, and algebraic geometry, the work provides a unified mathematical foundation that informs both analysis and practical computations in quantum chemistry.

Abstract

This article presents an in-depth educational overview of the latest mathematical developments in coupled cluster (CC) theory, beginning with Schneider's seminal work from 2009 that introduced the first local analysis of CC theory. We offer a tutorial review of second quantization and the CC ansatz, laying the groundwork for understanding the mathematical basis of the theory. This is followed by a detailed exploration of the most recent mathematical advancements in CC theory.Our review starts with an in-depth look at the local analysis pioneered by Schneider which has since been applied to analyze various CC methods. We then move on to discuss the graph-based framework for CC methods developed by Csirik and Laestadius. This framework provides a comprehensive platform for comparing different CC methods, including multireference approaches. Next, we delve into the latest numerical analysis results analyzing the single reference CC method developed by Hassan, Maday, and Wang. This very general approach is based on the invertibility of the CC function's Fréchet derivative. We conclude the article with a discussion on the recent incorporation of algebraic geometry into CC theory, highlighting how this novel and fundamentally different mathematical perspective has furthered our understanding and provides exciting pathways to new computational approaches.

Figures (9)

• Figure 1: Case study of lithium hydride comparing the linear parametrization (blue) and the exponential parametrization (red) for different values of $R$ in the AUG-cc-pVTZ basis set helgaker2014molecular.
• Figure 2: Full multi-reference excitation multigraph for five spin orbitals, labeled $\{1,...,5\}$ and two reference states, $\{1,2,3\}$ and $\{1,2,4\}$. The excitations w.r.t the references $\{1,2,3\}$ and $\{1,2,4\}$ are represented as edges. To distinguish the excitations from $\{1,2,3\}$ and $\{1,2,4\}$, the edges are colored in red and blue, respectively. See csirik2023coupled1 for more details.
• Figure 3: Numerically computed constants for the HF molecule at different bond lengths. The equilibrium bond length is 0.9168 Å. The figure on the right uses a log scale on the y-axis. For more details see hassan2023analysis
• Figure 4: Left: Newton fractal of $p(z) = z^3 -1$. The white dots correspond to the roots $z_1, z_{2}, z_{3}$. The colored regions, red, blue, and green, correspond to the basins of attraction of the roots $z_1$, $z_2,z_3$, respectively faulstich2023homotopy. Right: Energy trajectory of CCS solutions for a two-electron system. The overlap of the eigenstates with the reference state is steered by the parameter $\varepsilon$. The plot shows the $\varepsilon$-energy trajectory of all CCS solutions, where $\varepsilon$ was varied between zero and eight. For more details see faulstich2022coupled.
• Figure 5: A sketch of possible homotopy paths. The solid line shows a path with no finite limit as $\lambda \to 0$, the dashed lines have the same limit, and the dotted-dashed line has a unique limit.

Theorems & Definitions (20)

• Example 1
• Example 2
• Example 3
• Theorem 4
• proof
• Proposition 5
• proof
• Proposition 6
• proof
• Theorem 7