Analysis of the Single Reference Coupled Cluster Method for Electronic Structure Calculations: The Discrete Coupled Cluster Equations

TL;DR

This work develops a rigorous numerical analysis for the single-reference coupled cluster method with a Hartree–Fock reference, establishing local well-posedness and computable error estimates for discrete CC equations. The authors formulate CC via excitation and cluster operators, relate zeros of the CC function to intermediately normalised eigenfunctions of the electronic Hamiltonian, and prove invertibility of the CC Fréchet derivative using inf-sup arguments. They introduce two discretisation strategies—Full-CC and excitation-rank truncated CC—and show that discrete inf-sup stability holds under structural Assumptions B.I or B.II, enabling quasi-optimality and residual-based a posteriori estimates. Numerical considerations indicate the structural assumptions hold for several small molecules and yield improved constants over prior analyses, strengthening the reliability and certification of CC-based electronic-structure computations.

Abstract

Coupled cluster methods are widely regarded as the gold standard of computational quantum chemistry as they are perceived to offer the best compromise between computational cost and a high-accuracy resolution of the ground state eigenvalue of the electronic Hamiltonian -- an unbounded, self-adjoint operator acting on a Hilbert space of antisymmetric functions that describes electronic properties of molecular systems. The present contribution is the second in a series of two articles where we introduce a new numerical analysis of the single-reference coupled cluster method based on the invertibility of the coupled cluster Fréchet derivative. In this contribution, we study discretisations of the single-reference coupled cluster equations based on a prior mean-field (Hartree-Fock) calculation. We show that under some structural assumptions on the associated discretisation spaces and assuming that the discretisation is fine enough, the discrete coupled cluster equations are locally well-posed, and we derive a priori and residual-based a posteriori error estimates for the discrete coupled cluster solutions. Preliminary numerical experiments indicate that the structural assumptions that we impose for our analysis can be expected to hold for several small molecules and the theoretical constants that appear in our error estimates are an improvement over those obtained from earlier approaches.

Figures (2)

• Figure 1: Graphical depiction of the coupled cluster discretisation parameters. Informally, the continuous coupled cluster equations should appear at the upper right corner of this graphic.
• Figure 2: Graphical depiction of region of validity of Structure Assumption B.I (in green) and Structure Assumption B.II (in magenta). Note that the exact size of the magenta region of validity, i.e., whether it begins at the double-zeta level or quadruple-zeta level etc., or whether it includes quadruples or triples etc., depends on the properties of the mean-field operator (c.f., the proof of Lemma \ref{['lem:inf-sup']}).

Theorems & Definitions (50)

• Definition 2: Reference Determinant for an Occupied Space
• Remark 3: Uniqueness up to sign of Reference Determinant
• Definition 4: Complementary Decomposition of $N$-particle Function Space
• Remark 5: Properties of Complementary Decomposition of $N$-particle Function Space
• Definition 6: Slater Determinant Basis for $N$-particle function space $\widehat{\mathcal{H}}^1$
• Definition 8: Slater Determinant Basis for $\widehat{\mathcal{H}}^{1, \perp}_{\Psi_0}$
• Definition 9: Excitation Index Sets
• Definition 10: Excitation Operators
• Definition 11: De-excitation Operators
• Theorem 12: Properties of Excitation and De-excitation Operators