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

TL;DR

This paper provides a rigorous numerical analysis of the single-reference coupled cluster method in quantum chemistry, establishing conditions for well-posedness and deriving error estimates for discretized solutions based on Hartree-Fock calculations.

Contribution

It introduces a new numerical analysis framework for the coupled cluster method using the invertibility of the Fréchet derivative, with error estimates and validation for small molecules.

Findings

Discrete coupled cluster equations are locally well-posed under certain conditions.

Error estimates for the discretized solutions are derived and validated.

Numerical experiments suggest the assumptions hold for small molecules.

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 Fr\'echet 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.