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

TL;DR

This paper introduces a new well-posedness analysis for the single reference coupled cluster method in electronic structure calculations, extending understanding beyond perturbative regimes and providing improved error estimates.

Contribution

It develops a novel well-posedness framework based on CC derivative invertibility, applicable under minimal assumptions, and offers enhanced residual-based error estimates for discretized equations.

Findings

Continuous CC equations are always locally well-posed under minimal assumptions.

Discrete Full-CC equations are locally well-posed with sufficiently fine discretization.

Numerical experiments show improved constants in error estimates over previous approaches.

Abstract

The central problem in electronic structure theory is the computation of the eigenvalues of the electronic Hamiltonian -- an unbounded, self-adjoint operator acting on a Hilbert space of antisymmetric functions. Coupled cluster (CC) methods, which are based on a non-linear parameterisation of the sought-after eigenfunction and result in non-linear systems of equations, are the method of choice for high accuracy quantum chemical simulations but their numerical analysis is underdeveloped. The existing numerical analysis relies on a local, strong monotonicity property of the CC function that is valid only in a perturbative regime, i.e., when the sought-after ground state CC solution is sufficiently close to zero. In this article, we introduce a new well-posedness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Under the minimal assumption that the sought-after eigenfunction is intermediately normalisable and the associated eigenvalue is isolated and non-degenerate, we prove that the continuous (infinite-dimensional) CC equations are always locally well-posed. Under the same minimal assumptions and provided that the discretisation is fine enough, we prove that the discrete Full-CC equations are locally well-posed, and we derive residual-based error estimates with guaranteed positive constants. Preliminary numerical experiments indicate that the constants that appear in our estimates are a significant improvement over those obtained from the local monotonicity approach.