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

• URL: http://arxiv.org/abs/2212.12788v1
• Date: Sat, 24 Dec 2022 17:29:43 GMT
• Title: Analysis of the Single Reference Coupled Cluster Method for Electronic Structure Calculations: The Full Coupled Cluster Equations
• Authors: Muhammad Hassan, Yvon Maday and Yipeng Wang
• Abstract summary: We introduce a new well-posedness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Preliminary numerical experiments indicate that the constants that appear in our estimates are a significant improvement over those obtained from the local monotonicity approach.
• Score: 2.3271703838711972
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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