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

Muhammad HassanYvon MadayYipeng Wang
Dec 2022
摘要
The central problem in electronic structure theory is the computation of theeigenvalues of the electronic Hamiltonian -- an unbounded, self-adjointoperator acting on a Hilbert space of antisymmetric functions. Coupled cluster(CC) methods, which are based on a non-linear parameterisation of thesought-after eigenfunction and result in non-linear systems of equations, arethe method of choice for high accuracy quantum chemical simulations but theirnumerical analysis is underdeveloped. The existing numerical analysis relies ona local, strong monotonicity property of the CC function that is valid only ina perturbative regime, i.e., when the sought-after ground state CC solution issufficiently close to zero. In this article, we introduce a new well-posednessanalysis for the single reference coupled cluster method based on theinvertibility of the CC derivative. Under the minimal assumption that thesought-after eigenfunction is intermediately normalisable and the associatedeigenvalue is isolated and non-degenerate, we prove that the continuous(infinite-dimensional) CC equations are always locally well-posed. Under thesame minimal assumptions and provided that the discretisation is fine enough,we prove that the discrete Full-CC equations are locally well-posed, and wederive residual-based error estimates with guaranteed positive constants.Preliminary numerical experiments indicate that the constants that appear inour estimates are a significant improvement over those obtained from the localmonotonicity approach.
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