Towards the development of a posteriori error estimates for the coupled cluster equations

Muhammad Hassan
Sorbonne Université
Laboratoire Jacques-Louis Lions

Coupled cluster (CC) methods are widely regarded as among the most effective algorithms for high precision resolution of the time-independent electronic Schrodinger equation in the non-static correlation regime. Despite their ubiquitous usage as \gold-standard" methods in quantum computational chemistry, the mathematical literature on the CC methodology is relatively scarce. Indeed, the numerical analysis of CC methods dates back only a few years to a pioneering article by Reinhold Schneider [1], and although the past decade has seen the emergence of further articles [2, 3, 4, 5] on the a priori numerical analysis of various flavours of CC methods, there remain several unresolved questions of fundamental mathematical importance, and practically no progress has been made towards deriving a posteriori estimates.
The goal of the current talk is to improve the state of the art on the numerical analysis
of the single reference coupled cluster method with the aim of laying out a potential
path towards a posteriori error estimates. We will begin by discussing the main limitation
of the existing a priori error analysis, namely, the diculty of establishing a guaranteed
local, strong monotonicity constant for the coupled cluster function. We will propose an
alternative strategy based on establishing direct invertibility of the coupled cluster Jacobian thereby showing that the coupled cluster function is locally invertible. We will show that this approach allows us to deduce explicitly characterisable constants which may be used for simple a posteriori error estimation, and we will briefly discuss possible strategies for the computation of such constants. Finally, we will describe challenges for deriving full, guaranteed error bars for the single reference CC method.

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