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FRINATEK-Fri prosj.st. mat.,naturv.,tek

Error estimates for coupled-cluster methods, ground states and excited states

Alternative title: Feilestimater for coupled-cluster metoder, grunntilstander og eksiterte tilstander

Awarded: NOK 8.0 mill.

The CCerror research project aims at improving the understanding of quantum chemistry by studying the coupled-cluster method with the aim to obtain (i) a mathematical analysis of excited states, and (ii) a detailed analysis of approximation errors since the exact solutions are never obtained in practice. The objective of the project is to mathematically study the coupled-cluster method and formulate a theory that can describe excited states as well as the ground state. The idea is to build on previous analyses and obtain quantitative error estimates that are accessible to the quantum chemistry community. This means having constants in the mathematical analysis that relate to properties of the system. So far we have worked on understanding truncations of the cluster operators better, since this is how approximations are introduced in practical calculations. Here a useful tool is the homotopy approach, where a more well-understood problem can be continued to the more difficult, targeted problem and where the process is tracked by a parameter. This can be used to connect a linear and variational formulation - and thereby simpler - version of the underlying Schrödinger equation to the coupled-cluster approach. We have analyzed the nonlinear equations of the single-reference coupled-cluster method using topological degree theory. Our results include existence proofs and qualitative information about the solutions of these equations that also sheds light on some of the numerically observed behavior. We have showed that the topological index of the coupled-cluster mapping is connected with the nonvariational property of the method and the eigenvalues of the Fock operator. Our result have now been published as a two-part series in the journal Mathematical Modelling and Numerical Analysis. During the last year we have again focused on a property called the (S+)-condition that we believe can shed new lights on the coupled-cluster method, especially when excited states are being considered. This property is a generalization of strong monotonicity, and where the latter is the key ingredient of Prof. Reinhold Schneider's pioneering work on mathematical analysis of coupled-cluster theory. However, a mathematical theory using strong monotonicity can only describe the (non-degenerate) ground state. We have again during the last year worked on a theory that establish a generalization to excited states and is based on the (S+)-condition. Our approach requires a slightly altered formulation of the standard theory and we are still working on a first publication outlining this approach. We have worked, in particular, with Mathias Oster on this topic. The PI of CCerror is planned to visit Oster during December 2023. We are still also working on extending the mathematical theory for the TCC method also to include excited states. Early in 2023, the PI of the project was invited to give a seminar at the Sanibel Symposium hosted by the University of Florida. This talk presented some recent mathematical advances in coupled-cluster theory and focused on CCerror’s result on the use of homotopies and topological degree theory. The PI also together with Dr. Fabian Faulstich wrote an article based on what was presented during the symposium. This article addressed the understanding of the multitude of solutions provided by the coupled-cluster polynomial equations using homotopy approaches. The article took a perspective from applied mathematics, where new interests in these approaches has emerged using both topological degree theory and algebraically oriented tools. This article provides an overview of describing this development. Furthermore, our analysis so far has also included the development of a criterion, or diagnostic, that guarantees that a quantum chemical computation gives a correct solution. Here the strong monotonicity of the coupled-cluster function that defines the problem has been studied. So far we have obtained a diagnostic that has better statistical correlation than previous suggested methods. One of the remaining issues of our diagnostic is to better understand its scaling with system size. The reason is that one wants a criterion that is independent of the number of electrons in the molecule, such that the diagnostic can provide general guidelines for quantum chemical calculations. During 2023 we published or results on this topic in the article “The S-diagnostic - an a posteriori error assessment for single-reference coupled-cluster methods” in Journal of Physical Chemistry A in the special issue “Early-Career and Emerging Researchers in Physical Chemistry Volume 2”. In this article, our suggested diagnostic is numerically scrutinized testing its performance for geometry optimizations, and electronic correlation computations for systems of varying numerical difficulty for single reference coupled-cluster methods. The research project is currently combining mathematics, physics and chemistry.

In quantum chemistry, the main problem is to solve the molecular Schrödinger equation that models interacting electrons in a molecule. The most stable configuration is called the ground state. However, there are solutions that correspond to higher energies as well. These are excited states. Both the ground state and excited states determine the properties of a molecule. Since the solutions depend on the position of each particle, the complexity of solving this equation increases as the system size increases. Approximations or truncations are therefore needed. One popular approach is the so-called coupled-cluster parametrization, a non-linear Galerkin approximation that makes use of an exponential ansatz. I intend to provide a mathematical analysis of this method in the following way; (i) put the coupled-cluster approach for excited states on firm mathematical ground, and (ii) since the exact solutions are never obtained in practice, provide a detailed analysis of the truncation error in the coupled-cluster approach. The objective is to go beyond existing a priori error estimates and establish quantitative error estimates that are more accessible to the quantum chemistry community. Furthermore, the second objective also includes the development of a criterion, or diagnostic, that guarantees that a quantum chemical computation using the coupled-cluster method gives a unique and correct solution. The importance of this project is that it would provide a sound mathematical foundation for widely applied approaches in quantum chemistry. It could also offer new insights concerning the practical use of these methods. In particular, successfully establishing a theoretically justified criterion that a coupled-cluster computation finds the correct solution would have large potential benefits and impact. The proposed research is an interesting example of interdisciplinary science.

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FRINATEK-Fri prosj.st. mat.,naturv.,tek