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.
In quantum chemistry, the main problem is to solve the molecular Schrödinger equation that models interacting electrons in a molecule. The most energetically stable configuration is called the ground state. However, there are solutions that correspond to higher energies as well. These are called excited states. Both ground state and excited states determine the properties of a molecule.
The most widely used approach to accurately solve the Schrödinger equation is the coupled-cluster method. The coupled-cluster method transforms the Schrödinger equation to a polynomial equation in the cluster operator, and where the cluster operator gives the state of the molecular system in the exact formulation.
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. We have also treated the degenerate case using the classic Leray reduction formula. In addition, motivated by the works of the chemists Kowalski and Piecuch, we have considered a homotopy that connects coupled-cluster mappings corresponding to different truncation levels. Using this, we have proved an existence result for the said homotopy, which also implies the existence of a truncated coupled-cluster solution under certain assumptions. For the truncated coupled-cluster method, we have derived an energy error bound for approximate eigenstates of the Schrödinger equation. This is, to the best of our knowledge, the first mathematical error bound for coupled-cluster theory not restricted to ground states but also applicable to excited states.
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.
The research project is currently combining mathematics, physics and chemistry and has so far been carried out at the Hylleraas Centre for Quantum Molecular Sciences including the principle investigator A. Laestadius and the post doc research fellow M. A. Csirik, as well as international (remote) collaboration with A. Krylov and P. Pokhilko.

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.