A quantum many-body system is called "strongly correlated" whenever the mean-field picture of the system is wrong, even on a qualitative level. In this case, the system is no longer accurately described by means of a single Slater Determinant (SD) from Hartree-Fock theory, but a (much) larger number of SDs is required to capture all quantum effects. Considering that the size of the Hilbert space grows exponentially with the size of the system, this can lead to problematic situations where the number of SDs required to accurately describe the system is way beyond reach of contemporary or even prospective computational resources. This situation is the case for many branches in physics or chemistry, for instance in magnetic spin systems, transition metal compounds, superconducting materials, atomic nuclei, etc... The task of the quantum many-body theorist is to pinpoint the essential degrees of freedom in the system, and design a quantum many-body method around these prerequisites. An important class of strong quantum correlations is given by pairing, in which each particle has a designated partner particle to which it is strongly coupled. Pairing correlations are ubiquitous in physics and chemistry, and have contributed to the understanding of, e.g. Lewis structures in molecules, Valence Bond dynamics in materials and Cooper pair condensation in superconducting materials.

The key observation here is that two paired fermions constitue a (hard-core) boson. This means that a system where pairing correlations are important is effectively bosonic by nature. Indeed, many of these systems can be accurately described by means of a mean-field theory where the fermion pairs are the elementary building blocks, rather than the fermions themselves. This is known as the theory of Geminals in molecular physics & quantum chemistry. Unfortunately, a Geminal theory is computationaly intractable in its most general formulation, and has therefore received little attention until very recently.

In the past years, it has been observed that the theory of Geminals has a strong connection with the theory of Richardson-Gaudin integrability. The peculiarity of integrable quantum systems is that they defy the exponential size explosion of the Hilbert space by virtue of a special symmetry in the system. As a result, they are exactly characterized by effectively factorisable (i.e. mean-field like) quantum states, despite the strong quantum correlations present in the system. In other words: integrable Richardson-Gaudin states can be interpreted as computationally tractable Geminal states, which has opened up a whole new research avenue in Geminal theory. Indeed, by means of a proper dressing of the Geminal state within the appropriate framework, our research unit (in collaboration with the Quantum Chemistry groups at UGent and McMaster University (Canada)) has been able to provide an accurate description of strongly correlated molecular systems at mean-field computational cost where Hartree-Fock theory failed.

However, as this research topic is very new, there are still many open research questions, including

. Benchmarking of the new Geminal methods for large (bio) molecular systems.
. Formulation of a variational Geminal theory for molecular systems.
. The incorporation of correlations that not in the pairing space

Goal

The goal of this thesis is to contribute to the development of the new Geminal theory for molecular systems. These contributions can be performed at many different levels, depending on the interests of the student. Until present, the majority of results have been obtained for systems that are within reach of exact diagonalisation methods (full Configuration Interaction) for benchmarking purposes. Obviously, the main potential asset of this new method is to provide an accurate quantum mechanical description of medium-sized molecular systems at low computational cost, so it is quintessential to confront the new method with (more expensive) established quantum chemistry methods, such as Coupled Cluster theory or Density Functional Theory. Technically inclined students will be more appealed by the variational formulation of the new Geminal method. This framework requires the development of a discrete-continuous optimization method on a set of disjoint manifolds. Considerable experience has been gained recently within our research unit for schematic quantum systems, however it is of utmost importance to transfer this experience to realistic molecular systems. Finally, a more mathematical angle is also possible for those students interested in theoretical extensions of the new Geminal theory. It is known that molecular systems are not completely characterized by pairing correlations only. Unfortunately, introducing non-paired correlations (such as triplet correlations) is reigniting the size explosion of the Hilbert space, and it is currently uncertain if a tractable Geminal theory including correlations outside the pairing space is possible. However, thanks to the link with Coupled Cluster theory, the possibility of an inclusive tractable Geminal theory may be real. Alternatively, missing correlations can always be captured by means a special adaptation of non-orthogonal perturbation theory. In both cases, good physical or chemical intuition will be of benefit for the development of applications of these methods.

Aspects
The engineering aspect of this thesis concerns the modeling, optimization, and computational investigation of physical and chemical systems, with direct application in (quantum) chemistry.

This research topic will be conducted in the framework of a strong international network and if possible the student will be actively involved in work discussions with collaborative partners.