White Paper: Advancing Quantum Mechanics with Mathematics and Statistics

The white paper summarizes the activities and outcomes of the Long Program “Advancing Quantum Mechanics with Mathematics and Statistics” which was held at the Institute of Pure and Applied Mathematics (IPAM) from March 7 to June 10, 2022. It also briefly explores some of the current open questions and future directions in the field of electronic structure theory and computational chemistry as well as related fields that were discussed during the program.

Field Theory Approaches: One challenge in quantum mechanics for large systems is connecting established methods across scales. Quantum Field Theory (QFT) provides a natural framework to derive existing methods from first principles, highlighting common approximations and a pathway to more accurate ones. This framework allows the description of collective and long-range effects in large complexes. In this context, the development of a rigorous effective field theory approach and the analysis of the light-matter interaction beyond the semiclassical approximations in large molecular systems have been identified as key open questions.

Excited and open quantum systems: Multiscale approaches to quantum mechanics need to describe collective degrees of freedom arising from large time and length scales. Open quantum system theory offers a productive way to describe excited electronic states, especially in the context of light-matter interactions. This accurate description is a key aspect both for cavity molecular quantum electrodynamics and large biomolecular complexes. The development of rigorous, accurate, and efficient methodologies for multiscale modeling across optical, electronic, and vibrational degrees of freedom has been envisaged as one of the main directions for future research.

Embedding theories: Embedding theories provide an efficient way to combine computational methods of different scales to treat large quantum systems. Within this program, a variety of embedding methods were discussed and scrutinized. Particular attention has been given to density matrix embedding theory, for which a numerical remedy for the chemical potential fitting problem has been developed. The implementation thereof is important as it extends the numerical applicability of embedding methods to larger and more complicated systems.

Response functions in molecules and solids: The number of materials and molecules grows combinatorially with the elements and building blocks considered. Dominating quantum effects can often be recovered with response functions describing how electronic structure changes under external influences. Participants devised new methods to obtain some of these response functions relevant to interacting systems and many-body interactions. Open challenges include obtaining high-quality response functions and the systematic improvement over current limitations of quantum mechanical calculations.

Non-covalent Interactions: Macroscopic properties of matter are chiefly determined by intermolecular interactions. In large systems, non-covalent interactions are highly non-local with van der Waals (vdW) dispersion being of particular importance. Due to its long-range nature and complex quantum many-body character, an accurate treatment of the vdW dispersion is currently challenging for large systems. During the program an interdisciplinary effort addressed the current lack of accurate and efficient methods capturing vdW dispersion interactions. In addition to the development of novel numerical approaches, this multidisciplinary perspective allows for a better understanding of the physical origins of non-covalent interactions.

Machine Learning for Quantum Mechanics: Quantum mechanical problems are high-dimensional and often nonlinear. If machine learning methods can yield models of relevant accuracy, more applications and systems become feasible. The program explored multiple complementary methods that aim to describe long-range interactions and improve transferability by introducing physical constraints directly. The participants identified interpretability of models, inclusion of functional derivatives, and treatment of large-scale many-body systems as some of the key open questions.

Additional Long Program Activity – 2D Moire Materials Workshop: Since its isolation by Geim, Novoselov, and collaborators in 2004, single layer graphene has drawn intense interest for its remarkable physical and theoretical properties. Graphene has the highest known tensile strength, and conductivity. More recently, theoretical studies have focused on the mathematical modeling of twisted bilayer graphene (TBG), a metamaterial constructed by stacking two layers of graphene and then rotating one layer with respect to the other. This has led to the development of models such as the Bistritzer-Mcdonald model (2011) and the Chiral model proposed by Tarnopolsky and coworkers (2019). The working group on 2D materials focused on reading and understanding the most recent mathematical results on the spectral properties of TBG and strained graphene at magic angles.