Interview with holders of the MaQuI high-risk, high-impact project

CNRS is one of five research organizations, along with CEA, Inrae, Inria and Inserm, to deploy a France 2030 “Research at Risk” program. This new program aims to identify and support bold scientific projects capable of bringing about major technological breakthroughs. Read our interview with the three leaders of the MaQuI project, which aims to accelerate the transfer of mathematical ideas to chemistry and physics, with notable impacts on quantum technologies and molecular-scale simulation.

Mathieu Lewin

Eric Cances

Julien Toulouse

● Hello, can you introduce yourself?

Mathieu Lewin : I'm a research director at the CNRS, and currently director of the Centre de Recherche en Mathématiques de la Décision (Ceremade) at Dauphine-PSL.

Éric Cancès: I'm a professor at the Ecole des Ponts - Institut Polytechnique de Paris, a researcher at CERMICS, the applied mathematics laboratory of the Ecole des Ponts, and a member of the Inria MATHERIALS project-team.

Julien Toulouse: I'm a professor at Sorbonne University (SU). I carry out my research at the Theoretical Chemistry Laboratory (SU/CNRS). I'm also in charge of the CNRS “GDR NBODY” Thematic Network.

● What are your areas of research?

Mathieu Lewin: I work on the theoretical aspects of quantum mechanics and statistical physics, mainly with tools derived from the calculus of variations and the theory of partial differential equations. Although I'm the most theoretical of the trio, I'm very attached to the physical meaning of the theorems I prove (or seek to prove). In my work, I almost always give priority to the initial physical question, over the mathematical tools that will need to be used to solve it. So, even though I'm mainly an analyst by training, I sometimes use different techniques, for example from probability theory.

Éric Cancès: I work mainly on electronic structure calculations for quantum chemistry, condensed matter physics and materials science. I'm interested in both theoretical and numerical aspects, and carry out interdisciplinary research with numerous collaborations with physicists and chemists. Electronic structure theory is an extremely well-developed field in physics and chemistry, but relatively unexplored by mathematicians, even though it abounds in exciting problems of a very varied nature, calling on tools from almost every branch of pure and applied mathematics. For example, while the Schrödinger equation describing electrons in a molecule can be written as a linear partial differential equation in very high dimension, its various approximations involve non-linear partial differential equations, optimization problems on Riemannian varieties, algebraic equations, or stochastic processes. To understand the electronic structure of a material, we sometimes need to use tools from algebraic topology, K-theory or dynamical systems theory.

Julien Toulouse: I'm working on the development of electronic structure calculation methods in quantum chemistry, to be able to quantitatively predict the properties of molecular systems. The challenge is to develop computational methods with controlled accuracy and computation time, applicable to all types of molecular systems and all types of properties. I work on various types of methods, but more particularly on density functional theory, which is one of the most widely used approaches. I'm the team's chemist, but my research is fairly theoretical, often at the interface between chemistry, physics and mathematics.

● Could you explain Schrödinger's equation in layman's terms, with an example if possible?

Schrödinger's equation will celebrate its hundredth anniversary next year, in 2025. It's an astonishing equation, because it fits on a single line and can, in principle, describe the precise microscopic behavior of any quantum system. The three of us are particularly interested in the electrons in atoms and molecules, as these are the lightest constituents, which we need to understand first. Solving Schrödinger's equation for electrons makes it possible to predict the precise behavior of a given molecule: its various equilibrium spatial configurations, the chemical reactions between them, its interactions with the environment and light, etc. Industrial applications are numerous, and include the pharmaceutical industry (discovery of new drugs) and new technologies (optimization of electronic components). The idea is to make predictions on the computer before embarking on costly experiments.

Unfortunately, it is often impossible to solve Schrödinger's equation directly, with the precision required for applications. Indeed, the number of variables to be optimized increases exponentially with the number of electrons, and we soon find ourselves having to manage an inordinate number of parameters. This means that, in most cases, we're forced to replace the Schrödinger equation with a simpler, less numerically costly model, while of course trying to lose as little predictive capability as possible.

To give a precise example, the water molecule_H2O contains 10 electrons, and we easily find ourselves working with billions of billions of parameters, even for such a small, perfectly known molecule.

● Could you popularize density functional theory? Could you share current examples of academic and industrial applications of density functional theory and examples of current patents?

Density functional theory is the technique most widely used in academic and industrial applications to replace the Schrödinger equation, as it provides the best compromise between accuracy and computational cost. It is a method that literally exploded in the 1990s and has now become indispensable. Its development was rewarded when Walter Kohn and John A. Pople were awarded the Nobel Prize in Chemistry in 1998.

To give an idea of the impact of this theory, the B3LYP model (derived from the work of Becke, Lee, Yang and Parr), which is among the most widely used in practice, is cited in over 350,000 research articles and 40,000 patents on Google Scholar. The applications are many and include lithium batteries, photovoltaic cells and energy materials in general, the design of new drugs, the study of porous materials, etc.

From a technical point of view, the idea is to get rid of the wave function, which is the unknown in the Schrödinger equation and a central element of quantum mechanics, and replace it with a simpler-to-handle unknown function called the electron charge density. The fact that this replacement is possible at all is astonishing, since it gives the impression of a loss of modeling capacity. It has been shown that this simplification is possible without any loss, when we seek to describe systems at equilibrium, i.e. in their “minimum energy state”.

● Could you give some concrete examples of possible future applications for each of the locks? What would be the scientific, technological or societal impact of each of the locks?

The aim of the project is to remove some famous locks in density functional theory, where this model malfunctions and does not provide predictions as reliable as in existing well-established applications, i.e. equilibrium situations.

The first concerns the case where the electrons move a little too much within the molecule, i.e. are far from equilibrium. This is what happens when they are excited by overly intense light, such as a laser. A fast-developing field is attosecond photochemistry, in which the movement of electrons is observed and controlled on the scale of a billionth of a billionth of a second. While experiments are in full development, numerical simulations are still imprecise in this field.

The second issue concerns the case where electrons reach high speeds, close to the speed of light. As is typically the case with heavy atoms, potential applications include heavy metal chemistry, toxicity and nuclear waste treatment. The challenge here is to take account of relativistic effects and interaction with the electromagnetic field of light.

Finally, the third challenge concerns the description of exotic materials that do not have a structure that repeats periodically in space. In particular, we want to study “moiré materials”, obtained by placing two periodic sheets one on top of the other, but rotated by an immeasurable angle. The case of graphene has recently received a great deal of attention, as it has been discovered that the double sheet possesses exceptional conduction properties (high-temperature superconduction, fractional quantum Hall effect, etc.). These are very difficult systems to model, as the interesting properties derive from very strong interactions between the electrons.

● How could the potential applications of this project be revolutionary and lead to major breakthroughs?

Our approach is to go back to the basics and try to remove the three locks described above with a mathematical approach.

It is a little-known fact that rigorous results have historically played a crucial role in the development of density functional theory. The most effective functionals (such as the B3LYP functional mentioned above) were developed on the basis of rather abstract mathematical results from the 1980s, in particular by Elliott Lieb. The Gauss Prize awarded to him at the last International Mathematical Congress mentions some of these results. Numerical analysis has also played a key role in optimizing calculation algorithms.

In all three cases, the mathematical structure is poorly understood, or has never even been studied. We believe that a complete overhaul of the mathematical formulation will enable us to build new, more efficient models. The expected advances are therefore far upstream, and mainly concern the modeling and simulation of these systems. If we are successful, however, we can hope to have a positive influence on real-life applications in the longer term.

● Why is the project risky? What difficulties have you identified? How do you perceive this risk?

This is a high-risk project because the problems addressed have largely resisted the efforts of physicists and chemists for decades. There is no guarantee that the mathematical results will be successful, nor that a more abstract approach will completely lift the existing barriers in physics and chemistry.

We have chosen to work with a highly interdisciplinary approach, which is quite innovative in this field of research. Any abstract idea will be immediately implemented numerically and tested on real systems, to determine its applicative potential. We won't wait until we've demonstrated that a given equation is mathematically sound before simulating it and testing its behavior on molecules. In this way, we hope to go back and forth between theory, numerical calculation and chemical results, in order to refine and adjust our approach. This organization is highly motivating, but also creates risk, due to the diversity of profiles that will be involved in the team. However, the project is designed so that the weight of interdisciplinarity falls mainly on the shoulders of the three team leaders. The young PhD students and post-docs will be working mainly in their respective fields, although their activities will be constantly compared, stimulated and adapted by the advances of others.

Mathieu Lewin: Risk is normally an integral part of research, and I personally don't hesitate to spend time on questions with a very uncertain outcome. Obviously, it's easier when you have a permanent position at CNRS... With the profusion of small calls for projects and the way in which individual research is evaluated (for example, when recruiting), it seems to me that we tend to favor research that produces results quickly, thus minimizing uncertainty. Risk is often seen as dangerous for young people, who have to publish in order to find a job. In my opinion, this has a negative impact on the quality of the research produced. The risk program therefore seems to me to be an excellent initiative. For this project, I don't see risk as a constraint, but rather as a stimulus.

Éric Cancès: I totally agree with Mathieu. Encouraging risk-taking in academic research seems to me to be a necessity. That said, the means to achieve this end are not obvious. As we saw at the round table to launch the “research at risk” program, the five institutes involved (CNRS, CEA, INRAE, Inria and Inserm) have opted for different - and in my view complementary - strategies to achieve this objective.

Julien Toulouse: Interdisciplinarity is often praised, but it can also be very risky, particularly mathematics/chemistry interdisciplinarity. The first risk for chemists is that of losing their way by trying to learn too much high-level mathematics. You need to know how to set the cursor at the right level to learn enough mathematics to be able to collaborate effectively with mathematicians, while remaining mainly within your own field of expertise. A second risk for chemists is that their research may be seen as too mathematical and too far upstream from concrete applications. In basic research, as elsewhere, time horizons are becoming shorter and shorter, which sometimes prevents us from seeing the long-term benefits of theoretical development upstream of applications.

● How did the team come together for this project? Have you worked together before, or on interacting or interdisciplinary math projects? What motivates you in this project?

We've known each other for a long time and have worked in pairs before, but never all three of us together. The project could be used to create a real interdisciplinary research team in Paris, one that would be visible and become a world leader in the field. The interactions between mathematics and quantum chemistry in Paris are already very well developed and, in fact, unique in the world. In fact, a genuine interdisciplinary community has been formed over the years, particularly within the “GDR NBODY” Thematic Network.

● How do you see mathematics in interaction?

Mathieu Lewin: There are many ways of doing mathematics in interaction. Personally, I spend a lot of time interacting and discussing with researchers from other fields, or reading articles from other disciplines. My personal work is strongly influenced by these interactions and I have published several articles in physics and chemistry journals. But little of this work has actually been carried out in direct collaboration with physicists or chemists. The project will therefore be an opportunity to take a further step in this direction.

Mathematical research is highly technical, and its products are often accessible to a very restricted community. It's very easy to get bogged down in a highly specialized subject that very few people can understand in detail. By interacting with other disciplines, we can help put mathematics back in the context of an open-minded approach to the outside world, which can only be beneficial. Many important theories have been inspired by applications.

Éric Cancès: I've been lucky enough to work with many physicists and chemists throughout my career. Around a third of my articles are the result of these collaborations and have been published in journals in disciplines other than mathematics. I very much appreciate the interactions with colleagues from other disciplines, which have introduced me to other ways of thinking and deepened my understanding of certain contemporary scientific issues (energy production and storage, drug design, quantum technologies, etc.).

I would like to stress, however, that setting up a fruitful interdisciplinary collaboration from scratch takes a great deal of time. It requires first clearing up misunderstandings about what it means to be a mathematician in the 21st century, building a common language, and identifying problems that are both interesting for both disciplines and not completely out of reach, which would lend themselves well to such a collaboration. One of the strong points of our trio is that, thanks to our regular interactions over the last 15 years, Julien, Mathieu and I have acquired this common language, which makes our exchanges considerably smoother.

Julien Toulouse: I'm convinced that great advances in quantum chemistry are possible thanks to mathematics. Indeed, the mathematical framework of quantum chemistry is subtle (infinite-dimensional spaces), and developments by quantum chemists are often slowed down and limited by a lack of mastery of various difficult mathematical aspects. In this context, collaborations between chemists and mathematicians can be very fruitful.

Although I've been interacting regularly with mathematicians for a long time, I really started collaborating with mathematicians, including Éric, about 5 years ago. This collaboration has made new developments in quantum chemistry possible, and has also encouraged me to think more mathematically, which has a very positive impact on my research. I'm delighted that this can continue and be greatly developed as part of the MaQuI project.

Like Eric, I believe that successful mathematics/chemistry collaboration often requires years of prior interaction to learn a common language. In the beginning, it's particularly difficult for a chemist by training, even one with a mathematical bent, to dialogue effectively with a mathematician. So I think it would be useful to develop mathematics/chemistry interactions early on in training.

● Do you have a message to get across? Is there a particular topic you'd like to address?

Mathieu Lewin: I'd like to thank the CNRS for selecting our project and thus supporting research at the interface between mathematics and quantum chemistry.

Éric Cancès: I'm delighted that three highly motivated young PhD students and post-docs have already joined our project. Training a new generation of researchers and teacher-researchers working at the interface between mathematics and quantum physics and chemistry, a prolific field that is central to applications, is a priority for me.

Julien Toulouse: I'd like to thank the CNRS for supporting this kind of mathematics/chemistry collaboration. A crucial point in developing this kind of project is to have young theoretical chemists who are sufficiently well trained in mathematics and physics. In this context, I'm concerned by the trend towards a reduction in the hourly volume of mathematics and chemistry-physics courses, including theoretical chemistry, in Bachelor's and Master's degree courses in Chemistry at French universities. There is a real risk, in the medium to long term, that we will no longer be able to train theoretical chemists in France, and that we will lose our mastery of the tools used in theoretical chemistry.