—Muhammad Ali
Our research group focuses on the intersection of mathematics and quantum chemistry, with a particular emphasis on fermionic many-body phenomena and the numerical methods used to simulate them. Due to the highly interdisciplinary nature of this field, we employ two distinct research philosophies:
Firstly, we use an applied mathematics approach that translates modern quantum chemistry methods into a mathematical framework, allowing for rigorous analysis from a numerical perspective.
Secondly, we adopt a computational mathematics approach to understand physical systems at the limits of computational complexity and to develop new numerical methods that extend these boundaries.
Applied Mathematics
While the underlying mathematical theory of the fermionic many-body problem is relatively well understood, solving its governing equation (the many-body Schrödinger equation) remains numerically intractable, presenting one of today's most formidable computational challenges.
This difficulty arises from the seemingly irreducible complexity of real-world chemical systems, which causes naive approximations to fail dramatically.
At the same time, brute-force approaches are computationally infeasible due to the problem's exponential scaling, known as the curse of dimensionality.
Consequently, state-of-the-art numerical methods in quantum chemistry are sophisticated, highly developed procedures that go well beyond basic numerical analysis paradigms.
These methods often comprise a series of nuanced ideas far removed from generic numerical recipes.
The number of approximative quantum chemistry methods that are at least somewhat mathematically understood is thus extremely small compared to the vast array of methods used in practice.
Within our research group we aim to define and analyze electronic structure methods in a rigorous mathematical framework, bridging the gap between theoretical understanding and practical application. To that end employ techniques from differential geometry as well as algebraic geometry.
Computational Mathematics
The increase in computational power and the significant scientific advances of recent years have elevated computational chemistry, particularly quantum chemistry, to a central branch of modern chemistry.
Today, quantum-chemical simulations are routinely conducted by thousands of researchers across chemistry and related fields, complementing and enhancing meticulous laboratory work.
Key applications include the design of new compounds for sustainable energy, green catalysis, and nanomaterials.
Over the past century, numerous approximate methods have been developed, with complexities ranging from high (such as density functional theory) to very high (such as configuration interaction).
Consequently, there is an urgent need for numerical methods with reduced complexity to simulate ever-larger realistic systems.
My research group has primarily focused on three high-accuracy numerical thursts:
Coupled-cluster theory
Coupled cluster (CC) theory is arguably one of the most widely used wavefunction methods in computational chemistry. It has played a revolutionary role in establishing a new level of accuracy in electronic structure calculations and quantum chemical simulations. Thirty years of active development have resulted in a variety of CC methods capable of achieving chemical accuracy (∼1 kcal/mol) in calculating the electronic correlation energy for chemical systems with up to hundreds of correlated electrons. Reliable geometry optimization, thermochemical and spectroscopic predictions, weak interaction modeling, and other theoretical problems can be accurately addressed using the existing CC framework.
Density matrix embedding theory
Density matrix embedding theory (DMET) is a quantum embedding method designed to handle strong correlation effects in large quantum systems while maintaining reasonable computational costs. The underlying concept of DMET is that, in complex systems, the region of interest often constitutes only a small part of a much larger system. Therefore, it is natural to approach the problem by treating the system with two different levels of calculation: high-level calculations for the active regions of interest and low-level calculations for their respective environments. The results are then `joined' together to provide an accurate overall description.
Quantum computing
Quantum computing is an emerging field poised to revolutionize the way we handle complex high-dimensional computations that are ultimately intractable for classical computers.
By leveraging the principles of quantum mechanics, quantum computing processes information in fundamentally new ways, opening new lines of research to explore innovative computational approaches.
This is particularly transformative for solving high-dimensional eigenvalue problems, such as those arising in electronic structure calculations, enabling us to tackle previously unsolvable challenges in this domain.
Our research involves creating advanced quantum algorithms that optimize these high-accuracy calculations, enabling more precise simulations of electronic structures. These algorithms are designed to be scalable, ensuring they can handle increasingly larger and more complex systems as quantum hardware continues to improve. As we push the boundaries of what is computationally feasible, we aim to bring about breakthroughs in quantum chemistry, materials science, and beyond, ultimately transforming how these fields approach simulation and problem-solving.