—Muhammad Ali
My area of research lies at the interface of mathematics and quantum chemistry. More precisely, I study fermionic many-body
phenomena and the numerical methods to simulate them.
Due to the highly interdisciplinary nature of this problem, I employ two different research philosophies:
First, is an applied mathematics approach that casts modern quantum chemistry methods into a mathematical framework that
can be rigorously analyzed from either a numerical analysis perspective.
Second, a computational mathematics approach where I aim to understand physical systems at the boundaries of accessible computational
complexity, and to devise new numerical methods to push these boundaries forward.
Applied Mathematics
While the underlying mathematical theory of the fermionic many-body problem is relatively well understood, its governing equation (the many-body Schrödinger equation) remains numerically intractable, posing one of today's most notorious computational challenges. The main reason is an apparently irreducible complexity of real-world chemical systems which yields naive approximations to fail dramatically. At the same time, brute-force approaches are computationally infeasible due to the problem's exponential scaling, also known as the curse of dimensionality. As a result, state-of-the-art numerical methods in quantum chemistry are sophisticated, highly evolved numerical procedures that are consequently very far from being mere applications of generic numerical analysis paradigms; on the contrary, they are almost always a series of subtle ideas reaching well beyond basic numerical recipes. The number of approximative quantum chemistry methods that are at least somewhat mathematically understood is therefore extremely small compared to the richness of practically employed methods. With my research I aim at defining and analyzing electronic structure methods in a sound mathematical way.
Computational Mathematics
The increase of computational power and the compelling scientific advances of the past years have promoted computational chemistry,
in particular quantum chemistry, to a central branch of modern chemistry.
Quantum-chemical simulations are today routinely performed by thousands of researchers not only in chemistry but also in related fields,
complementing and supplementing painstaking laboratory work.
Important examples are the design of new compounds for sustainable energy, green catalysis, and nanomaterials.
Over the past century, numerous approximative methods have been developed with complexities that range from high (like density
functional theory) to very high (like configuration interaction), and hence numerical methods with reduced complexity are urgently
needed in order to simulate ever larger realistic systems.
My research has been mainly focused on two high-accuracy numerical approaches:
Coupled-cluster theory
Coupled cluster (CC) theory is arguably the most weirdly used wavefunction method 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 providing ∼1 kcal/mol accuracy (chemical accuracy) in calculations of 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 treated with the existing CC machinery.
Density matrix embedding theory
Density matrix embedding theory (DMET) is a quantum embedding theory designed to treat strong correlation effects in large quantum systems while maintaining reasonable computation costs. The idea behind DMET is that in complex systems the region of interest often forms merely one (small) part of a much larger system. It is therefore natural to think about numerically treating the system with two different approaches, high-level calculations on the active regions of interest and low-level calculations on the respective environments, and then `join' the obtained results together.