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Density functional theory is constrained by a conundrum at its core: the universal exchange-correlation functional. U-M researchers are striving to unveil it

A fresh method for simulating molecules with quantum precision advances toward clarifying the equation at the heart of a widely-used simulation technique, employed in essential chemistry and materials science research.

The pursuit to understand materials and chemical reactions consumes approximately one-third of national laboratory supercomputer time in the United States. The benchmark for precision is the quantum many-body problem, which provides insight into occurrences at the level of individual electrons. This is crucial for understanding chemical and material behaviors since electrons govern chemical reactivity and bonding, electrical characteristics, and more. Nonetheless, quantum many-body calculations are so intricate that researchers can only apply them to compute atoms and molecules with a limited number of electrons at any moment.

Density functional theory, or DFT, is less complex—the computational demands for its calculations increase with the cube of the number of electrons, as opposed to escalating exponentially with each additional electron. Rather than tracking each separate electron, this theory computes electron densities—indicating where the electrons are most likely to be found in space. Consequently, it can be employed to simulate the dynamics of hundreds of atoms.

A significant challenge for DFT practitioners is the exchange-correlation functional, which outlines how electrons interact with one another, adhering to quantum mechanical principles. Until now, scientists have had to rely on approximations of the XC functional for their specific needs.

“We are aware that a universal functional exists—it remains consistent regardless of whether the electrons reside in a molecular structure, a chunk of metal, or a semiconductor. However, its precise form eludes us,” remarked Vikram Gavini, U-M mechanical engineering professor and corresponding author of the study in Science Advances.

Due to the significance of DFT for prospective materials alongside basic science, the Department of Energy allocated funding and supercomputer resources for the U-M team’s endeavor to approach that universal XC functional.

The researchers commenced by analyzing individual atoms and small molecules through quantum many-body theory to allow them to reverse-engineer the DFT issue. Instead of incorporating the approximate XC functional to model the electrons’ behavior in atoms and molecules, they determined—via machine learning—what XC functional would yield the behavior of electrons as derived from quantum many-body theory.

“Many-body theories provide accurate results for valid reasons, albeit at an exorbitant computational expense. Our team has adapted many-body findings into a more straightforward, quicker format that maintains most of its accuracy,” stated Paul Zimmerman, U-M chemistry professor, who led the quantum many-body calculations alongside chemistry Ph.D. student Jeffrey Hatch.

Zimmerman’s team assembled a training dataset consisting of five atoms and two molecules specifically: lithium, carbon, nitrogen, oxygen, neon, dihydrogen, and lithium hydride. They attempted to include fluorine and water, but these additions did not enhance the XC functional—the team is convinced that the quality was already at its peak by leveraging data from lighter atoms and molecules.

However, DFT computations utilizing that XC functional were already significantly superior to what was anticipated given its complexity level.

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DFT precision is illustrated as a series of steps within a ladder. In its simplest, initial-step form, the electrons are perceived as present within a uniform cloud. In the second-step version utilized by Gavini’s team, the electron cloud varies in density and is regarded as a gradient.

For the third step, investigators incorporate additional details about the electrons, such as their kinetic energies. This typically involves employing simplified representations of the intricate many-electron wavefunction, which can more effectively depict the behavior of the electrons. Nevertheless, by calculating a superior XC functional, Gavini’s team achieved third-step precisions.

“The application of an accurate XC functional is as varied as chemistry itself, particularly due to its material neutrality. It’s equally pertinent for researchers seeking enhanced battery materials as it is for those uncovering new pharmaceuticals or constructing quantum computers,” stated Bikash Kanungo, U-M assistant research scientist in mechanical engineering and the primary author of the study.

Researchers have the option to utilize the XC functional identified by the group directly or explore the team’s methodology. For example, Gavini mentions that they commenced with light atoms and molecules, and next, he intends to delve into solid materials.

Once more, the XC functional is anticipated to possess a universal form, yet the challenging aspect is determining what that form is. Will the XC functional discovered by his team perform effectively for solids? Would a newly computed functional for solids yield better results? And might they create a combined functional that operates efficiently for both categories of materials?

The additional enhancement the team aims to pursue is achieving higher precisions. This would entail not merely assessing electrons collectively, as electron densities, but rather incorporating the individual orbitals through which the electrons move. In that scenario, their technique of reversing the problem to obtain the XC functional becomes significantly more complex. Even with density gradients, they necessitated calculations on one of the largest supercomputers in the U.S., so this path would demand further computational time.

The study was financed by the Department of Energy (grant no. DE-SC0022241).
The Air Force Office of Scientific Research (grant no. FA9550-21-1-0302) provided supplementary support. Supercomputer time was granted by the National Energy Research Scientific Computing Center and Oak Ridge National Laboratory, DOE Office of Science User Facilities.

Gavini also holds a position as a professor of materials science and engineering.

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