Improving density functional theory one flaw at time

Density functional theory (DFT) is a cornerstone tool of modern physics, chemistry, and engineering used to explore the behavior of electrons. While essential in modeling systems with many electrons, it suffers from a well-known flaw called self-interaction error. A recent study has identified a new area where a correction for this error breaks down.

The research team includes Professor J Karl Johnson and his graduate student Priyanka Bholanath Shukla from the University of Pittsburgh, along with John Perdew, professor and renowned theoretical physicist, and his graduate student Rohan Maniar from Tulane University, and Professor Koblar Alan Jackson from Central Michigan University.

This research, which strengthens DFT and has real-world implications in areas such as catalytic conversion, is published in Proceedings of the National Academy of Sciences (“Atomic Ionization: sd energy imbalance and Perdew-Zunger self-interaction correction energy penalty in 3d atoms”  DOI: 10.1073/pnas.2418305122).

Since its inception in the 1970s, DFT has been an incomplete but essential tool for scientists. “The theory has been improved over the years, but it has some flaws that many researchers overlook,” said Shukla, a PhD student in chemical and petroleum engineering at Pitt’s Swanson School of Engineering. “One flaw is the self-interaction error, which occurs when an electron interacts with itself.”

Professor John Perdew likens the self-interaction error to billiards. Electrons in a material should behave somewhat like billiard balls—the motion of one ball should change only due to interactions or collisions with others. Self-interaction error is like a billiard ball colliding with itself.

The problem arises because DFT thinks that the electron is interacting with another electron, which is in fact itself. This error can create inaccurate modeling. To help correct the error, Perdew, with fellow theoretical physicist Alex Zunger, developed a computational correction in 1981.

This advancement improved DFT, but areas persist where the self-interaction correction (SIC) can, as Johnson said, “get it wrong. And when you can find out where it doesn’t work, you can fix it.”

Through a grant with the U.S. Department of Energy, Perdew and fellow researchers have developed the FLOSIC (Fermi-Löwdin Orbital Self-Interaction Correction) Center. Scholars from five universities are working to identify problems with the SIC and develop solutions to improve DFT. For the past eight years, Pitt’s Johnson has been part of this team, and in 2021 he brought on Shukla.

A novel approach to finding an imbalance in transition metals

Recent research in DFT and SIC has focused on transition metals, which are crucial for catalysts, electronics, and development of new materials. Specifically, researchers have looked at how DFT handles different types of electrons, those in the outermost "s" orbitals and the more tightly bound "d" orbitals in metals such as chromium, copper, and cobalt.

A well-known issue in DFT is the sd energy imbalance, which is the relative error that DFT makes for the energy of d electrons when compared with s electrons. DFT needs to provide a balanced description of s and d electrons to accurately describe the energetics of transition metals.

Previous methods for measuring this imbalance relied on calculations for excited states, which are outside the formal domain of DFT and therefore problematic. This research, though, introduced a novel way to assess this imbalance using ionization energies (the energy needed to remove electrons from an atom).

Through their computational research, conducted in part at the University of Pittsburgh Center for Research Computing and Data, the team discovered that the Perdew-Zunger self-interaction correction method struggles to find the correct energy balance for s and d electrons. They found that a local scaling of the correction offers a much better balance by paring down the correction in regions of space where it can be predicted that little or no correction is needed.

This research identifies failures in existing SIC methods and paves the way for refining DFT. “Transition metals are essential in our lives, and as we increase the accuracy of modeling through density function theory, we can improve catalysis. We can design better catalysts,” said Johnson. “So much in our world depends on catalysis. Uncovering these flaws and fixing them has a real impact on everything from the food we eat to the technology we use every day.”

###