New materials and catalysts have the potential to transform everything from food production and medicine to batteries and stainless steel. Computer simulations help scientists narrow down promising candidates before testing them in a lab, but those simulations can be unreliable, especially for molecules that contain multiple metals.
A team at the University of Pittsburgh has developed a practical fix that scientists can use to get better results. The work was led by first author Abdulrahman Zamani, a postdoctoral associate in Kay Carter-Fenk's group in the Department of Chemistry in the Kenneth P. Dietrich School of Arts and Sciences, along with undergraduate researcher Andrew Ricciuti. They collaborated with John Keith and his PhD student Barbaro Zulueta in the Swanson School of Engineering. The work could immediately improve the data used to train machine-learning and artificial intelligence (AI) models.
The team’s research “ Improving coupled cluster theory for strongly correlated molecules with Kohn-Sham density encoding ” (DOI: 10.1038/s41467-026-77540-x ) is published in the journal Nature Communications .
Imperfect approximations
“To predict how atoms interact and undergo reactions, we use techniques like coupled cluster [CC] theory and density functional theory [DFT],” said Keith, professor of chemical and petroleum engineering . “Both are useful, but they model problems differently.”
CC begins with a picture in which each electron moves through the average field created by all the others. CC then refines that picture step by step by adding back the missing energetics of how pairs of electrons interact, then groups of three, and so on. For many molecules, CC is considered the “gold standard” of quantum chemistry, but its use is restricted because its computational cost rises steeply the bigger the molecule.
DFT is a more efficient approach because it begins with a simpler picture of the overall density of the electrons. However, DFT makes several cost-cutting approximations and uses parameters that researchers have not yet perfected, making results sometimes inaccurate.
Unfortunately, both CC and DFT perform poorly with abundant transition metals such as iron, cobalt, and chromium. Chromium, an essential ingredient in stainless steel, is famously difficult to model accurately.
“Chromium is a fairly small metal atom. Its electrons are especially crowded into compact orbitals like residents in ‘tiny apartments,’ and modeling those crowded electrons is tricky,” said Keith, who also directs the Keith Lab in Computational Chemistry .
Keith and Zulueta were researching ways to model chromium to understand how to make better stainless steels for the Navy, and they went deep into the literature. Zulueta said, “The community loves using coupled cluster when it can, but it’s not used with chromium because the crowded electrons make the method break. DFT is used instead with mixed success, but we wanted something even more accurate.”
“We had all this empirical evidence around us, but like other researchers, we didn’t understand why,” Keith said. “Then we asked our colleagues what they thought, and things became beautiful.”
Where the electrons are matters
Zulueta and Keith contacted Kay Carter-Fenk and their team, who are experts in modeling crowded electrons, or what is called “the strong correlation problem.”
Together, they analyzed data and saw what others had seen: that DFT sometimes but not always brings a better starting picture of crowded transition metal electrons than what is normally used with CC. “It was puzzling that other researchers had claimed that this was because DFT gave better orbitals, but CC doesn’t care what the individual orbitals look like. It’s a bit like describing a room: you can use different sets of directions to say where things are, but the room itself doesn’t change,” said Carter-Fenk, an assistant professor of chemistry. “It had to be due to the underlying math.”
Once that clicked, the team found out how to make the approach reliable and achieve the high standard of “chemical accuracy” for many difficult combinations of atom pairs that standard CC and DFT model poorly. Getting reliably high accuracy on those systems normally requires weeks or months of super-computing time, but using CC in this way got almost as good accuracies in minutes to hours.
The key to bringing it together was a simple diagnostic developed in Carter-Fenk’s group that identified when a DFT method would be useful. “We created a mathematical model that finds the difference between the two starting densities and distills that difference into a number,” said Zamani, first author of the paper. “The magnitude of that number can indicate whether the DFT starting point is beneficial or not.”
Stronger data, better training
While the research could help to uncover new materials and catalysts in the long-term, it has immediate implications for developing machine-learning potentials.
“Most machine-learning models are trained on DFT data alone. And DFT data, especially for strongly correlated systems that contain chromium, is hit or miss. It misses more than it hits,” Carter-Fenk said. “Our approach helps ensure researchers will get more reliable data.”
“We now have a pathway to make coupled cluster accessible to metals, and that will lead to improved machine-learning methods and better models for complex systems like stainless steel and heavy lanthanides and actinides,” said Keith.
The team hopes that these improvements will help drive the discovery of new materials and catalysts that could lead to a cleaner, more sustainable energy future.
Nature Communications
Computational simulation/modeling
Not applicable
Improving coupled cluster theory for strongly correlated molecules with Kohn-Sham density encoding
11-Sep-2026
The authors declare no competing interests.