Density functional theory is limited by a mystery at its heart: the universal exchange-correlation functional. U-M researchers are trying to uncover it
Study: Learning local and semi-local density functionals from exact exchange-correlation potentials and energies (DOI: 10.1126/sciadv.ady8962)
A new trick for modeling molecules with quantum accuracy takes a step toward revealing the equation at the center of a popular simulation approach, which is used in fundamental chemistry and materials science studies.
The effort to understand materials and chemical reactions eats up roughly a third of national lab supercomputer time in the U.S. The gold standard for accuracy is the quantum many-body problem, which can tell you what's happening at the level of individual electrons. This is the key to chemical and material behaviors as electrons are responsible for chemical reactivity and bonds, electrical properties and more. However, quantum many-body calculations are so difficult that scientists can only use them to calculate atoms and molecules with a handful of electrons at a time.
Density functional theory, or DFT, is easier-the computing resources needed for its calculations scale with the number of electrons cubed, rather than rising exponentially with each new electron. Instead of following each individual electron, this theory calculates electron densities-where the electrons are most likely to be located in space. In this way, it can be used to simulate the behavior of many hundreds of atoms.
A key problem for DFT users is the exchange-correlation functional, which describes how electrons interact with each other, following quantum mechanical rules. So far, researchers have had to settle for approximating the XC functional for their particular application.
"We know that there exists a universal functional-it doesn't matter whether the electrons are in a molecular system, a piece of metal or a semiconductor. But we do not know what its form is," said Vikram Gavini, U-M professor of mechanical engineering and corresponding author of the study in Science Advances.
Because of the importance of DFT to future materials as well as to basic science, the Department of Energy provided funding and supercomputer time for the U-M team's quest to approach that universal XC functional.
The researchers started by studying individual atoms and small molecules with quantum many-body theory so that they could flip the DFT problem around. Instead of adding in the approximate XC functional to give the electrons' behavior in atoms and molecules, they figure out-using machine learning-what XC functional will give the electrons' behavior as calculated through quantum many-body theory.
"Many-body theories give us the right answer for the right reason, but at an unreasonable computational cost. Our team has translated many-body results into a simpler, faster form that retains most of its accuracy," said Paul Zimmerman, U-M professor of chemistry, who led the quantum many-body calculations with chemistry Ph.D. student Jeffrey Hatch.
Zimmerman's group created a training data set of five atoms and two molecules, specifically, lithium, carbon, nitrogen, oxygen, neon, dihydrogen and lithium hydride. They tried adding fluorine and water, but these additions did not improve the XC functional-the team believes that it was already as good as it was going to get by drawing from data on light atoms and molecules.
However, DFT calculations using that XC functional were already much better than expected for its level of complexity. DFT accuracy is described as a set of rungs in a ladder. In the most basic, first-rung form, the electrons are viewed as existing in a uniform cloud. In the second-rung version Gavini's team used, the electron cloud changes in density, viewed as a gradient.
For the third rung, researchers add more information about the electrons, such as their kinetic energies. This usually means bringing in simplified versions of the difficult many-electron wavefunction, that can better describe what is going on with the electrons. However, by calculating a better XC functional, Gavini's team was getting third-rung accuracies.
"The use of an accurate XC functional is as diverse as chemistry itself, precisely because it is material agnostic. It's equally relevant for researchers trying to find better battery materials to those discovering new drugs to those building quantum computers," said Bikash Kanungo, U-M assistant research scientist in mechanical engineering and first author of the study.
Researchers can use the XC functional discovered by the group directly or experiment with the team's approach. For instance, Gavini says that they started with light atoms and molecules, and next, he would like to explore solid materials.
Again, the XC functional is expected to have a universal form, but the tricky part is figuring out what it is. Will the XC functional his team discovered work well for solids? Would a new functional calculated for solids be more successful? And could they build a combined functional that worked well for both sets of materials?
The other improvement the team would like to pursue is higher accuracies. This would mean that instead of looking at electrons collectively, as electron densities, they would need to include the individual orbitals the electrons move in. In that case, their trick of inverting the problem to get the XC functional becomes a much harder calculation. Even with density gradients, they had to do the calculations on one of the biggest supercomputers in the U.S., so this avenue would require more computing time.
The study was funded by the Department of Energy (grant no. DE-SC0022241).
Air Force Office of Scientific Research (grant no. FA9550-21-1-0302) provided additional support. Supercomputer time was provided by the National Energy Research Scientific Computing Center and Oak Ridge National Laboratory, DOE Office of Science User Facilities.
Gavini is also a professor of materials science and engineering.
