An act as simple as stepping on the surface of the Earth without falling through the ground is owed to interactions taking place at scales nearly incomprehensible to the human mind. The atoms making up the ground—atoms roughly 10 billionths of a centimeter in diameter—contain electrons that push against each other and maintain its solid state. Beyond the ground, intricate interactions of matter at the molecular and subatomic (or quantum) levels inform the behavior of everything in the universe, from macroscopic-level properties such as solidity to more intricate ones such as conductivity, magnetism, stability, and optical properties.
Understanding the influence of electron interactions on material properties, especially under varied conditions, offers valuable insight into material behavior. This knowledge can be applied to comprehend the interiors of stars, fusion ignition, molecules optimized for drug development, pollutants, and plutonium research to bolster national security. Researching these systems on small scales can lead to big breakthroughs.
To do so, researchers lean on quantum mechanics. (See S&TR, July/August 2024, Revolutionizing Quantum Science) Quantum mechanics is a field of physics focused on mathematically describing how all subatomic particles behave, not only electrons. Notably, it explains that such particles have both wavelike and particlelike characteristics. Unlike matter at the macroscale experienced in everyday life, at atomic scales some measurable quantities—particularly the energies of bound electrons—have discrete values. Electrons surround a nucleus in a cloud (orbital) existing in a probabilistic set of locations at discrete energies. “Electrons are at such a small scale that they don’t have simple trajectories like planets orbiting a star,” says Livermore computational physicist John Pask. “They can only have certain energies, and quantum mechanics must be used to understand materials in terms of their electronic interactions.”
Simplifying Complexity
In 1926, Erwin Schrödinger developed the concept of a “wave function,” which still defines the fundamental aspects of quantum mechanics. According to this concept, the wave function acts as a mathematical “probability map” for a particle’s energy state, position, and momentum, representing a fundamental mathematical description of the particle. This description can be used to determine a particle’s physical state, which in turn determines how particles such as electrons arrange within a material, ultimately determining that material’s properties.
Although this and other quantum mechanical equations are generally well understood, calculating wave functions is extremely computationally intensive due to the exponential increase in complexity with the number of atoms and electrons in a system. In this way, density functional theory (DFT) has had a tremendous effect on quantum mechanical calculations. Formulated by Walter Kohn in 1965 after first ideations began in the 1920s, DFT simplifies quantum calculations by formulating interactions in terms of the electron probability density, or electron density for short. Instead of accounting for the spatial coordinates of each electron, DFT determines a density function that describes the probability of finding any of a system’s electrons at a given point in space. The theory is based on a Hohenberg–Kohn theorem positing a one-to-one correspondence between a system’s electron density and external potential. Electron density is a function of position, and DFT states that a system’s energy is a function of electron density. Therefore, determining electron density enables the determination of a range of energy-based properties—such as the forces acting between atoms—that researchers can use to simulate a system’s evolution in time. As a result, quantum mechanical interactions are now defined in terms such that their cost increases less rapidly—often cubically rather than exponentially.
DFT achieves the best balance between accuracy and cost of computation compared to the other existing approximations to the full quantum mechanical many-body problem, or the challenge of determining the behavior of a system with multiple interacting components. Although introduced to describe electrons in atoms, DFT is generally applicable to any multiparticle subatomic system, from electrons in atoms up to atoms in molecules and down to protons and neutrons within an atomic nucleus. “We’re interested in DFT because it’s close to the real thing in terms of quantum mechanics, so we’re not adding in so many assumptions,” says Nir Goldman, a Livermore computational physical chemist. “Consequently, we can do predictive calculations. We can construct and simulate a system, add temperature and pressure, and then DFT supplies the energies and forces on atoms, which are used in classical calculations of the material response over time.”
Kohn and chemist John Pople shared the 1998 Nobel Prize in Chemistry for developing DFT. Today, codes using DFT are the most-run codes on supercomputers, and the Laboratory has been leading the charge in supercomputing for years. Recently, El Capitan ushered in the exascale era (see S&TR, December 2024, Introducing El Capitan), which, along with ongoing developments in high-performance computing (HPC) and machine learning (ML), enables advanced use of DFT to tackle problems of national importance and more. “The reason for DFT’s wide use is its great predictive power combined with its optimal balance between computational cost and physical accuracy,” says Pask.
Among many applications, Lawrence Livermore computational physicists apply DFT to uncover more information about elusive systems, from ultrawide bandgap materials (see S&TR, December 2023, Ultrawide Bandgap Materials in the Spotlight) to predictive simulation tools for batteries and dielectric materials (see S&TR, March 2016, Understanding Materials at the Nanoscale) to superionic ice’s role in planets’ magnetic fields (see S&TR, December 2018, Defending the Vulnerable Power Grid). DFT calculations work in tandem with experimentation, providing an understanding of microscopic behavior to create a complete picture of a system. The simulations also offer a glimpse into systems that aren’t suitable for experimentation. Says Livermore physicist Sebastian Hamel, “DFT calculations are important for exploring conditions in which experiments are challenging. In certain cases, we can only describe material properties through numerical experiments with DFT.”
Out-of-This-World Applications
Searching for potentially life-supporting molecules on other planets (astrobiology), as well as studying the composition of planets themselves, are examples of research areas in which extreme conditions make experimentation challenging. Humans cannot visit these planets to look for signs of life, much less go inside them to study their internal processes in real time. A high-power laser at a facility such as the National Ignition Facility (NIF) can replicate internal planetary conditions here on Earth for experimental probing. DFT, however, enables the prediction of atomic behaviors in planets using the quantum mechanical properties of the materials involved. “Suppose we have a liquid mixture impacting a planet, and the liquid mixture contains raw materials,” says Goldman. “Could that impact form life-building compounds? We’ve been able to investigate questions like that with DFT.”
Hamel and his team have been investigating exotic phases of matter under planetary conditions, particularly the superionic phase in which matter remains solid even under extremely high pressures and temperatures. For example, planets such as Uranus and Neptune each contain roughly 50,000 times as much water internally as Earth’s oceans. In 1998, researchers used DFT to predict the existence of a superionic phase of water under the pressure and temperature conditions inside these ice giants. Under such conditions, water’s oxygen atoms remain solidlike on a crystalline lattice while the hydrogen atoms flow freely. This prediction guided experiments in 2018 by Livermore’s Marius Millot, Federica Coppari, and their team to finally measure the structure of superionic water. “DFT predicts the properties of a phase and provides the means to tailor experiments to reach those conditions,” says Hamel. Adds Pask, “We can do many more computational experiments in a given amount of time than we can perform physical experiments, so DFT is a powerful screening tool.”
Yet, the work to understand superionic water was not finished. Hamel recently incorporated ML into DFT codes to better simulate large system sizes and more accurately determine the phase boundaries of this material, seeking improved understanding of the ice giant planets in Earth’s solar system and beyond. His new simulations shed light on the interior structures, evolution, and magnetic fields of the planets while also guiding further experimentation to corroborate the models. Says Hamel, “Physics often works the other way, right? Experiments usually lead the way, and then DFT is used to try to explain what’s going on. This example shows how that method was flipped.” Adds Goldman, “In astrobiology, we’ve seen a similar phenomenon. We have on occasion made predictions for experiments that led to experimental discovery. Since using first principles makes relatively few assumptions about the chemistry and physics of the problem, these numerical experiments are well-connected to reality and can inform laboratory experiments.”
In addition to predicting never-before-seen phenomena, DFT’s screening power narrows down experiments in astrobiology as researchers look for water and life-building compounds such as ammonia and methanol on other planets. Shock compression is the usual technique to simultaneously pressurize and heat a system and form products, but Goldman wondered if any specific pressures and temperatures might yield a prebiotic result. Without simulations to show which conditions are best suited to generate the desired outcome, finding prebiotic products could take ages. DFT simulations, however, can monitor a system’s chemistry at multiple temperatures and pressures and quickly define a set at which relevant results may be yielded. “Say we simulate the shock of a specific mixture up to 300,000 atmospheres and 3,000 Kelvin, and we find that amino acids form. That’s the experimental hook right there,” says Goldman. “Then, experimentalists can try to replicate what we’ve simulated as closely as possible.”
Exploring Elusive Elements
Critical to Livermore’s mission is the study of plutonium and other stockpile-relevant metals, such as uranium—specifically, how they behave under different conditions, succumb to radiation damage, age, and corrode. Since these materials are extremely difficult and impractical to handle for experimentation, DFT is an appealing solution for researching them in a fraction of the time and effort. The path to applying DFT to these elements and other actinides has not been straightforward, however, as they are notoriously complex. “For a long time, people discarded DFT as being simply wrong for plutonium because of some experimental evidence that was not replicated exactly as DFT calculations would expect,” says Livermore physicist Per Söderlind. “This work with plutonium was extremely controversial for years and is mostly resolved now.”
A remnant sticking point in DFT for plutonium has been as simple as magnetism. For ferromagnetic materials that exhibit strong, permanent attraction such as iron, cobalt, and nickel, DFT is remarkably effective at predicting magnetic properties as observed in experiments. However, plutonium is not ferromagnetic—in fact, scientists long assumed plutonium was not magnetic at all based on experimental probing that yielded weak or no signs of magnetism. Calculations with DFT firmly predicted magnetism in plutonium, which was not confirmed with neutron scattering experiments until 2015, decades after first proposed. “This example demonstrates how powerful a tool DFT is,” says Söderlind. “Even if researchers think they understand the scenario, they may not. The theory may point to something different.”
In 2024, Livermore researchers studied plutonium experimentally in a magnetic field with a conclusion that was later verified by the DFT model. With magnetism confirmed, Söderlind has used DFT to incorporate this property into new delta-phase plutonium models. Delta-phase plutonium—one of six different structural forms of the element—is stable between 310 and 452°C and preferred for applications due to its improved softness and ductility over the room-temperature alpha phase. The inclusion of dynamic magnetism, that is, rapidly fluctuating magnetic moments, as suggested by the 2015 neutron scattering experiments is critical to accurately predict plutonium’s thermodynamic properties and better explain existing anomalies in thermal expansion and bulk modulus of the material. This information provides greater confidence in modeling and a more accurate prediction of how plutonium reacts under conditions of Laboratory interest and persists over its lifetime.
Such breakthroughs motivate continuing the evolution of DFT approximations to remain at the theoretical cutting edge. The computational boon provided by ML has enabled improvement to exchange correlation functionals—the only approximations in DFT, which are used to account for parts of electronic energy. The result is more accurate modeling, which is important for plutonium and other systems that are historically difficult to predict. “Per (Söderlind) has done so much work to be able to use theory to elucidate what is going on in plutonium and other actinides at the lowest atomic level,” says Pask. “He did so, as he alluded to, amid quite a lot of controversy and while getting a lot of grief from experimentalists. That he was right in the end and that he used DFT as the means to get there was a triumph.”
Yet to be resolved is the inability for DFT to describe rare-earth elements, which encompass all the lanthanides plus scandium and yttrium. Rare earths are crucial for national security and consumer electronic purposes. (See S&TR, April/May 2024, Advancing Rare-Earth Biomining for a Secure Supply) Continued evaluation of potential DFT calculation constraints may help untangle the mysteries of rare earths and enable researchers to separate and refine them more effectively, with potential to support a strong national rare-earth materials supply.
Aiming Higher
Pask specializes in achieving more accuracy with DFT codes while minimizing computational demand—crucial because DFT calculations use a significant amount of HPC resources daily—and enabling their application in systems beyond those with just a few atoms. The Laboratory hosts one of the world’s most powerful supercomputers in El Capitan, which can support calculations far larger and more complex than other computers. However, the code must scale to match. “John’s (Pask’s) work allowing DFT codes to scale to one of the world’s largest computers is a significant advancement unique to the Laboratory,” says Goldman.
DFT improvements proceed along the two major axes of its methodology: accuracy and complexity. Pask and collaborators at the Georgia Institute of Technology (Georgia Tech) have been working to improve the accuracy of DFT calculations by improving the exchange-correlation functionals, which account for uncaptured complicated many-electron effects to better represent quantum mechanics in the models. Improving these approximations seems like an obvious solution, but one that comes with major increases in computational cost. The team’s SPARC code enables efficient and accurate solution of the DFT equations, making them capable of incorporating many exchange-correlation functionals and amenable to calculations for systems of different spins, sizes, and symmetries. “We’ve been able to make higher levels of exchange and correlation practical and accessible to use, bringing a higher accuracy of DFT to more and more realistic systems,” says Pask.
Improvement on the system complexity front seeks to model as many atoms as possible. This work required breaking down the DFT equations and addressing a particular matrix step that scales as the cube of the number of atoms in the system being modeled, rapidly becoming prohibitively expensive as the number of atoms in the system increases. A major advance occurred with the advent of order-N methods for large-scale, complex simulations, which exploit the locality of electronic interactions to reduce the scaling from cubic with the number of atoms to linear.
For a time, these methods were fragile and limited to insulating systems that do not enable the flow of electrical current. Pask and Georgia Tech collaborators developed and implemented the Spectral Quadrature method to generalize and further break down the calculations such that they can now accurately and efficiently calculate the electronic structure of any system: metallic, semiconducting, and insulating alike. “Not only can we calculate for metals or insulators, but we can do it in linear scaling time, and that allows us to scale DFT all the way from hundreds of atoms to a million at a fraction of the computational cost,” says Pask. “Once we achieve that scale, more work than we could have ever dreamed becomes possible with DFT.”
Spectral Quadrature enables the accurate evaluation of electronic density, energy, and atomic forces in both insulating and metallic systems. Computing these parameters enables computation of a range of materials properties, including diffusivity, viscosity, and conductivity. The ability to compute such properties in both metallic and insulating systems is crucial; both types of materials are used in applications such as lithium-ion batteries, which, with further understanding and improvement, have the potential to make alternative energy sources more practical in the future.
Nuclear Know-How
Beyond studying the electronic behavior of systems, some research applies DFT to scales even tinier than the diameter of an atom, looking within a nucleus at its protons and neutrons to understand what characterizes each element and isotope. Electronics research operates at sizes around 10-10 meter. Studying the structure of the nucleus, however, works at a scale of about five orders of magnitude smaller. “Something like seven to eight thousand atomic nuclei are supposed to exist in nature,” says Nicolas Schunck, the deputy group leader of Livermore’s Nuclear Data and Theory group. “Only about 50 percent of them, perhaps even less, have been observed either in nature or in laboratory experiments. The other ones are predicted by theory.”
Theoretical models for nuclei follow the rules of quantum mechanics while also incorporating information about the forces between protons and neutrons. Therefore, researchers can predict properties such as nuclear energy and mass, radius, decay rates, discrete energy levels, and more. The problem with standard theory is that the direct methods for extracting this information from a nuclear system are only scalable to the lightest of nuclei—roughly up to carbon or oxygen—representing a tiny subset of the nuclei observed within the periodic table and an even smaller fraction of those expected to exist.
While experiments can determine force information for larger nuclei, they are expensive and difficult to bring to fruition. DFT once again comes to the rescue to scale the calculations of nuclear systems and bolster the computational research side of the coin, which validates and produces higher quantities of results than experimentation. Improved computational capabilities have allowed realistic DFT calculations across the breadth of isotopes, including even the heaviest ones.
The benefits of studying nuclei are twofold, having impacts on both fundamental science and Laboratory mission applications. As studies improve, researchers use DFT to describe superheavy nuclei beyond what has been observed within the periodic table. Some seek physics outside of the currently understood standard model. Nuclear DFT also has applications in astrophysics, offering information about the structure of neutron stars and the complex nuclear reactions that take place inside stars and in star explosions. For Livermore specifically, DFT can model the fission process more accurately, potentially aiding in nuclear material signatures detection, nuclear reactors design, and stockpile modernization, proving valuable in plutonium work. Says Schunck, “The potential of nuclear DFT to provide detailed information about the structure and reaction properties of atomic nuclei is still largely untapped.”
DFT has been systematically applied in nuclear systems only within the last 15 years, making up a small fraction of the community applying the theory. However, it has already enabled the prediction of quantities that were only modeled phenomenologically (not derived from first principles) until now. The combination of raw computational power in the exascale era with advances in AI and ML will accelerate these calculations to enable more rapid discovery in the many areas of nuclear systems research. “The devil is in the details. The mathematical methods in DFT are similar whether used for electronic systems, for nuclei, or for atoms and molecules,” says Schunck. “I think that is the appeal of the whole idea—we have one DFT to rule them all.”
Faster for the Future
Whether DFT is an effective tool is no longer in question. The technique has been brought to bear on the most extreme and crucial of systems, both electronic and nuclear, with great success. With the exascale era only beginning, computational power will continue pushing DFT—and the researchers developing and employing it—to new heights. Application of the theory is particularly poised for improvement with ML, which is taking hold across disciplines and, unsurprisingly, can be applied to DFT calculations to lower computational cost and improve accuracy. For instance, a quantum molecular dynamics simulation models a system of atoms moving and calculates forces at each step in the evolution. A machine can learn and eventually predict what forces are associated with each configuration, enabling these simulations to take place a hundred to a thousand times faster while maintaining chemical accuracy. “Machine learning increases both the length and the time scales we can access,” says Pask. “We can think in terms of billions of atoms instead of thousands or millions. We can simulate in hours instead of months. DFT calculations also remain necessary to train ML models for particular materials and conditions of interest and to provide checks that required accuracies are achieved.”
AI can also add functionality and lower the barrier to DFT calculations for users with less knowledge in code, enabling the theory to enhance the work of even noncomputational scientists. “Someone knowledgeable about the theory but uncertain about how to use a specific code can almost certainly dive in with the help of an AI agent, learn a lot, and possibly even make modifications for their work,” says Goldman.
Equipped with higher computational power and ever-evolving codes, the computational physicists writing and applying DFT codes show no signs of slowing down. DFT’s trajectory is unlike that at any previous point in the theory’s history, battling back from controversies across fields and proving itself to be an impactful technology with much more yet to be uncovered. “I would say that all of us are sold on DFT,” says Söderlind. Adds Goldman, “The technique is anchored in reality enough to not be going anywhere at this point. DFT will remain the gold standard for condensed matter physics and condensed matter chemistry.”
—Lilly Ackerman
For further information contact John Pask (925) 422-8392 (pask1 [at] llnl.gov (pask1[at]llnl[dot]gov)).
