Better Electron Models Boost Materials and Machine Learning

Scientists are continually refining density functional theory, a cornerstone of modern electronic structure calculations used across chemistry, physics, and materials science. Fabien Tran from VASP Software GmbH, Susi Lehtola from the University of Helsinki, Stefano Pittalis from Istituto Nanoscienze, Consiglio Nazionale delle Ricerche, and Miguel A. L. Marques from Ruhr University Bochum, working in collaboration with all four institutions, have comprehensively reviewed semi-local exchange-correlation functionals, vital approximations used to model electron interactions within this theory.

This research is significant because, despite decades of development yielding hundreds of approximations, a clear and consistently organised overview of these semi-local functionals has been lacking, hindering both accessibility for newcomers and advanced application by experienced researchers. Their work consolidates historical developments and recent advances, offering a unified framework to guide future improvements in density functional approximations and broaden their application to complex scientific challenges. Six decades of effort have yielded hundreds of approximations to understand how electrons interact within materials — despite this extensive work, accurately modelling these interactions remains a central challenge in chemistry and physics. This review offers a unified framework to guide future development of these essential computational tools, and scientists have long relied on density functional theory (DFT) as a cornerstone of modern electronic structure calculations. A method with broad applications spanning chemistry, physics, materials science, and biochemistry. At the core of DFT lies the exchange-correlation functional, a mathematical quantity that accounts for the complex interactions between electrons within a material. To determine the exact form of this functional remains an unsolved problem, necessitating the use of approximations to make practical calculations feasible. Over the last six decades, hundreds of these approximations have been proposed, each offering a different balance between accuracy and computational demand.

Scientists have undertaken a detailed survey of semi-local functionals, a specific class of approximations including local density approximations, generalised gradient approximations, and meta-generalised gradient approximations, to consolidate existing knowledge and chart a path forward for future development. By beginning with the fundamental principles of Kohn-Sham DFT, this effort carefully examines the construction of these semi-local functionals, paying particular attention to the physical reasoning behind their design, the mathematical principles guiding their formulation, and the practical considerations influencing their use with diverse materials. Constructing accurate approximations is not simply a matter of mathematical elegance. The development of these functionals has involved a careful interaction between theoretical rigor and empirical adjustments, progressively refining their ability to predict material properties. Successive improvements, often described as rungs on “Jacob’s ladder”, have incorporated additional ingredients to enhance accuracy while preserving computational efficiency. This detailed review intends to serve as both an accessible introduction for those new to the field and a thorough reference for experienced practitioners, aiming to stimulate further advances in DFT approximations and their application to complex scientific challenges. Understanding the limitations of current approximations is vital for accurate modelling. While DFT has proven remarkably successful, its performance can vary markedly depending on the system under investigation and the chosen functional. By providing a unified framework for understanding the construction and application of semi-local functionals, this effort seeks to address these challenges and enable the development of even more accurate and reliable computational methods. For instance, the ability to accurately predict fundamental gaps, the minimum energy required to excite an electron. Remains a persistent challenge for many standard functionals. Here, this review thoroughly examines how these approximations incorporate information about the electron density — unlike simpler methods that only consider the density at a single point in space, semi-local functionals also account for its gradient and. In more advanced cases, the kinetic energy density. These additional ingredients allow for a more accurate description of the electron distribution and its impact on material properties. Beyond the theoretical foundations, The team also address practical considerations, such as the computational cost associated with different functionals and their suitability for various types of systems. Evolution of semi-local exchange-correlation functionals and limitations of historical benchmark datasets Through beginning with the foundational concepts of Kohn-Sham density functional theory, research details the construction of semi-local exchange-correlation functionals. Initial assessments of functional performance were often conducted on limited datasets, potentially restricting their generality from a modern viewpoint. These datasets were rather small, covering only a restricted portion of chemical space. Sometimes did not accurately reflect the performance of certain functionals. The project consolidates historical developments and recent advances in the field, providing a consistently organized discussion. At the core of The effort lies the exchange-correlation functional, a quantity encapsulating many-body effects arising from electron interactions. Meanwhile, the precise form of this functional remains unknown, necessitating computationally manageable approximations for practical applications. Through tracing the conceptual foundations back to The project of Thomas and Fermi — the electron density can serve as the fundamental variable for describing quantum systems. Slater’s application of Hartree-Fock exchange to atoms, molecules, and solids further advanced this idea. Here, the breakthrough arrived in 1964 with Hohenberg and Kohn. Demonstrating that the ground-state energy could be determined solely as a functional of the electronic density. Shortly after, Kohn and Sham proposed a method to determine this density and energy in a way that is, in principle, exact. In turn, the KS scheme, resembling the Hartree-Fock approximation, maps an interacting many-electron problem onto an effective single-particle problem defined by a local potential, expressed by the KS equation: −1/2∇² + vs(r) ψi(r) = εiψi(r). Here, the particle density for the ground state, n(r), is obtained from a single Slater determinant, where n(r) = Σσ Σi fiσ|ψiσ(r)|² — the ground state energy is then expressed as E[n] = Ts[n]+ ∫ d³r n(r)vext(r)+EHxc[n]. Meanwhile, the kinetic energy of the non-interacting KS system, Ts[n] = −1/2 Σσ Σi ∫ d³r fiσψ∗iσ(r)∇²ψiσ(r), constitutes the first term, and the notation [n] denotes a functional of the density n. At the same time, the solutions to the KS equation carry a self-consistent dependence on n. Evolution of exchange-correlation approximations within density functional theory Density functional theory calculations rely heavily on the exchange-correlation functional, a component representing many-body electron interactions. Since the precise form of this functional remains unknown, approximations are necessary for practical computation. Work began with the local density approximation (LDA) in the 1960s. This estimates the exchange-correlation energy at each point by referencing the homogeneous electron gas (HEG) at that local density. Despite its simplicity, LDA continues to be useful in certain applications. Then, generalised gradient approximations (GGAs) were proposed in the 1980s, incorporating the electron density gradient to account for variations in electron density. Early tests showed GGAs provided good agreement with experimental harmonic vibrational frequencies and atomization energies, establishing their potential. The implementation of DFT within the GAUSSIAN program broadened access to the method within the chemistry community, accelerating its adoption. Further refinement led to the development of meta-GGA (MGGA) functionals, which include kinetic-energy density alongside density and its gradient. These functionals aim to improve the description of electronic structure by considering the energy of motion of the electrons. Beyond meta-GGAs, research expanded to range-separated hybrids and double hybrid functionals, each introducing more complex elements to enhance accuracy. This effort focuses specifically on semi-local functionals, LDA, GGA. MGGA, as these remain essential for many calculations, particularly for large systems — by consolidating historical developments and recent advances, this effort intends to provide a unified framework for understanding the construction and application of these approximations. To support continued progress in computational chemistry and condensed matter physics. Systematic analysis clarifies strengths and weaknesses of density functionals For decades, the pursuit of better density functional approximations has felt like refining a ghost image. While density functional theory remains the dominant method for modelling materials, its accuracy hinges on a single, elusive component: the exchange-correlation functional. This recent review of semi-local functionals doesn’t promise a sudden breakthrough. But instead offers a carefully organised consolidation of existing knowledge, a task that is valuable in itself. The problem isn’t a lack of functionals, hundreds have been proposed. But a lack of systematic understanding of their strengths, weaknesses, and underlying principles. Once a tool largely confined to theoretical chemistry, these calculations now underpin materials discovery, drug design, and increasingly, machine learning applications. To achieve reliable predictions across diverse systems demands a functional that balances accuracy with computational cost. Many existing approximations struggle with strongly correlated materials or systems where van der Waals forces are important, necessitating more complex and expensive methods. The sheer number of options can be overwhelming, making it difficult for researchers to select the most appropriate functional for a given problem.

The team’ work isn’t simply a cataloguing exercise. By focusing on the physical motivations and mathematical underpinnings of functional development — they provide a framework for understanding why certain approximations work well in some cases and fail in others. Limitations remain in accurately describing excited states and charge-transfer processes, and the field needs more than incremental improvements; it requires a fundamental rethinking of how exchange-correlation effects are modelled. Future efforts will focus on combining semi-local functionals with machine learning techniques, while or on developing functionals that are explicitly designed for specific classes of materials. 👉 More information 🗞 Semi-Local Exchange-Correlation Approximations in Density Functional Theory 🧠 ArXiv: https://arxiv.org/abs/2602.17333 Tags: