Scientists are continually refining density functional theory, the basis 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 and Consiglio Nazionale delle Ricerche from the Nanoscience Institute, and Miguel AL Marques from Ruhr University Bochum, in collaboration with all four institutions, provided a comprehensive review of the semilocal exchange-correlation functional, a key approximation used to model electronic interactions within this theory. This work is important because, despite decades of development resulting in hundreds of approximations, a clear and consistently organized overview of these semilocal functions is lacking, hindering both their accessibility for novices and their advanced applications by experienced researchers. Their work integrates historical developments and recent advances, providing a unified framework to guide future improvements in density function approximations and broaden their applications to complex scientific problems.
Six decades of effort have resulted in hundreds of approximations to understanding how electrons interact in materials. Despite this extensive research, accurately modeling these interactions remains a central challenge in chemistry and physics. This review provides a unifying framework to guide the future development of these important computational tools, and scientists have long relied on density functional theory (DFT) as the basis for modern electronic structure calculations. A method with a wide range of applications spanning chemistry, physics, materials science, and biochemistry.
At the core of DFT is the exchange-correlation function, a mathematical quantity that describes the complex interactions between electrons in a material. Determining the exact form of this function remains an open problem, and approximations must be used to make practical calculations feasible. Over the past 60 years, hundreds of these approximations have been proposed, each offering a different balance between accuracy and computational demands.
Scientists have undertaken a detailed investigation of semilocal functionals, a particular class of approximations that includes local density approximations, generalized gradient approximations, and metageneralized gradient approximations, in order to consolidate existing knowledge and chart paths for future development. In this effort, we carefully consider the construction of these semilocal functionals by starting with the fundamental principles of the Kohn-Sham DFT, paying particular attention to the physical reasoning behind their design, the mathematical principles guiding their formulation, and the practical considerations that influence their use in diverse materials.
Constructing accurate approximations is not simply a matter of mathematical elegance. The development of these functions involves a careful interplay between theoretical rigor and empirical tuning, gradually refining the ability to predict material properties. Continuous improvement, often described as the rungs of “Jacob’s ladder,” has incorporated additional elements to increase accuracy while maintaining computational efficiency.
This detailed review is intended to serve as an accessible introduction for beginners in the field and as a thorough reference for experienced practitioners, with the aim of promoting further advances in DFT approximation and its application to complex scientific problems. Understanding the limitations of the current approximation is essential for accurate modeling.
Although DFT has proven to be a notable success, its performance can vary widely depending on the system under investigation and the features selected. This effort aims to address these challenges by providing a unified framework for understanding the construction and application of semilocal functions, enabling the development of more accurate and reliable computational methods.
For example, the ability to accurately predict the fundamental gap, the minimum energy required to excite an electron. It remains a persistent challenge for many standard functions. Here, in this review, we thoroughly examine how these approximations incorporate information about electron density. Unlike simpler methods that only consider the density of a single point in space, semilocal functionals also consider its gradient. In more advanced cases, kinetic energy density.
These additional components allow a more accurate description of the electron distribution and its influence on material properties. Beyond the theoretical foundations, the team is also addressing practical considerations such as the computational costs associated with different features and suitability for different types of systems.
Evolution of semi-local exchange correlation function and limitations of past benchmark datasets
Starting from the basic concepts of Kohn-Sham density functional theory, we study in detail the construction of semilocal exchange-correlation functionals. Initial assessments of functional performance are often performed on limited datasets, which can limit generality from a modern perspective. These datasets were fairly small and covered only a limited portion of chemical space. In some cases, it did not accurately reflect the performance of certain features.
This project integrates historical developments and recent advances in the field and provides a coherent and organized discussion. At the core of this effort is the exchange-correlation functional, a quantity that encapsulates the many-body effects resulting from electronic interactions. On the other hand, the exact form of this function remains unknown, and practical applications require computationally manageable approximations.
Tracing its conceptual foundations back to the Thomas and Fermi project, electron density serves as a fundamental variable for describing quantum systems. Slater’s application of Hartree-Fock exchange to atoms, molecules, and solids further advanced this idea. Here, a breakthrough was made by Hohenberg and Cohn in 1964. We show that the ground state energy can only be determined as a function of the electron density.
Shortly thereafter, Kohn and Shum proposed a method to determine this density and energy in an in-principle accurate manner. A KS scheme similar to the Hartree-Fock approximation then maps the interacting many-electron problem to a valid single-particle problem defined by the local potential expressed by the KS equation −1/2∇² + vs(r) ψi(r) = εiψi(r).
Here, the ground state particle density n(r) is obtained from a single Slater determinant. where n(r) = Σσ Σi fiσ|ψiσ(r)|² — The ground state energy is expressed as E.[n] = one[n]+ ∫ d³r n(r)vext(r)+EHxc[n]. On the other hand, the kinetic energy Ts of the non-interacting KS system[n] = −1/2 Σσ Σi ∫ d³r fiσψ∗iσ(r)∇²ψiσ(r) constitutes the first term and has the following notation: [n] represents a function of density n. At the same time, the solution to the KS equation has a consistent dependence on n.
Development of exchange-correlation approximation in density functional theory
Density functional theory calculations rely heavily on the exchange-correlation functional, a component that represents many-body electron interactions. The exact form of this function remains unknown, so approximations are required for actual calculations. Research began in the 1960s with local density approximation (LDA). It estimates the exchange-correlation energy at each point by reference to the homogeneous electron gas (HEG) at that local density.
Despite its simplicity, LDA remains useful in certain applications. Subsequently, the generalized gradient approximation (GGA), which incorporates an electron density gradient to account for variations in electron density, was proposed in the 1980s. Initial tests showed that GGA was in good agreement with experimental harmonic frequencies and atomization energies, establishing its feasibility.
The implementation of DFT within the GAUSSIAN program has broadened the access and accelerated the adoption of this technique within the chemical community. Further improvements led to the development of a meta-GGA (MGGA) functional that includes the kinetic energy density in addition to the density and its gradient. These functions aim to improve the description of the electronic structure by taking into account the kinetic energy of the electrons.
Beyond meta-GGA, research has been extended to range-separated hybrid and double-hybrid functionals, each of which introduces more complex elements to increase accuracy. This effort specifically focuses on semilocal functions, LDA, and GGA. Since MGGA remains essential for many computations, especially for large-scale systems, this effort aims to provide a unified framework for understanding the construction and application of these approximations by integrating historical developments and recent advances. Supporting continued advances in computational chemistry and condensed matter physics.
Systematic analysis reveals strengths and weaknesses of density functionals
For decades, the pursuit of better density function approximations has felt like refining ghost images. Density functional theory remains the dominant method for modeling materials, but its accuracy relies on a single, elusive component: the exchange-correlation functional. This recent review of semilocal functionals does not promise sudden progress. However, in return, existing knowledge can be carefully organized and integrated, which is a valuable task in itself.
The problem isn’t a lack of features; there are hundreds of features proposed. However, a systematic understanding of its strengths, weaknesses, and underlying principles is lacking. Once a tool primarily limited to theoretical chemistry, these calculations now support applications in materials discovery, drug discovery, and increasingly machine learning. Achieving reliable predictions across diverse systems requires a function that balances accuracy and computational cost.
Many existing approximations have difficulty dealing with highly correlated materials and systems where van der Waals forces are important, requiring more complex and expensive techniques. The sheer number of options can make it difficult for researchers to select the most appropriate function for a particular problem. The team’s job is more than just cataloging.
By focusing on the physical motivations and mathematical foundations of functional development, we provide a framework for understanding why certain approximations work well in some cases and fail in others. There remain limitations in accurately describing excited states and charge transfer processes, and more than incremental improvements are needed in this area. We need to fundamentally rethink how we model currency correlation effects. Future work will focus on the combination of semi-local functions and machine learning techniques, and at the same time on the development of functions explicitly designed for specific classes of materials.
