Density functional theory is the workhorse of modern electronic structure calculations, with wide-ranging applications in chemistry, physics, materials science, and machine learning. At its heart lies the exchange-correlation functional, a quantity which exactly encapsulates the many-body effects stemming from the quantum mechanical interactions between the electrons. Yet, the exact functional is unknown, and computationally tractable approximations are therefore necessary for practical applications. Over the past six decades, hundreds of density functional approximations have been proposed with varying degrees of accuracy and computational efficiency. This review surveys the theoretical foundations of semi-local functionals, including local density approximations, generalized gradient approximations, and meta-generalized gradient approximations. We provide a comprehensive, consistently organized discussion that consolidates both historical developments and recent advances in this field. Beginning with the essential concepts of Kohn-Sham density functional theory, we present the construction principles of semi-local exchange-correlation functionals. Special attention is given to the physical motivations underlying functional development, the mathematical properties that guide their construction, and the practical considerations that determine their applicability across different chemical and physical systems. This work is intended to serve as both a introduction for newcomers to the field and a comprehensive reference for practitioners. By consolidating the extensive literature on semi-local functionals and providing a unified framework for understanding their construction and application, we aim to facilitate further developments in density functional approximations and their use in tackling the diverse challenges of modern computational chemistry and condensed matter physics.
Density functional theory has become the workhorse of modern electronic structure calculations, with wide-ranging applications in chemistry, physics, materials science, biochemistry, etc. At its heart lies the exchangecorrelation functional, a quantity which exactly encapsulates the many-body effects stemming from the quantum mechanical interactions between the electrons. Yet, the exact functional is unknown, and computationally tractable approximations are therefore necessary for practical applications. Over the past six decades, hundreds of density functional approximations have been proposed with varying degrees of accuracy and computational efficiency.
This review surveys the theoretical foundations of semi-local functionals, including local density approximations, generalized gradient approximations, and meta-generalized gradient approximations. We provide a comprehensive, consistently organized discussion that consolidates both historical developments and recent advances in this field. Beginning with the essential concepts of Kohn-Sham density functional theory, we present the construction prin-ciples of semi-local exchange-correlation functionals. Special attention is given to the physical motivations underlying functional development, the mathematical properties that guide their construction, and the practical considerations that determine their applicability across different chemical and physical systems. For each class of functionals, we trace their evolution from early prototypes to modern sophisticated forms, highlighting key innovations and the interplay between theoretical rigor and empirical fitting. We examine how successive rungs of Jacob’s ladder, from the local density approximation through generalized gradient approximations to meta-generalized gradient approximations, incorporate additional ingredients to improve accuracy while maintaining computational efficiency.
This work is intended to serve as both a introduction for newcomers to the field and a comprehensive reference for practitioners. By consolidating the extensive literature on semilocal functionals and providing a unified framework for understanding their construction and application, we aim to facilitate further developments in density functional approximations and their use in tackling the diverse challenges Density functional theory (DFT) 1,2 has emerged as one of the most powerful and widely used computational methods in quantum chemistry and condensed matter physics due to its remarkable success in calculating ground-state properties, while maintaining a favorable balance between computational cost and accuracy. DFT has enabled unprecedented insights into electronic structure across an extensive range of systems. It is nowadays routinely used to guide experimental design and provide a fundamental understanding of chemical bonding and reactivity. The profound impact of DFT is evidenced by its widespread adoption across disciplines ranging from materials science to biochemistry.
The exchange-correlation (xc) energy functional 2,3 is at the heart of DFT. This is a functional of the electron density that embodies the quantum mechanical many-body effects of the electron-electron interaction. Although the exact form of the xc functional remains unknown, it can be successfully approximated by assuming a semi-local dependence on the electron density. Advancements in the development of such approximations have expanded the applicability and accuracy of DFT calculations. The careful balance between physical insight and (more or less) empirical parameterization in these functionals has enabled systematic improvements in the description of electronic structure, which have given rise to the celebrated status of DFT as an indispensable tool in modern computational science.
The development of xc functionals began with the local density approximation (LDA) in the 1960s, 2 relying on the known properties of the homogeneous electron gas (HEG). 4,5 The LDA is the simplest and most fundamental type of functional: It approximates the xc energy at each point in space by that of the HEG of that local density. LDAs perform remarkably well for many properties despite their conceptual simplicity, and still remain used nowadays in some applications.
Generalized gradient approximations (GGAs), which incorporate the electron density gradient to better describe inhomogeneous systems, were proposed in the 1980s. 6 GGAs were found early on to lead to good agreement with experiment for harmonic vibrational frequencies as well as excellent agreement for atomization energies, 7,8 laying the ground for their later popularity. Crucially, Pople’s implementation of DFT in the GAUSSIAN program 9 around the same time 10 made DFT accessible to the broader chemistry community, catalyzing its widespread adoption. This combination of improved accuracy through GGAs and increased availability through user-friendly software established DFT as a practical tool for
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