Kinetic energy density functional: advances, challenges, and future directions

Density functional theory (DFT) is widely used for electronic structure calculations, primarily utilizing the Kohn–Sham scheme. However, this approach reintroduces orbitals, resulting in a high computational cost. In contrast, the original orbital-free Hohenberg–Kohn DFT (OF-DFT) offers the potential for linear-scaling computations suitable for large systems. The advancement of OF-DFT relies on the development of an accurate and universal kinetic energy density functional (KEDF). This review surveys the progress, challenges, and prospects in KEDF development. It presents the essential physical and mathematical constraints that any KEDF must comply with, tracing the evolution from early models like Thomas–Fermi and von Weizsäcker to contemporary semi-local, nonlocal, and machine-learned approaches. While the developed KEDFs have improved the treatment of metals and some semiconductors, achieving transferable accuracy for molecules and systems with considerable density inhomogeneities remains a critical challenge. We highlight two emerging paradigms; the use of physics-guided machine learning to identify accurate KEDFs and information-theoretic approaches that provide deep insights. The path forward requires a renewed focus on fundamental physical constraints, steering the field away from purely empirical fitting toward a universal, computationally efficient KEDF that maximizes the advantages of OF-DFT. The main KEDFs are listed, including gradient expansions, enhancement-factor strategies, density-decomposition methods, and nonlocal KEDFs guided by linear-response theory.