## Introduction

Kohn-Sham density functional theory (DFT) provides a formally exact treatment of the ground-state electronic structure of many-electron systems. Practical DFT calculations must approximate the exchange-correlation (XC) density functional that captures all many-body effects.1 Development of accurate and computationally tractable approximate XC functionals has made DFT the preëminent electronic structure approximation in computational chemistry and solid-state physics. However, approximate XC functionals must always trade off accuracy versus computational cost. This perspective discusses our efforts toward a new compromise: functionals accurate enough for molecular thermochemistry and kinetics, and inexpensive enough to model solids and surfaces.

Perdew and Schmidt classify approximate XC functionals as “rungs” on a “Jacob's Ladder,” extending from Hartree theory to the “heaven” of chemical accuracy (Fig. 1).1 Higher rungs contain more complicated ingredients and can generally give higher accuracy. The first rung is the local spin density approximation (LSDA), in which the XC energy density at each point **r** is taken as that of a uniform electron gas (UEG) with electron density ρ(**r**) (spin dependence is suppressed for conciseness. All orbitals, densities, and related quantities are assumed to be spin). Semilocal generalized gradient approximations2, 3 (GGAs) incorporate . Meta-GGAs4 also include the density Laplacian and/or the kinetic energy density of the Kohn-Sham reference system . This reference system places *N* noninteracting Fermions in orbitals , with equal to the real system's electron density. “Semilocal” is used here to distinguish GGAs and meta-GGAs from the LSDA. Other authors use “local” to denote first- through third-rung functionals. Semilocal functionals satisfy many exact constraints5, 6 and are computationally cheap enough to model solids and surfaces. But they systematically over-delocalize electrons,7, 8 giving overestimated heats of formation9 and molecule-surface adsorption energies, underestimated band gaps,10, 11 underestimated reaction barriers in molecules12, 13 and surfaces, overestimated magnetic exchange couplings,14 and related errors.

Fourth-rung “hybrid” XC functionals compensate these errors by incorporating a fraction of nonlocal exact exchange15–17

γ is the one-particle density matrix of the Kohn-Sham reference system, an implicit density functional obeying . Admixture of a fraction of Eq. (1) tunes semilocal functionals' systematic overbinding.7, 8 This gives hybrid XC functionals a dramatically improved treatment of many properties.9, 12, 14

An instructive connection between semilocal and hybrid XC functionals comes from expressing a semilocal exchange functional in terms of a semilocal model for γ,

The semilocal exchange hole , is localized by construction around the reference point **r**. This approximates localization of the XC hole in chemical bonds. Such localization ensures that, for example, the electrons in singlet H_{2} have a high probability being on different H atoms.7, 8 Admixing a fraction of delocalized exact exchange tunes this approximation.7 Semilocal exchange functionals are typically expressed in terms of models for the exchange energy density or exchange hole, rather than the one-particle density matrix itself.3 However, Eq. (3) is useful in what follows.

Fourth-rung hybrids have (at least) two important limitations in applications to solids and surfaces. First, the optimal admixture of exact exchange differs for different properties: for thermochemistry,6, 15, 18 for kinetics,5 and 100% in one-electron regions where Eq. (1) is the exact XC functional. Second, slow decay of in makes evaluation of Eq. (1) computationally expensive19, 20 and formally problematic21 in metals. These limitations have motivated recent work on improved semilocal functionals,22–25 screened hybrids26–28 that cut off the long-range part of Eq. (1), algorithms for evaluating exact exchange,29 and hyper-GGAs30–33 incorporating different fractions of exact exchange in different regions.