REVIEW pubs.acs.org/CR Challenges for Density Functional Theory Aron J. Cohen,* Paula Mori-Sanchez,* and Weitao Yang* Department of Chemistry, Lensfield Road, University of Cambridge, Cambridge, CB2 1EW, United Kingdom Departamento de Química, Universidad Autonoma de Madrid, 28049 Madrid, Spain Department of Chemistry, French Family Science Center, Duke University, Durham, North Carolina, 27708, United States CONTENTS 3.1. Adiabatic Connection 300 1. Introduction 289 3.2. Methods for Minimizing Energy Functionals 300 1.1. What Are the Challenges for Density Functional 3.3. KohnÀSham and Generalized KohnÀSham Theory? 291 Eigenvalues 302 1.1.1. Challenge 1: To Develop a Functional That 3.3.1. Janak’s Theorem 302 Performs Uniformly Better Than B3LYP 291 4. Insight into Large Systematic Errors of Functionals 303 + 1.1.2. Challenge 2: The Need To Improve the 4.1. Stretched H2 and Delocalization Error 303 Description of Reaction Barriers and 4.1.1. Self-Interaction 303 Dispersion/van der Waals Interactions 291 4.1.2. Many-Electron Self-Interaction Error 303 1.1.3. Challenge 3: To Understand the 4.1.3. Fractional Charges 304 Significance of E[F]vsE[{ϕi,ɛi}], OEP, 4.1.4. Chemical Potential and Physical Meaning and Beyond 292 of the Frontier KS and GKS Eigenvalues 305 1.1.4. Challenge 4: Delocalization Error and 4.1.5. Fractional Occupations vs Ensemble 307 Static Correlation Error 292 4.1.6. Delocalization Error 308 1.1.5. Challenge 5: The Energy of Two Protons 4.2. Stretched H2 and Static Correlation Error 308 Separated by Infinity with One and Two 4.2.1. Static Correlation and Degeneracies 309 Electrons: Strong Correlation 292 4.2.2. Fractional Spins 309 2. The Entrance of DFT into Chemistry 292 4.3. Coming in from Infinity 310 2.1. ExchangeÀCorrelation Functionals 292 5. Strong Correlation 310 + 2.1.1. LDA 292 5.1. Errors for H2 and H2 at the Same Time 311 2.1.2. GGA 293 5.2. Fractional Charges and Fractional Spins 312 2.1.3. Meta-GGA 293 5.3. Derivative Discontinuity and Mott Insulators 313 2.1.4. Hybrid Functionals 293 5.4. Integer Nature of Electrons and the Right Form E 2.1.5. Recent Developments in for xc 314 Functionals 294 5.5. Contrast between DFT and Ab Initio Quantum 2.2. Performance with Respect to Chemistry 315 Chemistry 296 6. Conclusions 315 2.2.1. Thermochemical Data Sets 296 Author Information 315 2.2.2. Geometries 296 Biographies 316 2.2.3. Kinetics and Reaction Barriers 296 Acknowledgment 316 References 316 2.2.4. Hydrogen Bonding 297 2.2.5. Other Sets 297 1. INTRODUCTION 2.2.6. Response Properties 298 Density functional theory (DFT) of electronic structure has made 2.2.7. Performance of Density Functional an unparalleled impact on the application of quantum mechanics to Approximations 298 interesting and challenging problems in chemistry. As evidenced by 2.3. Dispersion and van der Waals Forces 298 some recent reviews,1À11 the number of applications is growing 6 2.3.1. C6/R Corrections and Other Simple rapidly by the year and some of the latest and most significant studies Corrections 298 2.3.2. Explicit van der Waals Functionals 299 Special Issue: 2012 Quantum Chemistry 3. Constructing Approximate Functionals and Received: April 5, 2011 Minimizing the Total Energy 299 Published: December 22, 2011 r 2011 American Chemical Society 289 dx.doi.org/10.1021/cr200107z | Chem. Rev. 2012, 112, 289–320 Chemical Reviews REVIEW include the following: the understanding and design of catalytic with a specified set of nuclei with charges and positions r and ZA A processes in enzymes and zeolites, electron transport, solar energy number of electrons N. The task is to simply minimize the energy harvesting and conversion, drug design in medicine, as well as many over all possible antisymmetric wave functions, Ψ(x ,x ,x ... x ), 1 2 3 N other problems in science and technology. where x contains the spatial coordinate r and spin coordinate σ . i i i The story behind the success of DFT is the search for the This enables us to find the minimizing Ψ and hence the ground exchangeÀcorrelation functional that uses the electron density state energy, E. However, technically, this is far from trivial, and to describe the intricate many-body effects within a single particle has been summarized by Paul Dirac in the following quote: formalism. Despite the application and success of DFT in many “The fundamental laws necessary for the mathematical branches of science and engineering, in this review we want to treatment of a large part of physics and the whole of focus on understanding current and future challenges for DFT. If chemistry are thus completely known, and the difficulty the exchange-correlation functional that is used was exact, then DFT would correctly describe the quantum nature of matter. lies only in the fact that application of these laws leads to equations that are too complex to be solved.” Indeed, it is the approximate nature of the exchangeÀcorrelation functional that is the reason both for the success and the failure of In DFT, the problem is reformulated in a philosophically and DFT applications. Early developments of DFT focused on the computationally different manner. The basic foundation of DFT most basic challenges in chemistry, in particular, the ability to is the Hohenberg Kohn theorem,12 which states that the have functionals that could give a reasonable description of both À external potential is a functional of the ground-state density. In the geometries and dissociation energies of molecules. The next other words, the density (an observable in 3D space) is used to major challenge for DFT arose from the need to accurately describe the complicated physics behind the interactions be- predict reaction barrier heights in order to determine the kinetics tween electrons and, therefore, determines everything about the of chemical reactions as well as to describe van der Waals interactions. Whether DFT can predict the small energy differ- system. As Kohn noted in his Nobel lecture, DFT “has been most useful for systems of very many electrons where wave function ences associated with van der Waals interactions or if additional methods encounter and are stopped by the exponential wall”.13 corrections or nonlocal functionals of the density are needed has 14 been the subject of much debate and current research. This In KohnÀSham (KS) theory, this is formulated as a simple expression for the ground state energy interaction, although one of the weakest, is key to the accurate understanding of the biological processes involved in many E½F¼Ts½FþVne½FþJ½FþExc½Fð3Þ drugÀprotein and proteinÀprotein interactions. All these challenges have been well-addressed by current where the forms of some of the functionals are explicitly known. developments, as described in the literature. However, it is our The kinetic energy for the KS noninteracting reference system is contention that there are even more significant challenges that 1 2 DFT, and specifically the exchange correlation functional, must s F ∑ Æϕ ∇ ϕ æ 4 À T ½ ¼ ij2 j i ð Þ overcome in order to fulfill its full promise. New and deeper i theoretical insights are needed to aid the development of new in terms of {ϕ }, the set of one electron KS orbitals. The electron i functionals. These are essential for the future development of density of the KS reference system is given by DFT. One way to facilitate this advance, as we will try to illustrate F r ϕ r 2 5 in this review, is to understand more deeply those situations ð Þ¼ ∑ j ið Þj ð Þ where DFT exhibits important failures. i An intriguing aspect of DFT is that even the simplest systems The other two known energy components are the nucleus can show intricacies and challenges reflecting those of much electron potential energy, expressed in terms of the external larger and complex systems. One example of this is the under- potential due to the nuclei, v(r)=À∑ (Z /|r À R |) standing encompassed in the widely used term, “strong correla- Z A A A tion” found in the physics literature. Strong correlation is meant ne½F¼ FðrÞ ðrÞ dr to refer to the breakdown of the single-particle picture, perhaps V v even of DFT itself, which is based on a determinant of single- and the classical electronÀelectron repulsion energy is particle KohnÀSham orbitals. However, it is essential to see it ZZ only as a breakdown of the currently used density functional 1 FðrÞ Fðr0Þ approximations. Strongly correlated systems offer significant new J½F¼ dr dr0 2 jr À r0j challenges for the functional. In this review we hope to demon- strate that the challenge of strong correlation for density func- Much is known about the key remaining term, the exchangeÀ tionals can be illustrated by the behavior of the energy of a single correlation functional, Exc[F], although no explicit form is avail- hydrogen atom. This understanding will help to realize the able. It can be expressed in the constrained search formulation for enormous potential of DFT. density functionals15 The Schr€odinger equation that describes the quantum nature Exc½F¼ min ÆΨjT þ VeejΨæ À Ts½FÀJ½F of matter is Ψ f F F F F F H^ Ψ ¼ EΨ ð1Þ ¼ðT½ ÀTs½ Þ þ ðVee½ ÀJ½ Þ It can also be expressed elegantly through the adiabatic where the Hamiltonian, H^ , for a Coulombic system is given by connection16,17 1 2 Z 1 Z H^ ¼∑ ∇ À ∑ A þ ∑ ð2Þ 1 2 i r r Exc½F¼ ÆΨλjVeejΨλæ dλ À J½F i iA j i À Aj i>j rij 0 290 dx.doi.org/10.1021/cr200107z |Chem.