Status and Challenges of Density Functional Theory

Density functional theory is in principle exact, but its success depends on improvement and refinement of the exchange-correlation functionals. This is being accomplished by using improved functional forms with more flexibility and more ingredients and by optimization against broader databases. The goal has been to obtain more universally applicable functionals that are simultaneously accurate for as many properties as possible. Recent optimizations also place a premium on smoothness that diminishes problems with self-consistent-field iterations and grid-size convergence.Density functional theory for ground-state properties has been mainly improved in recent work by optimizing functionals with simultaneously good performance for both main-group and transition-series chemistry, including both bond energies and barrier heights. This can be especially important, for example, when treating catalysis in metal–organic frameworks.Density functional theory for excited-state properties has been mainly improved in recent work by optimizing functionals with simultaneous good performance for valence excitations, Rydberg excitations, and charge-transfer excitations. We discuss some of the challenges facing density functional theory (DFT) and recent progress in DFT for both ground and excited electronic states. We discuss key aspects of the results we have been able to obtain with the strategy of designing density functionals to have various ingredients and functional forms that are then optimized to accurately predict various types of properties and systems with as much universality as possible. Finally, we make specific recommendations of approximate density functionals that are well suited for particular kinds of applications. We discuss some of the challenges facing density functional theory (DFT) and recent progress in DFT for both ground and excited electronic states. We discuss key aspects of the results we have been able to obtain with the strategy of designing density functionals to have various ingredients and functional forms that are then optimized to accurately predict various types of properties and systems with as much universality as possible. Finally, we make specific recommendations of approximate density functionals that are well suited for particular kinds of applications. Recent advances in the development of Kohn–Sham DFT (KS-DFT) (see Glossary) have focused on obtaining more accurate and universal density functionals. KS-DFT has been successful in describing numerous properties of atoms, molecules, and solids, and very large systems can be treated accurately at an affordable computational expense. Nevertheless, KS-DFT suffers from limitations. In practical applications of the original theory, these may all be considered to arise from the need to approximate the exchange-correlation functional. Most approximate functionals suffer from self-interaction error, delocalization error, or both and their task is made particularly difficult by the need to make up for the representation of the density as a single-configuration reference wave function that corresponds to noninteracting electrons. Probably the single most influential tactic for improving DFT has been the addition of new ingredients, with each new ingredient leading to improved or broader accuracy. Some historical examples of this have been the introduction of density gradients, Hartree–Fock (HF) exchange, kinetic energy density, various ways to include nonlocal correlation, and range separation. We have attempted to use a combination of ingredients and flexible functional forms to design universal density functionals and we have also proposed some specific-purpose functionals with the intent of understanding what ingredients of a functional affect which properties. In this review, we summarize some recent progress we have made in improving DFT and advancing toward the goal of achieving universality. Because universality is only partially achieved, we also summarize recommendations for functionals well adapted to specific kinds of applications. Solving the Schrödinger equation for an atom, a molecule, or a solid yields its wave function, which can be used to determine various properties of the system. The two quantum chemistry methods widely used to calculate the properties are DFT and wave function theory (WFT). In WFT, one works explicitly with the wave function, which depends on 3n coordinates for an n-electron system with fixed nuclear coordinates. In contrast, the basic premise of DFT is that, for a given set of nuclear coordinates, the energy density at a point in space is a functional of the 3D electron density. Therefore, the original formulation of DFT involves only three dimensions. For open-shell systems, we need the up-spin density and the down-spin density, which are each functions of only three spatial coordinates. For this reason, DFT is practical even for large molecules and complicated materials, whereas accurate WFT is an impractical method for large systems, although it is capable of high accuracy when it is affordable for small systems. Because it is impossible to solve the Schrödinger equation exactly for a many-body system, approximations are made in both WFT and DFT methods, and the type of approximation one enforces determines the usefulness of these methods for various properties. In practice, much progress in DFT has involved adding additional ingredients to the spin densities. The accurate treatment of electron correlation constitutes a central problem in both DFT and WFT. It is often defined with respect to a HF wave function, which is a single-configuration wave function obtained by using the variational principle to find the orbitals that lead to the lowest energy. Electron correlation energy is often quantified somewhat arbitrarily as the difference between the exact energy and the HF energy of a system at the complete basis set limit. Electron correlation energy can be broadly divided into dynamic and static correlation energy [1.Mok D.K.W. et al.Dynamical and nondynamical correlation.J. Phys. Chem. 1996; 100: 6225-6230Crossref Scopus (131) Google Scholar]. The correlation energy that can be treated well by adding a small number of nearly degenerate configuration state functions (CSFs) to a single-configuration starting point is usually called static correlation energy [2.Handy N.C. Cohen A.J. Left–right correlation energy.Mol. Phys. 2001; 99: 403-412Crossref Scopus (1402) Google Scholar, 3.Hollett J.W. Gill P.M.W. The two faces of static correlation.J. Chem. Phys. 2011; 134: 114111Crossref PubMed Scopus (90) Google Scholar, 4.Cremer D. et al.Implicit and explicit coverage of multi-reference effects by density functional theory.Int. J. Mol. Sci. 2002; 3: 604-638Crossref Scopus (110) Google Scholar]. A prominent example is left–right correlation energy associated with the near degeneracy of bonding and antibonding orbitals when a bond is broken. The remaining correlation energy is then called dynamic. The dynamic correlation energy converges very slowly with respect to the addition of more CSFs to a reference CSF and this is the main reason for the high cost of well-converged WFT calculations. Systems with high static correlation energy are called strongly correlated systems or single-reference (SR) systems and other systems are called weakly correlated or multireference (MR) systems. Independently of the language, the border between strongly correlated and weakly correlated is imprecise. Nevertheless, we can make two generalizations: (i) that dynamic correlation is expected to have a very similar character in all states, while static correlation is expected to be very system-dependent, state-dependent, and even geometry-dependent; and (ii) that systems with high static correlation are hard to treat accurately if one uses a single CSF as a reference function. KS-DFT, like HF theory, is based on a mean-field formalism and is an SR method, but in KS theory the reference wave function is not a zero-order wave function for perturbation theory or a starting point for adding CSFs by excitations (single, double, triple, etc.) in configuration interaction theory or coupled cluster theory, but rather it is a Slater determinant corresponding to noninteracting electrons that have the same one-electron density as the exact solution to the Schrödinger equation. Thus, this Slater determinant is not a wave function for the real system. Because it corresponds to the same density as the real system, it can be used like a wave function for calculating properties like dipole moments that depend only on the density, but – except for the energy – it should not be used for properties that depend on correlated electron distributions, which are described by the two-particle density. (For example, the many-electron spin S is not a one-electron property, so the KS-DFT determinant should not be used to calculate the expectation value of S2.) Although KS-DFT has a mean-field formulation and represents the density with a single Slater determinant, it is in principle an exact theory. The energy in KS-DFT is not calculated by adding more CSFs but rather by using an exchange-correlation functional, which is a functional of the density. Although there is an existence theorem for an exact exchange-correlation functional, even for strongly correlated systems, in practice we have to approximate the exchange-correlation functional, and currently available functionals are more accurate for weakly correlated systems than for strongly correlated ones. This is understandable because, in a strongly correlated system, the exchange-correlation functional must account not just for the generic dynamic correlation but also for the very state-dependent static correlation. One obtains the orbitals of a KS-DFT determinant by self-consistent-field variational calculations and the variational principle sometimes accounts for static correlation (as well as it can) by using a Slater determinant that does not have the same symmetry properties as the exact wave function of that state. For example, even in the absence of spin-orbit coupling, the variationally best Slater determinant for an open-shell system will be spin polarized (sometimes called spin unrestricted or just unrestricted), which means that the spin-up orbitals are not the same as the spin-down ones, and the Slater determinant will not be an eigenfunction of S2 [5.Jacob C.R. Reiher M. Spin in density-functional theory.Int. J. Quantum Chem. 2012; 112: 3661-3684Crossref Scopus (168) Google Scholar]; this is not entirely unexpected since S2 is a not a one-electron property, but it can cause problems in determining whether the KS-DFT calculation has actually approximated the state of interest, and in fact, the state produced by a KS-DFT calculation with presently available functionals for strongly correlated systems is sometimes best interpreted as an approximation to an ensemble of states rather than to a single state. One way to alleviate the problem with strongly correlated systems is to develop density functional methods that employ a multiconfigurational reference function; a promising approach of this type is multiconfiguration pair DFT (MC-PDFT) [6.Li Manni G. et al.Multi-configuration pair-density functional theory.J. Chem. Theory Comput. 2014; 10: 3669-3690Crossref PubMed Scopus (272) Google Scholar]. This approach shows great promise and is typically much less computationally demanding than wave function methods of comparable accuracy, but it does raise the cost compared with KS-DFT. This review is therefore restricted to the less-expensive approach based on a single Slater determinant as the reference wave function. In addition to suffering from the difficulty of treating strongly correlated systems, another important difficulty of the original KS-DFT is self-interaction error. This comes about as follows. A key element of the original KS-DFT method is that the SCF equations for the orbitals involve each electron moving in a local potential. The interaction energy with this potential is the classical Coulomb energy. In contrast, in the SCF equations of HF theory, the field in which the electrons move is nonlocal because it includes the HF exchange potential, which is an integral operator; that is, at a given point in space, the HF exchange potential involves integration over all space. In both HF theory and KS-DFT, the potential field includes the Coulomb potential, which is the interaction of the electron with the entire electron density of the atom, molecule, or material. That is physically incorrect, because an electron does not interact with itself. In HF theory, the exchange potential cancels the self-interaction part of the Coulomb potential. In KS-DFT this cannot be done exactly because a local potential cannot exactly replace an integral operator. The exchange-correlation potential mimics the exchange potential to some extent (as well as approximating the correlation energy), but the fact that it does not completely cancel self-interaction error is usually considered to be the most important source of error in exchange-correlation functionals, especially for weakly correlated systems. A specific fundamental source of error in KS-DFT is delocalization, which is sometimes considered to be the same as self-interaction error (because the self-interaction of electrons promotes excessive spreading over multiple centers), but it should probably be considered to be different [7.Li C. et al.Localized orbital scaling correction for systematic elimination of delocalization error in density functional approximations.Natl. Sci. Rev. 2018; 5: 203-215Crossref Scopus (87) Google Scholar]. Delocalization error can be analyzed in terms of charge delocalization (e.g., in the dissociation of H2+) and spin delocalization (e.g., in the dissociation of H2) [8.Cohen A.J. et al.Challenges for density functional theory.Chem. Rev. 2012; 112: 289-320Crossref PubMed Scopus (1627) Google Scholar]. However, we do not use this kind of analysis in this review. One strategy to alleviate the self-interaction error is to replace part of the local exchange-correlation functional by nonlocal HF exchange [9.Parr R.G. Yang W. Density–Functional Theory of Atoms and Molecules. Oxford University Press, 1989Google Scholar, 10.Becke A.D. A new mixing of Hartree–Fock and local density functional theories.J. Chem. Phys. 1993; 98: 1372-1377Crossref Scopus (13786) Google Scholar, 11.Becke A.D. Density–functional thermochemistry. III. The role of exact exchange.J. Chem. Phys. 1993; 98: 5648-5653Crossref Scopus (89711) Google Scholar]. This was originally introduced in an ad hoc manner; Parr and Yang call it the Hartree–Fock–Kohn–Sham method [9.Parr R.G. Yang W. Density–Functional Theory of Atoms and Molecules. Oxford University Press, 1989Google Scholar]. Most often it is now called hybrid KS-DFT. However, the unrealized possibilities of this method are better appreciated by noting that it is a special case of generalized KS (GKS) [12.Seidl A. et al.Generalized Kohn–Sham schemes and the band-gap problem.Phys. Rev. B. 1996; 53: 3764-3774Crossref Scopus (923) Google Scholar] DFT. In the GKS theory [12.Seidl A. et al.Generalized Kohn–Sham schemes and the band-gap problem.Phys. Rev. B. 1996; 53: 3764-3774Crossref Scopus (923) Google Scholar], the reference wave function, although still a single Slater determinant, represents the density of a partially interacting system rather than a noninteracting one. The most common choice of the partial interaction is to put some portion of the two-electron energy of the Slater determinant into the reference system. For example, if we include X percent of the two-electron interaction, the resulting self-consistent-field equations contain X percent of the nonlocal HF exchange, and in fact it is identical to hybrid KS-DFT. One important aspect of the GKS derivation is that it shows that, just as in the original KS-DFT, that there exists in principle an exact exchange-correlation functional; it is different for each choice of interaction included in the reference (e.g., it is different for each X). Both HF theory and the original KS-DFT may be considered special cases of GKS theory. Properties that can be improved with the GKS theory can be as simple as the charge distribution of a molecule [13.Verma P. Truhlar D.G. Can Kohn–Sham density functional theory predict accurate charge distributions for both single-reference and multi-reference molecules?.Phys. Chem. Chem. Phys. 2017; 19: 12898-12912Crossref PubMed Google Scholar] and can be as challenging as the excitation energies of molecules [14.Verma P. et al.Revised M11 exchange-correlation functional for electronic excitation energies and ground-state properties.J. Phys. Chem. A. 2019; 123: 2966-2990Crossref PubMed Scopus (52) Google Scholar] and band gaps of solids [15.Heyd J. et al.Energy band gaps and lattice parameters evaluated with the Heyd–Scuseria–Ernzerhof screened hybrid functional.J. Chem. Phys. 2005; 123: 174101Crossref PubMed Scopus (1438) Google Scholar]. In this overview, we focus on our group’s work on two kinds of problems: (i) designing new density functionals for ground states that are accurate for both weakly correlated and strongly correlated atoms and molecules; and (ii) designing density functionals that are accurate for both excitation energies of molecules and band gaps of periodic solids. A key aspect in designing density functionals is the development of broad databases [14.Verma P. et al.Revised M11 exchange-correlation functional for electronic excitation energies and ground-state properties.J. Phys. Chem. A. 2019; 123: 2966-2990Crossref PubMed Scopus (52) Google Scholar,16.Goerigk L. et al.A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions.Phys. Chem. Chem. Phys. 2017; 19: 32184-32215Crossref PubMed Google Scholar]. Compilation of accurate data from accurate experimental measurements or high-level quantum mechanical calculations (in the absence of accurate experimental data) is important in testing the accuracy of existing and newly designed density functionals. For widely testing the performance of density functionals, one would like to include data that are disparate and data that are seemingly incongruent; for example, data on: molecular and solid-state properties; ground-state and excited-state properties; various types of excitation energies such as valence, Rydberg, and charge transfer (CT); short-range and long-range CT; SR and MR species; and so on. In our recent work, we put together 56 diverse databases to make Minnesota Database 2019 [14.Verma P. et al.Revised M11 exchange-correlation functional for electronic excitation energies and ground-state properties.J. Phys. Chem. A. 2019; 123: 2966-2990Crossref PubMed Scopus (52) Google Scholar,17.Verma P. Truhlar D.G. Geometries for Minnesota Database 2019. Data Repository for the University of Minnesota, 2019Google Scholar]. Minnesota Database 2019 can be broadly divided into subdatabases for ground-state and excited-state properties; see Figure 1 for properties represented in this database. Within ground-state properties, it includes both geometries and energies. The geometric data include transition-state geometries, transition-metal dimer bond lengths, and main-group bond lengths. The energetic data include bond energies, reaction energies, proton affinities, electron affinities, ionization potentials, noncovalent interaction energies, reaction barrier heights, and total atomic energies. In various tests performed by us and by others, one finds that there are properties that correlate with each other and that there are properties that do not correlate with each other. For instance, the properties that might not correlate with each other could be: (i) properties of the ground state versus those for an excited state; (ii) energies versus geometries; (iii) energies of SR systems versus energies of MR systems; and (iv) atomic versus molecular versus solid-state properties. Therefore, finding a balanced density functional that provides good results for as many properties as possible is a formidable challenge. In the remainder of this review we discuss how well we have been able to achieve this goal with our most recently developed functionals and we discuss which functionals perform best for various properties. Although our discussion here is focused rather tightly on functionals developed at Minnesota, we have presented extensive comparisons to other functionals in the papers where we published new functionals. The reader is also referred to the recent work of Grimme and coworkers in testing a large set of density functionals against main-group [16.Goerigk L. et al.A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions.Phys. Chem. Chem. Phys. 2017; 19: 32184-32215Crossref PubMed Google Scholar] and transition-metal [18.Dohm S. et al.Comprehensive thermochemical benchmark set of realistic closed-shell metal organic reactions.J. Chem. Theory Comput. 2018; 14: 2596-2608Crossref PubMed Scopus (143) Google Scholar] databases, as well as the review by Laurent and Jacquemin [19.Laurent A.D. Jacquemin D. TD-DFT benchmarks: a review.Int. J. Quantum Chem. 2013; 113: 2019-2039Crossref Scopus (795) Google Scholar] of tests of DFT for electronic excitation energies. An early forerunner of DFT was the density-based Thomas–Fermi model [20.Thomas L.H. The calculation of atomic fields.Proc. Camb. Philos. 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Self-consistent equations including exchange and correlation effects.Phys. Rev. 1965; 140: A1133-A1138Crossref Scopus (47924) Google Scholar]. The unknown quantity in KS-DFT is the exchange-correlation functional, and the exchange-correlation energy contribution to KS equations can be approximated in more than one way as discussed next. In what follows, we will often divide exchange-correlation functionals into two classes, local and nonlocal. A local functional depending only on the spin densities and their gradients is called a gradient approximation (GA). Functionals depending also on the local kinetic energy density are called meta functionals. Any local functional may be augmented with nonlocal elements, the most common of which are nonlocal HF exchange (which has already been discussed), nonlocal orbital-dependent correlation [24.Zhao Y. et al.Doubly hybrid meta DFT: new multi-coefficient correlation and density functional methods for thermochemistry and thermochemical kinetics.J. Phys. Chem. A. 2004; 108: 4786-4791Crossref Scopus (300) Google Scholar,25.Goerigk L. Grimme S. Double-hybrid density functionals.Wiley Interdiscip. Rev. Comput. Mol. Sci. 2014; 4: 576-600Crossref Scopus (260) Google Scholar], nonlocal density-dependent correlation [26.Román-Pérez G. Soler J.M. Efficient implementation of a van der Waals density functional: application to double-wall carbon nanotubes.Phys. Rev. Lett. 2009; 103: 096102Crossref PubMed Scopus (1230) Google Scholar], or so-called rung-3.5 terms [27.Janesko B.G. Rung 3.5 density functionals.J. Chem. 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Based on their percentage (called X) of HF exchange, hybrid functionals may be subclassified into various classes; in this review we consider four such classes: global-hybrid functionals in which X is a constant; long-range-corrected (LRC) hybrid functionals in which X depends on the interelectronic separation and increases to 100% at large separation; Coulomb-attenuated-hybrid functionals in which X increases with interelectronic separation but to less than 100% at large separation; and screened-exchange-hybrid functionals in which X decreases to zero with increasing interelectronic separation. This review does not include functionals with nonlocal correlation terms other than rung-3.5 terms. Most (but not all) density functionals have been developed with the developers’ eyes on ground-state properties and the functionals were only later tested for molecular excitation energies and solid-state band gaps. 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