REVIEW ARTICLE www.rsc.org/pccp | Physical Chemistry Chemical Physics Density functional theory for transition metals and transition metal chemistry Christopher J. Cramer* and Donald G. Truhlar* Received 8th April 2009, Accepted 20th August 2009 First published as an Advance Article on the web 21st October 2009 DOI: 10.1039/b907148b We introduce density functional theory and review recent progress in its application to transition metal chemistry. Topics covered include local, meta, hybrid, hybrid meta, and range-separated functionals, band theory, software, validation tests, and applications to spin states, magnetic exchange coupling, spectra, structure, reactivity, and catalysis, including molecules, clusters, nanoparticles, surfaces, and solids. 1. Introduction chemical systems, in part because its cost scales more favorably with system size than does the cost of correlated WFT, and yet Density functional theory (DFT) describes the electronic states it competes well in accuracy except for very small systems. This of atoms, molecules, and materials in terms of the three- is true even in organic chemistry, but the advantages of DFT dimensional electronic density of the system, which is a great are still greater for metals, especially transition metals. The simplification over wave function theory (WFT), which involves reason for this added advantage is static electron correlation. a3N-dimensional antisymmetric wave function for a system It is now well appreciated that quantitatively accurate 1 with N electrons. Although DFT is sometimes considered the electronic structure calculations must include electron correlation. B ‘‘new kid on the block’’, it is now 45 years old in its modern It is convenient to recognize two types of electron correlation, 2 formulation (more than half as old as quantum mechanics the first called dynamical electron correlation and the second 3,4 itself), and it has roots that are almost as ancient as the called static correlation, near-degeneracy correlation, or non- Schro¨ dinger equation. In practical work, DFT is almost always dynamical correlation. Dynamical correlation is a short-range 5 applied in the form introduced by Kohn and Sham, including effect by which electrons avoid one another to reduce electron 6,7 its spin-polarized extension. The basic quantity in DFT is the repulsion. It is a very general effect for all finite systems many-electron spin density, r. The spin-polarized Kohn–Sham containing two or more electrons. Accounting for dynamical formalism involves a determinant formed from a set of N correlation by a configuration interaction wave function is fictitious single-particle spin-orbitals corresponding to a non- very slowly convergent and requires a very large number of interacting system of electrons with the same spin densities, ra configurations. Other correlation effects, which are very system and rb, as the real system, where r is the sum of ra and rb,the specific and can be either medium ranged or long ranged, can spin density ra is the 3-dimensional electron density of all be accounted for to a large extent by mixing a small number spin-up electrons, and rb is the same for spin-down electrons. (sometimes two, sometimes more) of configurations that are In the original Kohn–Sham formalism (applicable to closed- ‘‘nearly’’ degenerate.8,9 Such correlation effects are called shell molecules and nonmagnetic solids), ra equals rb.(The static or near-degeneracy correlation and systems exhibiting original Kohn–Sham formalism may also be labeled spin- significant static correlation effects are often called multi- restricted Kohn–Sham or restricted Kohn–Sham, and the reference systems;10 likewise, WFT methods based on spin-polarized version may be called spin-unrestricted or multi-configurational zero-order states are often called multi- unrestricted.) We note that ‘‘spin density’’ is a generic term for reference methods. Due to partially filled d subshells, and the density associated with the subset of electrons characterized nearly degenerate (n +1)s and nd subshells, systems containing by the same definite value of Sz, i.e.,eithera or b (thus one might transition metals often have a plethora of low-lying nearly say ‘‘spin-up density and spin-down density’’ rather than ‘‘spin degenerate states, and near-degeneracy correlation effects on densities’’). In a many-electron system comprised of both spin-up the ground-state structure, electron distribution, and energy of and spin-down electrons, the term spin-density is also sometimes transition metal systems can be very large. Even though static used to refer to the position-dependent difference between the up correlation can often be accounted for to a zero-order anddownspindensities,ortothevector analog of this quantity. approximation by a small number of configurations, it is often To avoid confusion, in this article we will refer to the difference very difficult to include correlation effects in a well balanced density as the spin polarization density. way in WFT calculations on multi-reference systems.11,12 Density functional theory (DFT) has now become the DFT, however, remains simple for such systems and is often preferred method for electronic structure theory for complex surprisingly accurate. This empirical fact adds to the advantage of computational efficiency in making DFT a preferred method for transition metal chemistry. Department of Chemistry and Supercomputing Institute, University of Minnesota, Minneapolis, MN 55455-0431, USA. The subject of transition-metal DFT is too large for any E-mail: [email protected], [email protected] single review to be complete. We selected recent papers that This journal is c the Owner Societies 2009 Phys.Chem.Chem.Phys., 2009, 11, 10757–10816 | 10757 best illustrate the promise of DFT in a number of very active includes only dynamic correlation, and DFT exchange areas of transition-metal chemistry, and we also include some includes not only exchange but also some static correlation, older references that help to put them in perspective. We although the latter is present in an unspecified and largely exclude biological applications, which deserve and uncontrolled way.18,19 In subsection 2.2 we introduce receive13–17 their own reviews. some exchange and correlation density functionals, and in Section 2 reviews the theory, section 3 contains some subsections 2.4 and 2.5 we describe a greater number of them. comments on methodology, and section 4 reviews validation Transition metal chemistry often involves open-shell systems studies. Section 5 reviews recent applications and section 6 and excited states. Kohn–Sham theory is not general enough to contains concluding remarks. treat all open-shell systems or excited states.20–22 Although generalized Kohn–Sham theories have been advanced to 2. Overview of DFT and functionals overcome this limitation, they are not in widespread use. Thus one encounters difficulties not just due to inaccurate density 2.1 Fundamental background functionals but due to Kohn–Sham theory itself. There is a rich theoretical literature on these questions, but it contains more Kohn–Sham spin-orbitals, c where s is a or b and j denotes js than one point of view, and much practical work in DFT the other quantum numbers, are obtained by a self-consistent involves using approximate functionals in a rough-and-ready field (SCF) calculation and are formally functions of the exact way (see, e.g., section 5.1). Despite such issues, Kohn–Sham density of the system. Then theory, even with approximate functionals, is often the most Xocc accurate available approach for practical work on a given r ¼ jc j2 ð1Þ s js system, especially for complex systems. j One type of generalized Kohn–Sham theory that is in where s is the spin component (a or b), the spin-orbitals are widespread use is often called hybrid DFT, and it involves normalized, and the sum is over occupied orbitals of a given combining Hartree–Fock exchange, which is orbital-dependent, spin component. The electronic energy of the system is with explicit functions of local spin densities and their gradients. approximated as a sum of four terms, Tn, ene, eee, and exc. (Note that one can also treat Hartree–Fock exchange within Tn is the kinetic energy of a system of noninteracting electrons the ungeneralized Kohn–Sham framework by using the 23,24 with the same spin densities as the real system; ene is the ‘‘optimized effective potential’’ (OEP) method, but we interaction of the electron distribution with the nuclear frame- shall not consider this formalism in detail within the present work; eee is the classical Coulomb energy of the spin densities review.) Further generalizing hybrid functionals to include interacting with each other and with themselves; and exc, called dependences on the local Laplacians of the spin densities or the exchange–correlation energy, is everything else (everything on the local spin kinetic energies computed from the spin- except Tn, ene, and eee). Therefore exc includes the interaction orbitals yields what are called hybrid meta functionals, and correction to Tn, the correction to Vee for the fact that real these are the most powerful functionals now available. electrons do not interact with themselves, the exchange energy Although one should distinguish between Kohn–Sham and (due to the indistinguishability of electrons exchanging their generalized Kohn–Sham methods, and making fundamental space and spin variables), and the correlation energy (due to progress at extending DFT to arbitrarily complex systems the fact that the many-electron spin densities are not certainly will require careful attention to the fundamentals 25 26 uncorrelated products of spin-orbital densities). exc is written of the theory, especially for magnetic properties, most as a functional, called the spin-density functional, of the spin practical algorithms in popular use ignore such issues and densities. Since the Kohn–Sham spin-orbitals are functions of the address the theory from a computational perspective with spin-densities, exc can depend explicitly on the spin densities and approximate methods and procedures being accepted or also implicitly on them by depending on the spin-orbitals.