Density Functional Theory of Open-Shell Systems. The 3d-Series Transition-Metal Atoms and Their Cations Sijie Luo, Boris Averkiev, † Ke R. Yang, Xuefei Xu, and Donald G. Truhlar* Department of Chemistry and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States ABSTRACT: The 3d-series transition metals (also called the fourth-period transition metals), Sc to Zn, are very important in industry and biology, but they provide unique challenges to computing the electronic structure of their compounds. In order to successfully describe the compounds by theory, one must be able to describe their components, in particular the constituent atoms and cations. In order to understand the ingredients required for successful computations with density functional theory, it is useful to examine the performance of various exchange-correlation functionals; we do this here for 4s N 3d N ′ transition-metal atoms and their cations. We analyze the results using three ways to compute the energy of the open-shell states: the direct variational method, the weighted-averaged broken symmetry (WABS) method, and a new broken-symmetry method called the reinterpreted broken symmetry (RBS) method. We find the RBS method to be comparable in accuracy with the WABS method. By examining the overall accuracy in treating 18 multiplicity-changing excitations and 10 ionization potentials with the RBS method, 10 functionals are found to have a mean-unsigned error of <5 kcal/mol, with ωB97X- D topping the list. For local density functionals, which are more practical for extended systems, the M06-L functional is the most accurate. And by combining the results with our previous studies of p-block and 4d-series elements as well as databases for alkyl bond dissociation, main-group atomization energies, and π-π noncovalent interactions, we find five functionals, namely, PW6B95, MPW1B95, M08-SO, SOGGA11-X, and MPWB1K, to be highly recommended. We also studied the performance of PW86 and C09 exchange functionals, which have drawn wide interest in recent studies due to their claimed ability to reproduce Hartree-Fock exchange at long distance. By combining them with four correlation functionals, we find the performance of the resulting functionals disappointing both for 3d transition-metal chemistry and in broader tests, and thus we do not recommend PW86 and C09 as components of generalized gradient approximations for general application. 1. INTRODUCTION Many systems involving transition-metal atoms have open-shell electronic structures with several low-energy states differing in spin and/or in orbital occupancy. It is common practice to apply Kohn-Sham (KS) density functional theory (DFT) 1 to the lowest-energy state of each total electronic spin component (M S ), 2,3 to identify the ground state, and to model the properties, spectra, and reactivity of each spin state. However, when 2M S is less than the number n SO of singly occupied orbitals, the state is intrinsically multideterminental, and one finds that KS-DFT, with its single-determinantal noninteracting reference state for computing the dominant portion of the kinetic energy, is often less accurate for the low-spin open-shell states than for the high-spin and closed-shell states (M S = n SO / 2) for which a single Slater determinant is a good reference function. This is mainly due to the fact that presently available exchange-correlation (xc) functionals, which are the only approximations in KS-DFT, usually cannot treat the multi- reference systems and single-reference ones equally well. 2-4 The difficulty is compounded when the different spin states differ in their s and d occupancies (e.g., ns N (n - 1)d N′ in one spin state and ns N-1 (n - 1)d N′+1 in the other), since the approximate xc functionals may have different accuracies for ns electrons than for (n-1)d electrons, which tend to be closer to the nucleus in a region of higher electron density. For the reasons above, it is worthwhile to test currently available xc functionals’ performance in these difficult cases. Testing the functionals requires care to be sure that the conclusions are not clouded by the following complications: (a) incomplete basis sets or inadequate effective core potentials; (b) incomplete treatment of relativistic effects, including spin- orbit coupling; (c) errors in reference data; and (d) cancellation of errors in treating the transition metal with errors in treating ligands or the ligand field exerted on the transition-metal site. All of these possible complications are minimized or even eliminated by studying unligated atoms M and monatomic cations M + of the 3d-series. In particular, because these systems are small, one can afford to use nearly complete all-electron basis sets without effective core potentials. Also, the experimental excitation energies and ionization potentials are well established. Furthermore, the relativistic effects are small enough that Douglas-Kroll-Hess (DKH) second-order calculations should account well for scalar relativistic effects, and spin-orbit effects can be largely removed from experiment by considering the weighted average of the configuration. Finally, by studying atoms and monatomic ions, the treatment of electronic structure is decoupled from geometry optimiza- Received: August 9, 2013 Published: November 5, 2013 Article pubs.acs.org/JCTC © 2013 American Chemical Society 102 dx.doi.org/10.1021/ct400712k | J. Chem. Theory Comput. 2014, 10, 102-121