We present a new theoretical framework, called Multiconfiguration Pair-Density Functional Theory (MC-PDFT), which combines multiconfigurational wave functions with a generalization of density functional theory (DFT). A multiconfigurational self-consistent-field (MCSCF) wave function with correct spin and space symmetry is used to compute the total electronic density, its gradient, the on-top pair density, and the kinetic and Coulomb contributions to the total electronic energy. We then use a functional of the total density, its gradient, and the on-top pair density to calculate the remaining part of the energy, which we call the on-top-density-functional energy in contrast to the exchange-correlation energy of Kohn-Sham DFT. Because the on-top pair density is an element of the two-particle density matrix, this goes beyond the Hohenberg-Kohn theorem that refers only to the one-particle density. To illustrate the theory, we obtain first approximations to the required new type of density functionals by translating conventional density functionals of the spin densities using a simple prescription, and we perform post-SCF density functional calculations using the total density, density gradient, and on-top pair density from the MCSCF calculations. Double counting of dynamic correlation or exchange does not occur because the MCSCF energy is not used. The theory is illustrated by applications to the bond energies and potential energy curves of H2, N2, F2, CaO, Cr2, and NiCl and the electronic excitation energies of Be, C, N, N(+), O, O(+), Sc(+), Mn, Co, Mo, Ru, N2, HCHO, C4H6, c-C5H6, and pyrazine. The method presented has a computational cost and scaling similar to MCSCF, but a quantitative accuracy, even with the present first approximations to the new types of density functionals, that is comparable to much more expensive multireference perturbation theory methods.

Systematic evaluation of the accuracy of exchange-correlation functionals is essential to guide scientists in their choice of an optimal method for a given problem when using density functional theory. In this work, accuracy of one Generalized Gradient Approximation (GGA) functional, three meta-GGA functionals, one Nonseparable Gradient Approximation (NGA) functional, one meta-NGA, and three hybrid GGA functionals was evaluated for calculations of the closest interatomic distances, cohesive energies, and bulk moduli of all 3d, 4d, and 5d bulk transition metals that have face centered cubic (fcc), hexagonal closed packed (hcp), or body centered cubic (bcc) structures (a total of 27 cases). Our results show that including the extra elements of kinetic energy density and Hartree-Fock exchange energy density into gradient approximation density functionals does not usually improve them. Nevertheless, the accuracies of the Tao-Perdew-Staroverov-Scuseria (TPSS) and M06-L meta-GGAs and the MN12-L meta-NGA approach the accuracy of the Perdew-Burke-Ernzerhof (PBE) GGA, so usage of these functionals may be advisable for systems containing both solid-state transition metals and molecular species. The N12 NGA functional is also shown to be almost as accurate as PBE for bulk transition metals, and thus it could be a good choice for studies of catalysis given its proven good performance for molecular species.

In this work the ISOL24 database of isomerization energies of large organic molecules presented by Huenerbein et al. [Phys. Chem. Chem. Phys., 2010, 12, 6940] is updated, resulting in the new benchmark database called ISOL24/11, and this database is used to test 50 electronic model chemistries. To accomplish the update, the very expensive and highly accurate CCSD(T)-F12a/aug-cc-pVDZ method is first exploited to investigate a six-reaction subset of the 24 reactions, and by comparison of various methods with the benchmark, MCQCISD-MPW is confirmed to be of high accuracy. The final ISOL24/11 database is composed of six reaction energies calculated by CCSD(T)-F12a/aug-cc-pVDZ and 18 calculated by MCQCISD-MPW. We then tested 40 single-component density functionals (both local and hybrid), eight doubly hybrid functionals, and two other methods against ISOL24/11. It is found that the SCS-MP3/CBS method, which is used as benchmark for the original ISOL24, has an MUE of 1.68 kcal mol(-1), which is close to or larger than some of the best tested DFT methods. Using the new benchmark, we find ωB97X-D and MC3MPWB to be the best single-component and doubly hybrid functionals respectively, with PBE0-D3 and MC3MPW performing almost as well. The best single-component density functionals without molecular mechanics dispersion-like terms are M08-SO, M08-HX, M05-2X, and M06-2X. The best single-component density functionals without Hartree-Fock exchange are M06-L-D3 when MM terms are included and M06-L when they are not.

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.

A notorious failing of approximate exchange-correlation functionals when applied to problems involving catalysis has been the inability of most local functionals to predict the correct adsorption site for CO on metal surfaces or to simultaneously predict accurate surface formation energies and adsorption energies for transition metals. By adding the kinetic energy density τ to the density functional, the revTPSS density functional was shown recently to achieve a balanced description of surface energies and adsorption energies. Here, we show that the older M06-L density functional, also containing τ, provides improved surface formation energies and CO adsorption energies over revTPSS for five transition metals and correctly predicted the on-top/hollow site adsorption preferences for four of the five metals, which was not achieved by most other local functionals. Because M06-L was entirely designed on the basis of atomic and molecular energies, its very good performance is a confirmation of the reasonableness of its functional form. Two GGA functionals with an expansion in the reduced gradient that is correct through second order, namely, SOGGA and SOGGA11, were also tested and found to produce the best surface formation energies of all tested GGA functionals, although they significantly overestimate the adsorption energies.