The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals

We present two new hybrid meta exchange- correlation functionals, called M06 and M06-2X. The M06 functional is parametrized including both transition metals and nonmetals, whereas the M06-2X functional is a high-nonlocality functional with double the amount of nonlocal exchange (2X), and it is parametrized only for nonmetals.The functionals, along with the previously published M06-L local functional and the M06-HF full-Hartree–Fock functionals, constitute the M06 suite of complementary functionals. We assess these four functionals by comparing their performance to that of 12 other functionals and Hartree–Fock theory for 403 energetic data in 29 diverse databases, including ten databases for thermochemistry, four databases for kinetics, eight databases for noncovalent interactions, three databases for transition metal bonding, one database for metal atom excitation energies, and three databases for molecular excitation energies. We also illustrate the performance of these 17 methods for three databases containing 40 bond lengths and for databases containing 38 vibrational frequencies and 15 vibrational zero point energies. We recommend the M06-2X functional for applications involving main-group thermochemistry, kinetics, noncovalent interactions, and electronic excitation energies to valence and Rydberg states. We recommend the M06 functional for application in organometallic and inorganometallic chemistry and for noncovalent interactions. Electronic Supplementary Material The online version of this article (doi:) contains supplementary material, which is available to authorized users. Contribution to the Mark S. Gordon 65th Birthday Festschrift Issue.     

Double-hybrid functionals for thermochemical kinetics

We propose two new double-hybrid functionals, denoted B2K-PLYP and mPW2K-PLYP, which yield thermochemical performance comparable to existing double-hybrid functionals but offer superior performance for barrier heights of various kinds. We show that the new functionals yield excellent performance for all of the following: (a) main-group thermochemistry; (b) main-group thermochemical kinetics; (c) late transition metal reactions. In addition, B2K-PLYP performs well for weak interactions.     

Comparative assessment of density functional methods for 3d transition-metal chemistry

In the present study, we comparatively assessed the newly developed M05 functional against a data set of reaction energies for transition-metal chemistry. The functionals to which we compare are BLYP, B3LYP, B97-2, MPWLYP1M, TPSS, and TPSSh. We draw the following conclusions: (1) TPSS gives the best performance for calculating the binding energies of three transition-metal dimers (Sc(2), Ni(2), and V(2)) that have severe multireference character, (2) B97-2 gives the best performance for calculating the binding energies of the nine metal-ligand diatomics (three monohydrides, three monoxide, and three monofluorides), and (3) M05 gives the overall best performance for all 18 data in the assessment, and it has a mean unsigned error 55% lower than the popular B3LYP functional. Since the M05 functional also gives good performance for main-group thermochemistry, for noncovalent chemistry, and for calculating barrier heights, M05 can be applied to a wide range of problems where nonhybrid functionals or functionals designed for kinetics fail.     

Exchange-correlation functional with broad accuracy for metallic and nonmetallic compounds, kinetics, and noncovalent interactions

By incorporating kinetic-energy density in a balanced way in the exchange and correlational functionals and removing self-correlation effects, we have designed a density functional that is broadly applicable to organometallic, inorganometallic, and nonmetallic bonding, thermochemistry, thermochemical kinetics, and noncovalent interactions as well as satisfying the uniform electron gas limit. The average error is reduced by a factor of 1.3 compared with the best previously available functionals, but even more significantly, we find a functional that has a high accuracy for all four categories of interaction.     

Performance of the M11-L density functional for bandgaps and lattice constants of unary and binary semiconductors

The recently developed SOGGA11 and M11-L density functionals have been tested for the prediction of bandgaps and lattice constants by comparing to databases containing 31 bandgaps and 34 lattice constants. To make a comparative assessment we also test several other density functionals against the same databases; in particular, we test the local spin density approximation, PBE, PBEsol, SOGGA, TPSS, revTPSS, and M06-L local density functionals and the HSE screened-exchange hybrid nonlocal density functional; and for a subset of 13 lattice constants we also compare the mean errors to those of the AM05 and WC local density functionals and the HISS and HSEsol nonlocal density functionals. The tests show that, of the ten functionals tested against all 65 data, the SOGGA, PBEsol, and HSE functionals are the most accurate for lattice constants, whereas the HSE, M11-L, and M06-L density functionals are the most accurate for bandgaps. However, the SOGGA11 density functional is the most accurate generalized gradient approximation for bandgaps.     

An improved and broadly accurate local approximation to the exchange-correlation density functional: The MN12-L functional for electronic structure calculations in chemistry and physics

We report a new local exchange-correlation energy functional that has significantly improved across-the-board performance, including main-group and transition metal chemistry and solid-state physics, especially atomization energies, ionization potentials, barrier heights, noncovalent interactions, isomerization energies of large moleucles, and solid-state lattice constants and cohesive energies.     

Performance of the M06 family of exchange-correlation functionals for predicting magnetic coupling in organic and inorganic molecules

The performance of the M06 family of exchange-correlation potentials for describing the electronic structure and the Heisenberg magnetic coupling constant (J) is investigated using a set of representative open-shell systems involving two unpaired electrons. The set of molecular systems studied has well defined structures, and their magnetic coupling values are known experimentally. As a general trend, the M06 functional is about equally as accurate as B3LYP or PBE0. The performance of local functionals is important because of their economy and convenience for large-scale calculations; we find that M06-L local functional of the M06 family largely improves over the local spin density approximation and the generalized gradient approximation.     

Assessment of new meta and hybrid meta density functionals for predicting the geometry and binding energy of a challenging system: the dimer of H2S and benzene

Noncovalent interactions of a hydrogen bond donor with an aromatic pi system present a challenge for density functional theory, and most density functionals do not perform well for this kind of interaction. Here we test seven recent density functionals from our research group, along with the popular B3LYP functional, for the dimer of H 2S with benzene. The functionals considered include the four new meta and hybrid meta density functionals of the M06 suite, three slightly older hybrid meta functionals, and the B3LYP hybrid functional, and they were tested for their abilities to predict the dissociation energies of three conformations of the H 2S-benzene dimer and to reproduce the key geometric parameters of the equilibrium conformation of this dimer. All of the functionals tested except B3LYP correctly predict which of the three conformations of the dimer is the most stable. The functionals that are best able to reproduce the geometry of the equilibrium conformation of the dimer with a polarized triple-zeta basis set are M06-L, PWB6K, and MPWB1K, each having a mean unsigned relative error across the two experimentally verifiable geometric parameters of only 8%. The success of M06-L is very encouraging because it is a local functional, which reduces the cost for large simulations. The M05-2X functional yields the most accurate binding energy of a conformation of the dimer for which a binding energy calculated at the CCSD(T) level of theory is available; M05-2X gives a binding energy for the system with a difference of merely 0.02 kcal/mol from that obtained by the CCSD(T) calculation. The M06 functional performs well in both categories by yielding a good representation of the geometry of the equilibrium structure and by giving a binding energy that is only 0.19 kcal/mol different from that calculated by CCSD(T). We conclude that the new generation of density functionals should be useful for a variety of problems in biochemistry and materials where aromatic functional groups can serve as hydrogen bond acceptors.     

A thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions

A thorough energy benchmark study of various density functionals (DFs) is carried out with the new GMTKN30 database for general main group thermochemistry, kinetics and noncovalent interactions [Goerigk and Grimme, J. Chem. Theor. Comput., 2010, 6, 107; Goerigk and Grimme, J. Chem. Theor. Comput., 2011, 7, 291]. In total, 47 DFs are investigated: two LDAs, 14 GGAs, three meta-GGAs, 23 hybrids and five double-hybrids. Besides the double-hybrids, also other modern approaches, i.e., the M05 and M06 classes of functionals and range-separated hybrids, are tested. For almost all functionals, the new DFT-D3 correction is applied in order to consistently test the performance also for important noncovalent interactions; the parameters are taken from previous works or determined for the present study. Basis set and quadrature grid issues are also considered. The general aim of the study is to work out which functionals are generally well applicable and robust to describe the energies of molecules. In summary, we recommend on the GGA level the B97-D3 and revPBE-D3 functionals. The best meta-GGA is oTPSS-D3 although meta-GGAs represent in general no clear improvement compared to numerically simpler GGAs. Notably, the widely used B3LYP functional performs worse than the average of all tested hybrids and is also very sensitive to the application of dispersion corrections. We discourage its usage as a standard method without closer inspection of the results, as it still seems to be often done nowadays. Surprisingly, long-range corrected exchange functionals do in general not perform better than the corresponding standard hybrids. However, the ωB97X-D functional seems to be a promising method. The most robust hybrid is Zhao and Truhlar's PW6B95 functional in combination with DFT-D3. If higher accuracy is required, double-hybrids should be applied. The corresponding DSD-BLYP-D3 and PWPB95-D3 variants are the most accurate and robust functionals of the entire study. Additional calculations with MP2 and and its spin-scaled variants SCS-MP2, S2-MP2 and SOS-MP2 revealed that double-hybrids in general outperform those. Only SCS-MP2 can be recommended, particularly for reaction energies. We suggest its usage when a large self-interaction error is expected that prohibits usage of double-hybrids. Perdews' metaphoric picture of Jacob's Ladder for the classification of density functionals' performance could unbiasedly be confirmed with GMTKN30. We also show that there is no statistical correlation between a functional's accuracy for atomization energies and the performance for chemically more relevant reaction energies.     

Performance of the M11 and M11-L density functionals for calculations of electronic excitation energies by adiabatic time-dependent density functional theory

Adiabatic time-dependent density functional theory is a powerful method for calculating electronic excitation energies of complex systems, but the quality of the results depends on the choice of approximate density functional. In this article we test two promising new density functionals, M11 and M11-L, against databases of 214 diverse electronic excitation energies, and we compare the results to those for 16 other density functionals of various kinds and to time-dependent Hartree-Fock. Charge transfer excitations are well known to be the hardest challenge for TDDFT. M11 is a long-range-corrected hybrid meta-GGA, and it shows better performance for charge transfer excitations than any of the other functionals except M06-HF, which is a specialized functional that does not do well for valence excitations. Several other long-range-corrected hybrid functionals also do well, and we especially recommend M11, ωB97X, and M06-2X for general spectroscopic applications because they do exceptionally well on ground-state properties as well as excitation energies. Local functionals are preferred for many applications to extended systems because of their significant cost advantage for large systems. M11-L is a dual-range local functional and-unlike all previous local functionals-it has good performance for Rydberg states as well as for valence states. Thus it is highly recommended for excitation energy calculations on extended systems.     

Long-range corrected hybrid density functionals with damped atom-atom dispersion corrections

We report re-optimization of a recently proposed long-range corrected (LC) hybrid density functional [J.-D. Chai and M. Head-Gordon, J. Chem. Phys., 2008, 128, 084106] to include empirical atom-atom dispersion corrections. The resulting functional, omegaB97X-D yields satisfactory accuracy for thermochemistry, kinetics, and non-covalent interactions. Tests show that for non-covalent systems, omegaB97X-D shows slight improvement over other empirical dispersion-corrected density functionals, while for covalent systems and kinetics it performs noticeably better. Relative to our previous functionals, such as omegaB97X, the new functional is significantly superior for non-bonded interactions, and very similar in performance for bonded interactions.     

Systematic optimization of long-range corrected hybrid density functionals

A general scheme for systematically modeling long-range corrected (LC) hybrid density functionals is proposed. Our resulting two LC hybrid functionals are shown to be accurate in thermochemistry, kinetics, and noncovalent interactions, when compared with common hybrid density functionals. The qualitative failures of the commonly used hybrid density functionals in some "difficult problems," such as dissociation of symmetric radical cations and long-range charge-transfer excitations, are significantly reduced by the present LC hybrid density functionals.     

Improved description of nuclear magnetic resonance chemical shielding constants using the M06-L meta-generalized-gradient-approximation density functional

The performance of the M06-L density functional has been tested for four databases of NMR isotropic chemical shielding constants. Comparison with the B3LYP, BLYP, HCTH, KT1, KT2, LSDA, OPBE, OLYP, PBE, TPSS, and VSXC functionals shows that M06-L has improved performance for calculating NMR chemical shielding constants, especially for highly correlated systems. We also found that VSXC and M06-L have encouraging accuracy for calculating (13)C chemical shielding constants, and both functionals perform very well for the chemical shielding constants in the o-benzyne molecule.     

DFT,DLPNO-CCSD(T), and NEVPT2 benchmark study of the reaction between ferrocenium and trimethylphosphine

The reaction between ferrocenium and trimethylphosphine was studied using density functional theory (DFT), domain-based local pair natural orbital coupled cluster theory with single-, double-, and perturbative triple excitations (DLPNO-CCSD(T)), and N-electron valence state perturbation theory (NEVPT2). The accuracy of the DFT functionals decreases compared to the DLPNO-CCSD(T) level in the following order: M06-L > TPSS > M06, BLYP > PBE, PBE0, B3LYP > > PWPB95 > > DSD-BLYP. The roles of thermochemical, continuum solvation (SMD), and counterpoise corrections were evaluated. Grimme's D3 empirical dispersion correction is essential for all functionals studied except M06 and M06-L. The reliability of the frequency calculations performed directly within the SMD was confirmed. The systems showed no significant multireference character according to T₁ and T₂ diagnostics and the fractional occupation number (FOD) weighted electron density analysis. The multireference NEVPT2 calculations gave qualitatively valid conclusions about the reaction mechanism. However, a multireference approach is generally not recommended because it requires arbitrary chosen active spaces.     

Assessment of multicoefficient correlation methods, second-order Møller-Plesset perturbation theory, and density functional theory for H3O(+)(H2O)n (n = 1-5) and OH(-)(H2O)n (n = 1-4)

We have assessed the ability of 52 methods including 15 multicoefficient correlation methods (MCCMs), two complete basis set (CBS) methods, second-order M?ller-Plesset perturbation theory (MP2) with 5 basis sets, the popular B3LYP hybrid functional with 6 basis sets, and 24 combinations of local density functional and basis set to accurately reproduce reaction energies obtained at the Weizmann-1 level of theory for hydronium, hydroxide, and pure water clusters. The three best methods overall are BMC-CCSD, G3SX(MP3), and M06-L/aug-cc-pVTZ. If only microsolvated ion data is included, M06-L/aug-cc-pVTZ is the best method; it has errors only half as large as the other density functionals. The deviations between the three best performing methods are larger for the larger hydronium- and hydroxide-containing clusters, despite a decrease in the average reaction energy, making it impossible to determine which of the three methods is overall the best, so they might be ranked in order of increasing cost, with BMC-CCSD least expensive, followed by M06-L/aug-cc-pVTZ. However, the cost for M06-L will increase more slowly as cluster size increases. This study shows that the M06-L functional is very promising for condensed-phase simulations of the transport of hydronium and hydroxide ions in aqueous solution.     

Benchmarking the performance of time-dependent density functional methods

The performance of 24 density functionals, including 14 meta-generalized gradient approximation (mGGA) functionals, is assessed for the calculation of vertical excitation energies against an experimental benchmark set comprising 14 small- to medium-sized compounds with 101 total excited states. The experimental benchmark set consists of singlet, triplet, valence, and Rydberg excited states. The global-hybrid (GH) version of the Perdew-Burke-Ernzerhoff GGA density functional (PBE0) is found to offer the best overall performance with a mean absolute error (MAE) of 0.28 eV. The GH-mGGA Minnesota 2006 density functional with 54% Hartree-Fock exchange (M06-2X) gives a lower MAE of 0.26 eV, but this functional encounters some convergence problems in the ground state. The local density approximation functional consisting of the Slater exchange and Volk-Wilk-Nusair correlation functional (SVWN) outperformed all non-GH GGAs tested. The best pure density functional performance is obtained with the local version of the Minnesota 2006 mGGA density functional (M06-L) with an MAE of 0.41 eV.     

Parameterized local hybrid functionals from density-matrix similarity metrics

We recently proposed a real-space similarity metric comparing the Kohn-Sham one-particle density matrix to the local spin-density approximation model density matrix [Janesko and Scuseria, J. Chem. Phys. 127, 164117 (2007)]. This metric provides a useful ingredient for constructing local hybrid density functionals that locally mix exact exchange and semilocal density functional theory exchange. Here we present two lines of inquiry: An approximate similarity metric comparing exact versus generalized gradient approximation (GGA), exchange and parameterized mixing functions using these similarity metrics. This approach yields significantly improved thermochemistry, including GGA local hybrids whose thermochemical performance approaches GGA global hybrids.     

Benchmark results for empirical post-GGA functionals: difficult exchange problems and independent tests

Many of the most promising new density functionals have improved the treatment of non-local exchange effects with the help of semi-empirical information and more sophisticated recipes for combining Hartree-Fock and local exchange approximations. In order to quantify recent advancements and identify directions for improvement, we have examined a broad spectrum of test problems. We evaluate the performance of several new hybrid density functionals (ωB97, ωB97X, ωB97X-D, LRC-ωPBEh, M06, M06-2X, and M06-HF) on a variety of chemical problems, some sensitive to the treatment of exact exchange (which we have hoped to systematically improve) and some which require a balanced treatment of correlation. Since all of the functionals under consideration are parameterized with ground-state thermochemical data, the benchmark aims to determine the applicability of the new density functionals to cases that have not been considered in the optimization of the semi-empirical parameters. The first class of benchmarks includes the excitation energies of 21 molecules (83 states) primarily from a recent benchmark conducted by Tozer and co-workers, with some additional references from data made available from the groups of Thiel and Truhlar. We briefly examine the conformational preferences of a small peptide and complete our study with two recently published sets of data that have shown large, systematic errors in simple alkane thermochemistry. While our results indicate that the more general hybrids currently under development perform well for problems outside of their parameterization and improve over the standard hybrid density functionals in an essentially systematic way, there is still a significant self-interaction error in the more difficult cases. Functionals based on a range-separation of exchange and functionals depending on the kinetic-energy density both perform comparably, and there is evidence for complementary strengths.     

Assessment of the accuracy of density functionals for prediction of relative energies and geometries of low-lying isomers of water hexamers

Water hexamers provide a critical testing ground for validating potential energy surface predictions because they contain structural motifs not present in smaller clusters. We tested the ability of 11 density functionals (four of which are local and seven of which are nonlocal) to accurately predict the relative energies of a series of low-lying water hexamers, relative to the CCSD(T)/aug'-cc-pVTZ level of theory, where CCSD(T) denotes coupled cluster theory with an interative treatment of single and double excitations and a quasi-perturbative treatment of connected triple excitations. Five of the density functionals were tested with two different basis sets, making a total of 16 levels of density functional theory (DFT) tested. When single-point energy calculations are carried out on geometries obtained with second-order M?ller-Plesset perturbation theory (MP2), only three density functionals, M06-L, M05-2X, and M06-2X, are able to correctly predict the relative energy ordering of the hexamers. These three functionals predict that the range of energies spanned by the six isomers is 3.2-5.6 kcal/mol, whereas the other eight functionals predict ranges of 1.0-2.4 kcal/mol; the benchmark value for this range is 3.1 kcal/mol. When the hexamers are optimized at each level of theory, all methods are able to reproduce the MP2 geometries well for all isomers except the boat and bag isomers, and DFT optimization changes the energy ordering for seven of the 16 methods tested. The addition of zero-point energy changes the energy ordering for all of the density functionals studied except for M05-2X and M06-2X. The variation in relative energies predicted by the different methods highlights the necessity for exercising caution in the choice of density functionals used in future studies. Of the 11 density functionals tested, the most accurate results for energies were obtained with the PWB6K, MPWB1K, and M05-2X functionals.