Assessment of theoretical procedures for hydrogen-atom abstraction by chlorine, and related reactions

We have examined a number of hydrogen-abstraction reactions and assessed various theoretical procedures with regard to their performance for geometry optimization and for calculating barriers and reaction energies. We find that the BH&H-LYP and M05-2X procedures with the 6-31+G(d,p) basis set provide reasonable predictions for the geometries of the transition structures and also yield reasonable imaginary frequencies when compared with our benchmark QCISD/6-31+G(d,p) and CCSD(T)/6-311+G(3df,2p) values. For the calculation of barriers and reaction energies, M05-2X appears to be the most accurate of the hybrid functionals. The double-hybrid functionals, B2K-PLYP, UB2-PLYP-09, ROB2-PLYP, and DSD-B-LYP-D3, when used in combination with an augmented triple-zeta basis set, give very good agreement with the benchmark URCCSD(T)/aug-cc-pVQZ energies. We find that for wavefunction procedures, use of CCSD(T) in combination with an augmented triple-zeta quality basis set is required for the accurate prediction of barriers and reaction energies for these reactions.     

Benchmark study of DFT functionals for late-transition-metal reactions

The performance of a wide variety of DFT exchange-correlation functionals for a number of late-transition-metal reaction profiles has been considered. Benchmark ab-initio reference data for the prototype reactions Pd + H2, Pd + CH4, Pd + C2H6 (both C-C and C-H activation), and Pd + CH3Cl are presented, while ab-initio data of lesser quality were obtained for the catalytic hydrogenation of acetone and for the low-oxidation-state and high-oxidation-state mechanisms of the Heck reaction. "Kinetics" functionals such as mPW1K, PWB6K, BB1K, and BMK clearly perform more poorly for late-transition-metal reactions than for main-group reactions, as well as compared to general-purpose functionals. There is no single "best functional" for late-transition-metal reactions, but rather a cluster of several functionals (PBE0, B1B95, PW6B95, and TPSS25B95) that perform about equally well; if main-group thermochemical performance is additionally considered, then B1B95 and PW6B95 emerge as the best performers. TPSS25B95 and TPSS33B95 offer attractive performance compromises if weak interactions and main-group barrier heights, respectively, are also important. In the ab-initio calculations, basis set superposition errors (BSSE) can be greatly reduced by ensuring that the metal spd shell has sufficient radial flexibility in the high-exponent range. Optimal HF percentages in hybrid functionals depend on the class of systems considered, increasing from anions to neutrals to cations to main-group barrier heights; transition-metal barrier heights represent an intermediate situation. The use of meta-GGA correlation functionals appears to be quite beneficial.     

A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions

We present a new local density functional, called M06-L, for main-group and transition element thermochemistry, thermochemical kinetics, and noncovalent interactions. The functional is designed to capture the main dependence of the exchange-correlation energy on local spin density, spin density gradient, and spin kinetic energy density, and it is parametrized to satisfy the uniform-electron-gas limit and to have good performance for both main-group chemistry and transition metal chemistry. The M06-L functional and 14 other functionals have been comparatively assessed against 22 energetic databases. Among the tested functionals, which include the popular B3LYP, BLYP, and BP86 functionals as well as our previous M05 functional, the M06-L functional gives the best overall performance for a combination of main-group thermochemistry, thermochemical kinetics, and organometallic, inorganometallic, biological, and noncovalent interactions. It also does very well for predicting geometries and vibrational frequencies. Because of the computational advantages of local functionals, the present functional should be very useful for many applications in chemistry, especially for simulations on moderate-sized and large systems and when long time scales must be addressed.     

Highly accurate first-principles benchmark data sets for the parametrization and validation of density functional and other approximate methods. Derivation of a robust, generally applicable, double-hybrid functional for thermochemistry and thermochemical kinetics

We present a number of near-exact, nonrelativistic, Born-Oppenheimer reference data sets for the parametrization of more approximate methods (such as DFT functionals). The data were obtained by means of the W4 ab initio computational thermochemistry protocol, which has a 95% confidence interval well below 1 kJ/mol. Our data sets include W4-08, which are total atomization energies of over 100 small molecules that cover varying degrees of nondynamical correlations, and DBH24-W4, which are W4 theory values for Truhlar's set of 24 representative barrier heights. The usual procedure of comparing calculated DFT values with experimental atomization energies is hampered by comparatively large experimental uncertainties in many experimental values and compounds errors due to deficiencies in the DFT functional with those resulting from neglect of relativity and finite nuclear mass. Comparison with accurate, explicitly nonrelativistic, ab initio data avoids these issues. We then proceed to explore the performance of B2x-PLYP-type double hybrid functionals for atomization energies and barrier heights. We find that the optimum hybrids for hydrogen-transfer reactions, heavy-atoms transfers, nucleophilic substitutions, and unimolecular and recombination reactions are quite different from one another: out of these subsets, the heavy-atom transfer reactions are by far the most sensitive to the percentages of Hartree-Fock-type exchange y and MP2-type correlation x in an (x, y) double hybrid. The (42,72) hybrid B2K-PLYP, as reported in a preliminary communication, represents the best compromise between thermochemistry and hydrogen-transfer barriers, while also yielding excellent performance for nucleophilic substitutions. By optimizing for best overall performance on both thermochemistry and the DBH24-W4 data set, however, we find a new (36,65) hybrid which we term B2GP-PLYP. At a slight expense in performance for hydrogen-transfer barrier heights and nucleophilic substitutions, we obtain substantially better performance for the other reaction types. Although both B2K-PLYP and B2GP-PLYP are capable of 2 kcal/mol quality thermochemistry, B2GP-PLYP appears to be the more robust toward nondynamical correlation and strongly polar character. We additionally find that double-hybrid functionals display excellent performance for such problems as hydrogen bonding, prototype late transition metal reactions, pericyclic reactions, prototype cumulene-polyacetylene system, and weak interactions.     

B2-PPW91: a promising double-hybrid density functional for the electric response properties

A new double-hybrid density functional, termed B2-PPW91, is presented which includes the Becke88 (B88) exchange in conjunction with Perdew-Wang91 (PW91) gradient-corrected correlation functional. The fitting parameters are obtained by minimization of mean absolute error of the static dipole polarizability of 4d transition metal monohalides against the CCSD(T)∕aug-cc-pVTZ∕SDD results. The performance of proposed functional has been assessed for estimation of other response properties, such as dipole moment and excitation energy, for the same species. We then proceed to explore the validity of B2-PPW91 method for calculation of the dipole polarizability of some 5d transition metal monofluorides. In all cases, the improvement compared to common density functional methods and even previously reported double-hybrid functionals such as B2-PLYP and mPW2-PLYP has been observed. This indicates that the utility of double-hybrid density functional methods can be further extended to study linear and non-linear optical properties of transition metal containing molecules.     

Assessment of theoretical procedures for calculating barrier heights for a diverse set of water-catalyzed proton-transfer reactions

Accurate electronic barrier heights are obtained for a set of nine proton-transfer tautomerization reactions, which are either (i) uncatalyzed, (ii) catalyzed by one water molecule, or (iii) catalyzed by two water molecules. The barrier heights for reactions (i) and (ii) are obtained by means of the high-level ab initio W2.2 thermochemical protocol, while those for reaction (iii) are obtained using the W1 protocol. These three sets of benchmark barrier heights allow an assessment of the performance of more approximate theoretical procedures for the calculation of barrier heights of uncatalyzed and water-catalyzed reactions. We evaluate initially the performance of the composite G4 procedure and variants thereof (e.g., G4(MP2) and G4(MP2)-6X), as well as that of standard ab initio procedures (e.g., MP2, SCS-MP2, and MP4). We find that the performance of the G4(MP2)-type thermochemical procedures deteriorates with the number of water molecules involved in the catalysis. This behavior is linked to deficiencies in the MP2-based basis-set-correction term in the G4(MP2)-type procedures. This is remedied in the MP4-based G4 procedure, which shows good performance for both the uncatalyzed and the water-catalyzed reactions, with mean absolute deviations (MADs) from the benchmark values lying below the threshold of "chemical accuracy" (arbitrarily defined as 1 kcal mol(-1) ≈ 4.2 kJ mol(-1)). We also examine the performance of a large number of density functional theory (DFT) and double-hybrid DFT (DHDFT) procedures. We find that, with few exceptions (most notably PW6-B95 and B97-2), the performance of the DFT procedures that give good results for the uncatalyzed reactions deteriorates with the number of water molecules involved in the catalysis. The DHDFT procedures, on the other hand, show excellent performance for both the uncatalyzed and catalyzed reactions. Specifically, almost all of them afford MADs below the "chemical accuracy" threshold, with ROB2-PLYP and B2K-PLYP showing the best overall performance.     

Semiempirical double-hybrid density functional with improved description of long-range correlation

The recently proposed new family of "double-hybrid" density functionals [Grimme, S. J. Chem. Phys. 2006, 124, 34108] replaces a fraction of the semi-local correlation energy by a non-local correlation energy expression that employs the Kohn-Sham orbitals in second-order many-body perturbation theory. These functionals have provided results of high accuracy over a wide range of properties but fail to accurately describe long-range van der Waals interactions. In this work, a distance-dependent scaling factor for the non-local correlation energy is introduced to address this problem, and two new double-hybrid density functionals are proposed. The new functionals are optimized with the finite cc-pVTZ basis on training sets of atomization energies and intermolecular interaction energies. They are compared against (scaled) second-order M?ller-Plesset perturbation theories and popular density functionals including the hybrid-GGA functional B3-LYP and the first double-hybrid functional (B2-PLYP). Tests are performed on an extensive set including reaction energies, barrier heights, weakly interacting complexes, transition-metal systems, molecular geometries, and harmonic vibrational frequencies. Within the cc-pVTZ atomic orbital basis, we have demonstrated the ability to find a parametrization scheme which is simultaneously able to describe thermochemistry and weakly bound systems with a satisfactory degree of accuracy.     

Benchmark Study of the Performance of Density Functional Theory for Bond Activations with (Ni,Pd)-Based Transition-Metal Catalysts

The performance of 23 density functionals, including one LDA, four GGAs, three meta-GGAs, three hybrid GGAs, eight hybrid meta-GGAs, and ten double-hybrid functionals, was investigated for the computation of activation energies of various covalent main-group single bonds by four catalysts: Pd, PdCl⁻, PdCl₂, and Ni (all in the singlet state). A reactant complex, the barrier, and reaction energy were considered, leading to 164 energy data points for statistical analysis. Extended Gaussian AO basis sets were used in all calculations. The best functional for the complete benchmark set relative to estimated CCSD(T)/CBS reference data is PBE0-D3, with an MAD value of 1.1 kcal mol⁻¹ followed by PW6B95-D3, the double hybrid PWPB95-D3, and B3LYP-D3 (1.9 kcal mol⁻¹ each). The other tested hybrid meta-GGAs perform less well (M06-HF: 7.0 kcal mol⁻¹; M06-2X: 6.3 kcal mol⁻¹; M06: 4.9 kcal mol⁻¹) for the investigated reactions. In the Ni case, some double hybrids show larger errors due to partial breakdown of the perturbative treatment for the correlation energy in cases with difficult electronic structures (partial multi-reference character). Only double hybrids either with very low amounts of perturbative correlation (e.g., PBE0-DH) or that use the opposite-spin correlation component only (e.g., PWPB95) seem to be more robust. We also investigated the effect of the D3 dispersion correction. While the barriers are not affected by this correction, significant and mostly positive results were observed for reaction energies. Furthermore, six very recently proposed double-hybrid functionals were analyzed regarding the influence of the amount of Fock exchange as well as the type of perturbative correlation treatment. According to these results, double hybrids with <50–60 % of exact exchange and ∼30 % perturbative correlation perform best.     

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.     

Further evidences of the quality of double-hybrid energy functionals for π-conjugated systems

Despite numerous interesting efforts along decades to improve the accuracy of density functionals with broad applicability, such as B3LYP, there are still large sets of molecular systems where improvements are badly needed. We select π-conjugated systems as an example of the latter due to the subtle interplay between some physical effects affecting possibly most of the calculations: self-interaction or delocalization error, medium-range correlation signatures, dispersive-like weak interactions, and static correlation effects. We further assess a recently proposed modification of the B2-PLYP double-hybrid functional, called B2π-PLYP, that is expected to yield substantial progress for this kind of systems. Generally speaking, when compared with other more popular and older density functionals, double hybrids behave particularly accurate for π-conjugated systems without suffering the large errors that are common in former yet conventional methods.     

CO oxidation catalyzed by a single gold atom: benchmark calculations and the performance of DFT methods

Quantum chemical calculations were carried out on CO oxidation catalyzed by a single gold atom. To investigate the performance of density functional theory (DFT) methods, 42 DFT functionals have been evaluated and compared with high-level wavefunction based methods. It was found that in order to obtain accurate results the functionals used must treat long range interaction well. The double-hybrid mPW2PLYP and B2PLYP functionals are the two functionals with best overall performance. CAM-B3LYP, a long range corrected hybrid GGA functional, also performs well. On the other hand, the popular B3LYP, PW91, and PBE functionals do not show good performance and the performance of the latter two are even at the bottom of the 42 functionals. Our accurate results calculated at the CCSD(T)/aug-cc-pVTZ//mPW2PLYP/aug-cc-pVTZ level of theory indicate that Au atom is a good catalysis for CO oxidation. The reaction follows the following mechanism where CO and O(2) adsorb on Au atom forming an Au(OCOO) intermediate and subsequently O(2) transfer one oxygen atom to CO to form CO(2) and AuO. Then AuO reacts with CO to form another CO(2) to complete the catalytic cycle. The overall energy barrier at 0 K for the first CO oxidation step (Au + CO + O(2)→ AuO + CO(2)) is just 4.8 kcal mol(-1), and that for the second CO oxidation step (AuO + CO → Au + CO(2)) is just 1.6 kcal mol(-1).     

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 hybrids and time-dependent density functional theory: An implementation and benchmark on charge transfer excited states

In this paper we present the implementation and benchmarking of a Time Dependent Density Functional Theory approach in conjunction with Double Hybrid (DH) functionals. We focused on the analysis of their performance for through space charge-transfer (CT) excitations which are well known to be very problematic for commonly used functionals, such as global hybrids.Two different families of functionals were compared, each of them containing pure, hybrid and double-hybrid functionals.The results obtained show that, beside the robustness of the implementation, these functionals provide results with an accuracy comparable to that of adjusted range-separated functionals, with the relevant difference that for DHs no parameter is tuned on specific compounds thus making them more appealing for a general use. Furthermore, the algorithm described and implemented is characterized by the same computational cost scaling as that of the ground state algorithm employed for MP2 and double hybrids.     

Density functionals for inorganometallic and organometallic chemistry

We present a database of 21 bond dissociation energies for breaking metal-ligand bonds. The molecules in the metal-ligand bond energy database are AgH, CoH, CoO+, CoOH+, CrCH3+, CuOH2+, FeH, Fe(CO)5, FeO, FeS, LiCl, LiO, MgO, MnCH3NiCH2+, Ni(CO)4, RhC, VCO+, VO, and VS. We have also created databases of metal-ligand bond lengths and atomic ionization potentials. The molecules used for bond lengths are AgH, BeO, CoH, CoO+, FeH, FeO, FeS, LiCl, LiO, MgO, RhC, VO, and VS and the ionization potentials are for the following atoms: C, Co, Cr, Cu, Ni, O, and V. The data were chosen based on their diversity and expected reliability, and they are used along with three previously developed databases (transition metal dimer bond energies and bond lengths and main-group molecular atomization energies) for assessing the accuracy of several kinds of density functionals. In particular, we report tests for 42 previously defined functionals: 2 local spin density approximation (LSDA) functionals, 14 generalized gradient approximation (GGA) methods, 13 hybrid GGA methods, 7 meta GGA methods, and 8 hybrid meta GGA methods. In addition to these functionals, we also examine the effectiveness of scaling the correlation energy by testing 13 functionals with scaled or no gradient-corrected correlation energy, and we find that functionals of this kind are more accurate for metal-metal and metal-ligand bonds than any of the functionals already in the literature. We also present a readjusted GGA and a hybrid GGA with parameters adjusted for metals. When we consider these 57 functionals for metal-ligand and metal-metal bond energies simultaneously with main-group atomization energies, atomic ionization potentials, and bond lengths we find that the most accurate functional is G96LYP, followed closely by MPWLYP1M (new in this article), XLYP, BLYP, and MOHLYP (also new in this article). Four of these five functionals have no Hartree-Fock exchange, and the other has only 5%. As a byproduct of this work we introduce a convenient diagnostic, called the B1 diagnostic, for ascertaining the multireference character in a bond.     

Double-hybrid density-functional theory made rigorous

We provide a rigorous derivation of a class of double-hybrid approximations, combining Hartree-Fock exchange and second-order M?ller-Plesset correlation with a semilocal exchange-correlation density functional. These double-hybrid approximations contain only one empirical parameter and use a density-scaled correlation energy functional. Neglecting density scaling leads to a one-parameter version of the standard double-hybrid approximations. We assess the performance of these double-hybrid schemes on representative test sets of atomization energies and reaction barrier heights, and we compare to other hybrid approximations, including range-separated hybrids. Our best one-parameter double-hybrid approximation, called 1DH-BLYP, roughly reproduces the two parameters of the standard B2-PLYP or B2GP-PLYP double-hybrid approximations, which shows that these methods are not only empirically close to an optimum for general chemical applications but are also theoretically supported.     

A new database and benchmark of the bond energies of noble-gas-containing molecules

We have developed a new database of structures and bond energies of 59 noble-gas-containing molecules. The structures were calculated by CCSD(T)/aug-cc-pVTZ methods and the bond energies were obtained using the CCSD(T)/complete basis set method. Many wavefunction-based and density functional theory methods have been benchmarked against the 59 accurate bond energies. Our results show that the MPW1B95, B2GP-PLYP, and DSD-BLYP functionals with the aug-cc-pVTZ basis set excel in predicting the bond energies of noble-gas molecules with mean unsigned errors (MUEs) of 2.0 to 2.1 kcal/mol. When combinations of Dunning's basis sets are used, the MPW1B95, B2GP-PLYP, DSD-BLYP, and BMK functionals give significantly lower MUEs of 1.6 to 1.9 kcal/mol. Doubly hybrid methods using B2GP-PLYP and DSD-BLYP functionals and MP2 calculation also provide satisfactory accuracy with MUEs of 1.4 to 1.5 kcal/mol. If the Ng bond energies and the total atomization energies of a group of 109 main-group molecules are considered at the same time, the MPW1B95/aug-cc-pVTZ single-level method (MUE = 2.7 kcal/mol) and the B2GP-PLYP and DSD-PLYP functionals with combinations of basis sets or using the doubly hybrid method (MUEs = 1.9-2.2 kcal/mol) give the overall best result.     

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.     

Harmonic vibrational frequencies: scale factors for pure, hybrid, hybrid meta, and double-hybrid functionals in conjunction with correlation consistent basis sets

Scale factors for (a) low (<1000 cm(-1)) and high harmonic vibrational frequencies, (b) thermal contributions to enthalpy and entropy, and (c) zero-point vibrational energies have been determined for five hybrid functionals (B3P86, B3PW91, PBE1PBE, BH&HLYP, MPW1K), five pure functionals (BLYP, BPW91, PBEPBE, HCTH93, and BP86), four hybrid meta functionals (M05, M05-2X, M06, and M06-2X) and one double-hybrid functional (B2GP-PLYP) in combination with the correlation consistent basis sets [cc-pVnZ and aug-cc-pVnZ, n = D(2),T(3),Q(4)]. Calculations for vibrational frequencies were carried out on 41 organic molecules and an additional set of 22 small molecules was used for the zero-point vibrational energy scale factors. Before scaling, approximately 25% of the calculated frequencies were within 3% of experimental frequencies. Upon application of the derived scale factors, nearly 90% of the calculated frequencies deviated less than 3% from the experimental frequencies for all of the functionals when the augmented correlation consistent basis sets were used.