Small basis sets for calculations of barrier heights, energies of reaction, electron affinities, geometries, and dipole moments

Beginning with the MIDI! basis set (also called MIDIX), we introduce the MIDIX+, MIDIY, and MIDIY+ basis sets. Using correlated ab initio and hybrid density functional theory, we compare their performance to that of several existing basis sets for electronic structure calculations. The new basis sets are tested with databases of 358 energies of reactions, 44 barrier heights, 31 electron affinities, 18 geometries, and 29 dipole moments. The MIDI!, MIDIX+, MIDIY, and MIDIY+ basis sets are shown to be cost-efficient methods for calculating relative energies, geometries, and dipole moments. The MIDIX+ basis is shown to be particularly efficient for calculating electron affinities of large molecules.Contribution to the Jacopo Tomasi Honorary Issue     

Testing the performance of density functionals for the calculation of energetic properties of complex-forming radical-molecule reactions

The performance of some popular and some more recent density functional methods for the calculation of energies of stationary points on the potential surfaces of radical-molecule reactions was examined. The functionals studied are B3-LYP, BH&H, BH&H-LYP, MPW1K, MPWB1K, TPSS, TPSSh, BB1K, M05 and M05-2X, in conjunction with nine different AO basis sets. The reaction energies, barrier heights and the relative energies of the pre-and post-reaction complexes were compared with those obtained at the CCSD(T)/CBS limit for the reactions of OH radicals with HOOH and CH₃OOH. Very poor barrier heights are provided by the B3-LYP, TPSS and TPSSh functionals. The best overall performance was obtained with the BB1K, MPW1K and MPWB1K functionals. In these reactions all of the studied functionals provide converged results only if they are used with large basis sets like aug-cc-pVTZ and def2-TZVP. The data show that before relying on a functional for a specific reaction, it is desirable to make some test calculations on the performance. The same functional can predict some relative energies very well and some others very poorly even in systems including chemically similar reactants.     

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.     

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.     

The 6-31B(d) basis set and the BMC-QCISD and BMC-CCSD multicoefficient correlation methods

Three new multicoefficient correlation methods (MCCMs) called BMC-QCISD, BMC-CCSD, and BMC-CCSD-C are optimized against 274 data that include atomization energies, electron affinities, ionization potentials, and reaction barrier heights. A new basis set called 6-31B(d) is developed and used as part of the new methods. BMC-QCISD has mean unsigned errors in calculating atomization energies per bond and barrier heights of 0.49 and 0.80 kcal/mol, respectively. BMC-CCSD has mean unsigned errors of 0.42 and 0.71 kcal/mol for the same two quantities. BMC-CCSD-C is an equally effective variant of BMC-CCSD that employs Cartesian rather than spherical harmonic basis sets. The mean unsigned error of BMC-CCSD or BMC-CCSD-C for atomization energies, barrier heights, ionization potentials, and electron affinities is 22% lower than G3SX(MP2) at an order of magnitude less cost for gradients for molecules with 9-13 atoms, and it scales better (N6 vs N,7 where N is the number of atoms) when the size of the molecule is increased.     

Doubly hybrid density functional xDH-PBE0 from a parameter-free global hybrid model PBE0

Following the XYG3 model which uses orbitals and density from B3LYP, an empirical doubly hybrid (DH) functional is developed by using inputs from PBE0. This new functional, named xDH-PBE0, has been tested on a number of different molecular properties, including atomization energies, bond dissociation enthalpies, reaction barrier heights, and nonbonded interactions. From the results obtained, xDH-PBE0 not only displays a significant improvement with respect to the parent PBE0, but also shows a performance that is comparable to XYG3. Arguably, while PBE0 is a parameter-free global hybrid (GH) functional, the B3LYP GH functional contains eight fit parameters. From a more general point of view, the present work points out that reliable and general-purpose DHs can be obtained with a limited number of fit parameters.     

Infinite Basis set extrapolation for Double Hybrid Density Functional Theory 1: effect of applying various extrapolation functions

We applied the Infinite Basis (IB) set extrapolation and Double Hybrid Density Functional Theory (DHDF) to calculate the databases of atomization energies, ionization energies, electron affinities, reaction barrier heights, proton affinities, alkyl bond dissociation energies, and noncovalent interactions. The Complete Basis Set (CBS) limit is estimated by extrapolating the hybrid density functional theory and PT2 energies using extrapolation functions including exponential, inverse power, modified exponential, and the combination of the these functions. We found that the combination of B2KPLYP/cc-pV[D|T]Z (which is the extrapolation based on the energies calculated in cc-pVDZ and cc-pVTZ) gives results in quadruple-ζ quality. However, if we want to reach the ～2 kcal/mol chemical accuracy limit, the cc-pV[T|Q]Z is required. Similar results with various extrapolation functions obtained, because the IB parameters were determined by minimizing the averaged mean unsigned error of the calculated databases. We generalized the IB set extrapolation to include more than two basis sets, but we found that extrapolation with two basis sets is satisfactory to give reasonable results. The largest error occurred in the databases of the electron affinities and the weak interactions between the noble gas and the nonpolar molecules. We expect that performing the DHDF-IB scheme with the basis sets augmented by diffuse basis functions will further improve the results.     

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.     

Density functional study on the reaction mechanism of proton transfer in 2-pyridone: effect of hydration and self-association

The proton-transfer mechanism in the isolated, mono, dehydrated forms and dimers of 2-pyridone and the effect of hydration or self-assistance on the transition state structures corresponding to proton transfer from the keto form to the enol form have been investigated using B3LYP and BH-LYP hybrid density functional methods at the 6-311++G (2d, 2p) basis set level. The barrier heights for both H2O-assisted and self-assisted reactions are significantly lower than that of the bare tautomerization reaction from 2-pyridone to 2-hydroxypyridine, implying the importance of the superior catalytic effect of H2O and (H2O)2 and the important role of 2-pyridone itself for the intramolecular proton transfer. Long-range solvent effects have also been taken into account by using the continuum model (Onsager model and polarizable continuum model (PCM)) of water. The tautomerization energies and the potential energy barriers are increased both for the water-assisted and for the self-assisted reaction because of the bulk solvent, which imply that the tautomerization of PY becomes less favorable in the polar solvent.     

Benchmark calculations of reaction energies, barrier heights, and transition-state geometries for hydrogen abstraction from methanol by a hydrogen atom

We report benchmark calculations of reaction energies, barrier heights, and transition-state geometries for the reaction of CH(3)OH with H to produce CH(2)OH and H(2). Highly accurate composite methods, such as CBS, G2, G3S, G3X, G3SX, and multi-coefficient correlation methods (MCCMs), are used to calibrate lower-cost methods. We also performed single-level CCSD(T) calculations extrapolated to the infinite-basis limit on the basis of aug-cc-pVXZ (X = 3, 4) correlation consistent basis sets. The benchmark high-level calculations give consensus values of the forward reaction barrier height and the reaction energy of 9.7 kcal/mol and - 6.4 kcal/mol, respectively. To evaluate the accuracy of cost-efficient methods that are potentially useful for dynamics studies of the title reaction, we further include the results obtained by hybrid density functional theory methods and hybrid meta density functional theory methods that have recently been designed for chemical kinetics. Results obtained by popular semiempirical methods are also given for comparison. On the basis of the benchmark gas-phase results, we suggest MC-QCISD/3, MC3BB, and BB1K as reasonably accurate and affordable electronic structure methods for calculating dynamics for the title reaction.     

Kinetics and Thermodynamics of Reactions Involving Criegee Intermediates: An Assessment of Density Functional Theory and Ab Initio Methods Through Comparison with CCSDT(Q)/CBS Data

Reactions involving Criegee intermediates (CIs, R₁R₂COO) are important in atmospheric ozonolysis models. In recent years, density functional theory (DFT) and CCSD(T)-based ab initio methods are increasingly being used for modeling reaction profiles involving CIs. We obtain highly accurate CCSDT(Q)/CBS reaction energies and barrier heights for ring-closing reactions involving atmospherically important CIs (R₁/R₂ = H, Me, OH, OMe, F, CN, cyclopropene, ethylene, acetaldehyde, and acrolein). We use this benchmark data to evaluate the performance of DFT, double-hybrid DFT (DHDFT), and ab initio methods for the kinetics and thermodynamics of these reactions. We find that reaction energies are more challenging for approximate theoretical procedures than barrier heights. Overall, taking both reaction energies and barrier heights into account, only one of the 58 considered DFT methods (the meta-GGA MN12-L) attains near chemical accuracy, with root-mean-square deviations (RMSDs) of 3.5 (barrier heights) and 4.7 (reaction energies) kJ mol⁻¹. Therefore, MN12-L is recommended for investigations where CCSD(T)-based methods are not computationally feasible. For reaction barrier heights performance does not strictly follow Jacob's Ladder, for example, DHDFT methods do not perform better than conventional DFT methods. Of the ab initio methods, the cost-effective CCSD(T)/CBS(MP2) approach gives the best performance for both reaction energies and barrier heights, with RMSDs of 1.7 and 1.4 kJ mol⁻¹, respectively. All the considered Gaussian-n methods show good performance with RMSDs below the threshold of chemical accuracy for both reaction energies and barrier heights, where G4(MP2) shows the best overall performance with RMSDs of 2.9 and 1.5 kJ mol⁻¹, respectively. © 2019 Wiley Periodicals, Inc.     

Barrier heights of hydrogen‐transfer reactions with diffusion quantum monte carlo method

Hydrogen‐transfer reactions are an important class of reactions in many chemical and biological processes. Barrier heights of H‐transfer reactions are underestimated significantly by popular exchange–correlation functional with density functional theory (DFT), while coupled‐cluster (CC) method is quite expensive and can be applied only to rather small systems. Quantum Monte‐Carlo method can usually provide reliable results for large systems. Performance of fixed‐node diffusion quantum Monte‐Carlo method (FN‐DMC) on barrier heights of the 19 H‐transfer reactions in the HTBH38/08 database is investigated in this study with the trial wavefunctions of the single‐Slater–Jastrow form and orbitals from DFT using local density approximation. Our results show that barrier heights of these reactions can be calculated rather accurately using FN‐DMC and the mean absolute error is 1.0 kcal/mol in all‐electron calculations. Introduction of pseudopotentials (PP) in FN‐DMC calculations improves efficiency pronouncedly. According to our results, error of the employed PPs is smaller than that of the present CCSD(T) and FN‐DMC calculations. FN‐DMC using PPs can thus be applied to investigate H‐transfer reactions involving larger molecules reliably. In addition, bond dissociation energies of the involved molecules using FN‐DMC are in excellent agreement with reference values and they are even better than results of the employed CCSD(T) calculations using the aug‐cc‐pVQZ basis set. © 2017 Wiley Periodicals, Inc.     

Simultaneous benchmarking of ground- and excited-state properties with long-range-corrected density functional theory

We present benchmark calculations using several long-range-corrected (LRC) density functionals, in which Hartree-Fock exchange is incorporated asymptotically using a range-separated Coulomb operator, while local exchange is attenuated using an ansatz introduced by Iikura et al. [J. Chem. Phys. 115, 3540 (2001)]. We calculate ground-state atomization energies, reaction barriers, ionization energies, and electron affinities, each as a function of the range-separation parameter mu. In addition, we calculate excitation energies of small- and medium-sized molecules, again as a function of mu, by applying the LRC to time-dependent density functional theory. Representative examples of both pure and hybrid density functionals are tested. On the basis of these results, there does not appear to be a single range-separation parameter that is reasonable for both ground-state properties and vertical excitation energies. Reasonable errors in atomization energies and barrier heights are achieved only at the expense of excessively high excitation energies, at least for the medium-sized molecules, whereas values of mu that afford reasonable excitation energies yield some of the largest errors for ground-state atomization energies and barrier heights in small molecules. Notably, this conclusion is obscured if the database of excitation energies includes only small molecules, as has been the case in previous benchmark studies of LRC functionals.     

Proper gaussian basis sets for density functional studies of water dimers and trimers

The accuracy of the Perdew-Burke-Ernzerhof and Tao-Perdew-Staroverov-Scuseria density functionals for describing noncovalent interaction energies in small water clusters is studied by testing 11 basis sets on a reduced test set proposed by Dahlke and Truhlar (J. Phys. Chem. B 2005, 109, 15677). We have also tested variants of the Perdew-Burke-Ernzerhof functional and the Becke98 hybrid functional. While moderate basis sets give converged density functional theory results for covalent dissociation energies, this is not true for noncovalent interaction energies. Our results show that density functionals give converged interaction energies with aug-cc-pVTZ and aug-cc-pVQZ basis sets. Gradual simplification of the basis set introduces an increasing overbinding effect. The best agreement with the high-level result was obtained by the Perdew-Burke-Ernzerhof functional at the basis set limit. The converged Tao-Perdew-Staroverov-Scuseria interaction energies show a systematic underbinding effect that can be compensated by a somewhat systematic overbinding basis set effect of smaller basis sets such as 6-31+G(d,2p). The inclusion of the diffuse functions in the oxygen basis set is very important, while the inclusion of the f functions practically does not influence the results. Improvement can be obtained by adding more hydrogen p functions to the 6-31+G basis set.     

Generalized hybrid-orbital method for combining density functional theory with molecular mechanicals.

The generalized hybrid orbital (GHO) method has previously been formulated for combining molecular mechanics with various levels of quantum mechanics, in particular semiempirical neglect of diatomic differential overlap theory, ab initio Hartree-Fock theory, and self-consistent charge density functional tight-binding theory. To include electron-correlation effects accurately and efficiently in GHO calculations, we extend the GHO method to density functional theory in the generalized-gradient approximation and hybrid density functional theory (denoted by GHO-DFT and GHO-HDFT, respectively) using Gaussian-type orbitals as basis functions. In the proposed GHO-(H)DFT formalism, charge densities in auxiliary hybrid orbitals are included to calculate the total electron density. The orthonormality constraints involving the auxiliary Kohn-Sham orbitals are satisfied by carrying out the hybridization in terms of a set of L?wdin symmetrically orthogonalized atomic basis functions. Analytical gradients are formulated for GHO-(H)DFT by incorporating additional forces associated with GHO basis transformations. Scaling parameters are introduced for some of the one-electron integrals and are optimized to obtain the correct charges and geometry near the QM/MM boundary region. The GHO-(H)DFT method based on the generalized gradient approach (GGA) (BLYP and mPWPW91) and HDFT methods (B3 LYP, mPW1PW91, and MPW1 K) is tested-for geometries and atomic charges-against a set of small molecules. The following quantities are tested: 1) the C--C stretch potential in ethane, 2) the torsional barrier for internal rotation around the central C--C bond in n-butane, 3) proton affinities for a set of alcohols, amines, thiols, and acids, 4) the conformational energies of alanine dipeptide, and 5) the barrier height of the hydrogen-atom transfer between n-C4H10 and n-C4H9, where the reaction center is described at the MPW1 K/6-31G(d) level of theory.     

Computational study of the reactions of methanol with the hydroperoxyl and methyl radicals. 1. Accurate thermochemistry and barrier heights

The reactions of CH(3)OH with the HO(2) and CH(3) radicals are important in the combustion of methanol and are prototypes for reactions of heavier alcohols in biofuels. The reaction energies and barrier heights for these reaction systems are computed with CCSD(T) theory extrapolated to the complete basis set limit using correlation-consistent basis sets, both augmented and unaugmented, and further refined by including a fully coupled treatment of the connected triple excitations, a second-order perturbative treatment of quadruple excitations (by CCSDT(2)(Q)), core-valence corrections, and scalar relativistic effects. It is shown that the M08-HX and M08-SO hybrid meta-GGA density functionals can achieve sub-kcal mol(-1) agreement with the high-level ab initio results, identifying these functionals as important potential candidates for direct dynamics studies on the rates of these and homologous reaction systems.     

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.     

DFT study of the C-Cl bond dissociation enthalpies and electronic structure of substituted chlorobenzene compounds

Quantum chemical calculations were used to estimate the bond dissociation energies (BDEs) for 13 substituted chlorobenzene compounds. These compounds were studied by employing the hybrid density functional theory methods (B3LYP, B3PW1, B3P86) with 6-31G** and 6-311G** basis sets. It was demonstrated that B3P86/6-311G** method is the best method for computing the reliable BDEs for substituted chlorobenzene compounds which contain the C-Cl bond. It was found that the C-Cl BDE depends strongly on a computational method and basis set used. Substitution effect on the C-Cl BDE of substituted chlorobenzene compounds is further discussed. It is shown that the effects of substitution on the C-Cl BDE of substituted chlorobenzene compounds are very insignificant. Frontier orbital energy gap of studied compounds was also investigated. From the data on frontier orbital energies gap, we estimated the relative thermal stability of substituted chlorobenzene compounds.     

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 density functional theory study of the dimers of HX (X = F,Cl, and Br)

The geometries, interaction energies, and vibrational properties of (HF)₂, (HCl)₂, and (HBr)₂ have been investigated using a variety of hybrid density functional methods and the 6‐311+G(2df,p) basis set. Although most of the density functional methods predict geometries in reasonable agreement with experiment, methods containing the LYP correlation functional yield geometrical parameters in slightly better agreement. The (HF)₂ interaction energy, predicted by the hybrid density functional methods, is in reasonable agreement with experiment, but the (HCl)₂ and (HBr)₂ interaction energies are underestimated substantially. The frequency shift for the X? H donor bond correlates linearly with the elongation of the X? H bond upon complexation, and is predicted reasonably well by methods containing the LYP functional. Overall, the hybrid density functional methods adequately predict the properties of the halide dimers. It is interesting to note that the BHandHLYP and B1LYP density functional methods offer a competitive alternative to the popular B3LYP method. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1590–1597, 2001