A pseudopotential-based composite method: the relativistic pseudopotential correlation consistent composite approach for molecules containing 4d transition metals (Y-Cd)

The correlation consistent composite approach (ccCA) has proven to be an effective first-principles-based composite approach for main group and first-row transition metal species. By combining relativistic pseudopotentials and ccCA, accurate energetic and thermodynamic data for heavier elements, including transition metals, is obtainable. Relativistic pseudopotential ccCA (rp-ccCA) was formulated and tested on 25 molecules from the G3∕05 set that contain 4p elements (Ga-Kr). A 32.5% time savings was obtained using rp-ccCA, relative to ccCA employing all-electron basis sets. When implementing rp-ccCA to compute dissociation energies and enthalpies of formation for molecules from the 4p block, rp-ccCA results in a mean absolute deviation of 0.89 kcal?mol(-1) from experimental data. rp-ccCA was also applied to a set of 30 4d transition metal-containing molecules, ranging from diatomics to Mo(CO)(6), and enthalpies of formation for these species were obtained with a mean absolute deviation of 2.89 kcal mol(-1) in comparison to experimental data. Based on quality of the experimentally available enthalpies of formation, where the average value of reported experimental error bars is 3.43 kcal mol(-1), rp-ccCA is within transition metal chemical accuracy for the 4d molecule set. rp-ccCA is a pseudopotential-based composite method for transition metals and is shown to yield accurate thermodynamic results for molecules containing heavy elements Ga-Kr and Y-Cd.     

Assessment of Gaussian-3 and density-functional theories on the G3/05 test set of experimental energies

The G3/99 test set [L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys. 112, 7374 (2000)] of thermochemical data for validation of quantum chemical methods is expanded to include 78 additional energies including 14 enthalpies of formation of the first- and second-row nonhydrogen molecules, 58 energies of molecules containing the third-row elements K, Ca, and Ga-Kr, and 6 hydrogen-bonded complexes. The criterion used for selecting the additional systems is the same as before, i.e., experimental uncertainties less than +/- 1 kcal/mol. This new set, referred to as the G3/05 test set, has a total of 454 energies. The G3 and G3X theories are found to have mean absolute deviations of 1.13 and 1.01 kcal/mol, respectively, when applied to the G3/05 test set. Both methods have larger errors for the nonhydrogen subset of 79 species for which they have mean absolute deviations of 2.10 and 1.64 kcal/mol, respectively. On all of the other types of energies the G3 and G3X methods are very reliable. The G3/05 test set is also used to assess density-functional methods including a series of new functionals. The most accurate functional for the G3/05 test set is B98 with a mean absolute deviation of 3.33 kcal/mol, compared to 4.14 kcal/mol for B3LYP. The latter functional has especially large errors for larger molecules with a mean absolute deviation of 9 kcal/mol for molecules having 28 or more valence electrons. For smaller molecules B3LYP does as well or better than B98 and the other functionals. It is found that many of the density-functional methods have significant errors for the larger molecules in the test set.     

Quantitative computational thermochemistry of transition metal species

The correlation consistent Composite Approach (ccCA), which has been shown to achieve chemical accuracy (+/-1 kcal mol-1) for a large benchmark set of main group and s-block metal compounds, is used to compute enthalpies of formation for a set of 17 3d transition metal species. The training set includes a variety of metals, ligands, and bonding types. Using the correlation consistent basis sets for the 3d transition metals, we find that gas-phase enthalpies of formation can be efficiently calculated for inorganic and organometallic molecules with ccCA. However, until the reliability of gas-phase transition metal thermochemistry is improved, both experimentally and theoretically, a large experimental training set where uncertainties are near +/-1 kcal mol-1 (akin to commonly used main group benchmarking sets) remains an ambitious goal. For now, an average deviation of +/-3 kcal mol-1 appears to be the initial goal of "chemical accuracy" for ab initio transition metal model chemistries. The ccCA is also compared to a more robust but relatively expensive composite approach primarily utilizing large basis set coupled cluster computations. For a smaller training set of eight molecules, ccCA has a mean absolute deviation (MAD) of 3.4 kcal mol-1 versus the large basis set coupled-cluster-based model chemistry, which has a MAD of 3.1 kcal mol-1. However, the agreement for transition metal complexes is more system dependent than observed in previous benchmark studies of composite methods and main group compounds.     

Quantum monte carlo study of heats of formation and bond dissociation energies of small hydrocarbons

A quantum Monte Carlo (QMC) benchmark study of heats of formation at 298 K and bond dissociation energies (BDEs) of 22 small hydrocarbons is reported. Diffusion Monte Carlo (DMC) results, obtained using a simple product trial wavefunctions consisting of a single determinant and correlation function, are compared to experiment and to other theory including a version of complete basis set theory (CBS‐Q) and density functional theory (DFT) with the B3LYP functional. For heats of formation, the findings are a mean absolute deviation from experiment of 1.2 kcal/mol for CBS‐Q, 2.0 kcal/mol for B3LYP, and 2.2 kcal/mol for DMC. The mean absolute deviation of 31 BDEs is 2.0 kcal/mol for CBS‐Q, 4.2 kcal/mol for B3LYP, and 2.5 kcal/mol for DMC. These findings are for 17 BDEs of closed‐shell molecules that have mean absolute deviations from experiment of 1.7 kcal/mol (CBS‐Q), 4.0 kcal/mol (B3LYP), and 2.2 kcal/mol (DMC). The corresponding results for the 14 BDEs of open‐shell molecules studied are 2.4 kcal/mol (CBS‐Q), 4.3 kcal/mol (B3LYP), and 2.9 kcal/mol (DMC). The DMC results provide a baseline from which improvement using multideterminant trial functions can be measured. © 2005 Wiley Periodicals, Inc. Int J Chem Kinet 37: 583–592, 2005     

The C2H3 + O2 reaction revisited: is multireference treatment of the wave function really critical?

This letter revisits critical intermediates and transition states of the C2H3 + O2 reaction. To obtain their accurate relative energies, ab initio calculations are performed using sophisticated single and multireference theoretical methods with various basis sets. The energy difference between two crucial transition states, for ring opening in dioxiranylmethyl radical and its isomerization to C2H3OO, is calculated as approximately 2 kcal/mol both at multireference MRCI and at single-reference CCSD(T) levels extrapolated to the complete basis set limit. The deviation from the earlier G2M(RCC,MP2) value (approximately 7 kcal/mol) is caused by a deficiency of the 6-311+G(3df,2p) basis set as compared to correlation-consistent Dunning's basis sets.     

Probing the limits of accuracy in electronic structure calculations: is theory capable of results uniformly better than "chemical accuracy"?

Current limitations in electronic structure methods are discussed from the perspective of their potential to contribute to inherent uncertainties in predictions of molecular properties, with an emphasis on atomization energies (or heats of formation). The practical difficulties arising from attempts to achieve high accuracy are illustrated via two case studies: the carbon dimer (C2) and the hydroperoxyl radical (HO2). While the HO2 wave function is dominated by a single configuration, the carbon dimer involves considerable multiconfigurational character. In addition to these two molecules, statistical results will be presented for a much larger sample of molecules drawn from the Computational Results Database. The goal of this analysis will be to determine if a combination of coupled cluster theory with large 1-particle basis sets and careful incorporation of several computationally expensive smaller corrections can yield uniform agreement with experiment to better than "chemical accuracy" (+/-1 kcal/mol). In the case of HO2, the best current theoretical estimate of the zero-point-inclusive, spin-orbit corrected atomization energy (SigmaD0=166.0+/-0.3 kcal/mol) and the most recent Active Thermochemical Table (ATcT) value (165.97+/-0.06 kcal/mol) are in excellent agreement. For C2 the agreement is only slightly poorer, with theory (D0=143.7+/-0.3 kcal/mol) almost encompassing the most recent ATcT value (144.03+/-0.13 kcal/mol). For a larger collection of 68 molecules, a mean absolute deviation of 0.3 kcal/mol was found. The same high level of theory that produces good agreement for atomization energies also appears capable of predicting bond lengths to an accuracy of +/-0.001 A.     

The correlation consistent composite approach (ccCA): an alternative to the Gaussian-n methods

An alternative to the Gaussian-n (G1, G2, and G3) composite methods of computing molecular energies is proposed and is named the "correlation consistent composite approach" (ccCA, ccCA-CBS-1, ccCA-CBS-2). This approach uses the correlation consistent polarized valence (cc-pVXZ) basis sets. The G2-1 test set of 48 enthalpies of formation (DeltaHf), 38 adiabatic ionization potentials (IPs), 25 adiabatic electron affinities (EAs), and 8 adiabatic proton affinities (PAs) are computed using this approach, as well as the DeltaHf values of 30 more systems. Equilibrium molecular geometries and vibrational frequencies are obtained using B3LYP density functional theory. When applying the ccCA-CBS method with the cc-pVXZ series of basis sets augmented with diffuse functions, mean absolute deviations within the G2-1 test set compared to experiment are 1.33 kcal mol(-1) for DeltaHf,0.81 kcal mol(-1) for IPs, 1.02 kcal mol(-1) for EAs, and 1.51 kcal mol(-1) for PAs, without including the "high-level correction" (HLC) contained in the original Gn methods. Whereas the HLC originated in the Gaussian-1 method as an isogyric correction, it evolved into a fitted parameter that minimized the error of the composite methods, eliminating its physical meaning. Recomputing the G1 and G3 enthalpies of formation without the HLC reveals a systematic trend where most DeltaHf values are significantly higher than experimental values. By extrapolating electronic energies to the complete basis set (CBS) limit and adding G3-like corrections for the core-valence and infinite-order electron correlation effects, ccCA-CBS-2 often underestimates the experimental DeltaHf, especially for larger systems. This is desired as inclusion of relativistic and atomic spin-orbit effects subsequently improves theoretical DeltaHf values to give a 0.81 kcal mol(-1) mean absolute deviation with ccCA-CBS-2. The ccCA-CBS method is a viable "black box" method that can be used on systems with at least 10-15 heavy atoms.     

Fitting atomic correlation parameters for RECEP (rapid estimation of correlation energy from partial charges) method to estimate molecular correlation energies within chemical accuracy

The accuracy of the RECEP method [Chem Phys 1997, 224, 33 and Chem Phys Lett 1999, 307, 469] has been increased considerably by the use of fitted atomic correlation parameters. This method allows an extremely rapid, practically prompt calculation of the correlation energy of molecules after an HF‐SCF calculation. The G2 level correlation energy and HF‐SCF charge distribution of 41 closed‐shell neutral molecules (composed of H, C, N, O, and F atoms) of the G2 thermochemistry database were used to obtain the fitted RECEP atomic correlation parameters. Four different mathematical definitions of partial charges, as a multiple choice, were used to calculate the molecular correlation energies. The best results were obtained using the natural population analysis, although the other three are also recommended for use. For the 41 molecules, the G2 results were approached within a 1.8 kcal/mol standard deviation (the mean absolute difference was 1.5 kcal/mol). The RECEP atomic correlation parameters were also tested on a different, nonoverlapping set of other 24 molecules from the G2 thermochemistry database. The G2 results of these 24 molecules were approached within a 2.3 kcal/mol standard deviation (the mean absolute difference was 1.9 kcal/mol). This method is recommended to estimate total correlation energies of closed shell ground‐state neutral molecules at stationary (minimums and transition states) points on the potential surface. Extension of the work for charged molecules, radicals, and molecules containing other atoms is straightforward. Numerical example as a recipe is also provided. © 2000 John Wiley & Sons, Inc. J Comput Chem 22: 241–254, 2001     

The accuracy of diffusion quantum Monte Carlo simulations in the determination of molecular equilibrium structures

For a test set of 17 first-row small molecules, the equilibrium structures are calculated with Ornstein-Uhlenbeck diffusion quantum Monte Carlo simulations guiding by trial wave functions constructed from floating spherical Gaussian orbitals and spherical Gaussian geminals. To measure performance of the Monte Carlo calculations, the mean deviation, the mean absolute deviation, the maximum absolute deviation, and the standard deviation of Monte Carlo calculated equilibrium structures with respect to empirical equilibrium structures are given. This approach is found to yield results having a uniformly high quality, being consistent with empirical equilibrium structures and surpassing calculated values from the coupled cluster model with single, double, and noniterative triple excitations [CCSD(T)] with the basis sets of cc-pCVQZ and cc-pVQZ. The nonrelativistic equilibrium atomization energies are also presented to assess performance of the calculated methods. The mean absolute deviations regarding experimental atomization energy are 0.16 and 0.21 kcal/mol for the Monte Carlo and CCSD(T)/cc-pCV(56)Z calculations, respectively.     

Predicting binding affinities of host-guest systems in the SAMPL3 blind challenge: the performance of relative free energy calculations

Relative free energy calculations based on molecular dynamics simulations are combined with available experimental binding free energies to predict unknown binding affinities of acyclic Cucurbituril complexes in the blind SAMPL3 competition. The predictions yield root mean square errors between 2.6 and 3.2 kcal/mol for seven host-guest systems. Those deviations are comparable to results for solvation free energies of small organic molecules. However, the standard deviations found in our simulations range from 0.4 to 2.4 kcal/mol, which indicates the need for better sampling. Three different approaches are compared. Bennett's Acceptance Ratio Method and thermodynamic integration based on the trapezoidal rule with 12 λ-points exhibit a root mean square error of 2.6 kcal/mol, while thermodynamic integration with Simpson's rule and 11 λ-points leads to a root mean square error of 3.2 kcal/mol. In terms of absolute median errors, Bennett's Acceptance Ratio Method performs better than thermodynamic integration with the trapezoidal rule (1.7 vs. 2.9 kcal/mol). Simulations of the deprotonated forms of the guest molecules exhibit a poorer correspondence to experimental results with a root mean square error of 5.2 kcal/mol. In addition, a decrease of the buffer concentration by approximately 20 mM in the simulations raises the root mean square error to 3.8 kcal/mol.     

Basis set limit electronic excitation energies, ionization potentials, and electron affinities for the 3d transition metal atoms: Coupled cluster and multireference methods

Recently developed correlation consistent basis sets for the first row transition metal elements Sc-Zn have been utilized to determine complete basis set (CBS) scalar relativistic electron affinities, ionization potentials, and 4s(2)3d(n-2)-4s(1)d(n-1) electronic excitation energies with single reference coupled cluster methods [CCSD(T), CCSDT, and CCSDTQ] and multireference configuration interaction with three reference spaces: 3d4s, 3d4s4p, and 3d4s4p3d'. The theoretical values calculated with the highest order coupled cluster techniques at the CBS limit, including extrapolations to full configuration interaction, are well within 1 kcal/mol of the corresponding experimental data. For the early transition metal elements (Sc-Mn) the internally contracted multireference averaged coupled pair functional method yielded excellent agreement with experiment; however, the atomic properties for the late transition metals (Mn-Zn) proved to be much more difficult to describe with this level of theory, even with the largest reference function of the present work.     

Approaching chemical accuracy with quantum Monte Carlo

A quantum Monte Carlo study of the atomization energies for the G2 set of molecules is presented. Basis size dependence of diffusion Monte Carlo atomization energies is studied with a single determinant Slater-Jastrow trial wavefunction formed from Hartree-Fock orbitals. With the largest basis set, the mean absolute deviation from experimental atomization energies for the G2 set is 3.0 kcal/mol. Optimizing the orbitals within variational Monte Carlo improves the agreement between diffusion Monte Carlo and experiment, reducing the mean absolute deviation to 2.1 kcal/mol. Moving beyond a single determinant Slater-Jastrow trial wavefunction, diffusion Monte Carlo with a small complete active space Slater-Jastrow trial wavefunction results in near chemical accuracy. In this case, the mean absolute deviation from experimental atomization energies is 1.2 kcal/mol. It is shown from calculations on systems containing phosphorus that the accuracy can be further improved by employing a larger active space.     

Thermochemical kinetics of hydrogen-atom transfers between methyl, methane, ethynyl, ethyne, and hydrogen

Saddle point properties of three symmetric and one asymmetric hydrogen transfer and the energy of reaction of the asymmetric reactions are investigated in the present work. These reactions were calculated by various density functionals, many of which were developed in recent years, by coupled cluster theory, and by multicoefficient correlation methods based on wave function theory. Instead of comparing calculated results to "semi-experimental" values, we compared them to very accurate theoretical values (e.g., to values obtained by the Weizmann-1 method). Coupled cluster theory and the multicoefficient correlation methods MC-QCISD/3 and MCQCISD-MPW are very accurate for these reactions with mean unsigned errors below 0.94 kcal/mol. Diagnostics for multireference character add additional reliability to these results. The newly developed hybrid density functional M06-2X shows very good performance for these reactions with a mean unsigned error of only 0.77 kcal/mol; the BHandHLYP, MPW1K, and BB1K density functionals, can also predict these reactions well with mean unsigned errors less than 1.42 kcal/mol.     

Automerization reaction of cyclobutadiene and its barrier height: an ab initio benchmark multireference average-quadratic coupled cluster study

The problem of the double bond flipping interconversion of the two equivalent ground state structures of cyclobutadiene (CBD) is addressed at the multireference average-quadratic coupled cluster level of theory, which is capable of optimizing the structural parameters of the ground, transition, and excited states on an equal footing. The barrier height involving both the electronic and zero-point vibrational energy contributions is 6.3 kcal mol(-1), which is higher than the best earlier theoretical estimate of 4.0 kcal mol(-1). This result is confirmed by including into the reference space the orbitals of the CC sigma bonds beyond the standard pi orbital space. It places the present value into the middle of the range of the measured data (1.6-10 kcal mol(-1)). An adiabatic singlet-triplet energy gap of 7.4 kcal mol(-1) between the transition state (1)B(tg) and the first triplet (3)A(2g) state is obtained. A low barrier height for the CBD automerization and a small DeltaE((3)A(2g),(1)B(1g)) gap bear some relevance on the highly pronounced reactivity of CBD, which is briefly discussed.     

Performance of the spin-flip and multireference methods for bond breaking in hydrocarbons: a benchmark study

Benchmark results for spin-flip (SF) coupled-cluster and multireference (MR) methods for bond-breaking in hydrocarbons are presented. The nonparallelity errors (NPEs), which are defined as an absolute value of the difference between the maximum and minimum values of the errors in the potential energy along bond-breaking curves, are analyzed for (i) the entire range of nuclear distortions from equilibrium to the dissociation limit and (ii) in the intermediate range (2.5-4.5 A), which is the most relevant for kinetics modeling. For methane, the spin-flip and MR results are compared against full configuration interaction (FCI). For the entire potential energy curves, the NPEs for the SF model with single and double substitutions (SF-CCSD) are slightly less than 3 kcal/mol. Inclusion of triple excitations reduces the NPEs to 0.32 kcal/mol. The corresponding NPEs for the MR-CI are less than 1 kcal/mol, while those of multireference perturbation theory are slightly larger (1.2 kcal/mol). The NPEs in the intermediate range are smaller for all of the methods. The largest errors of 0.35 kcal/mol are observed, surprisingly, for a spin-flip approach that includes triple excitations, while MR-CI, CASPT2, and SF-CCSD curves are very close to each other and are within 0.1-0.2 kcal/mol of FCI. For a larger basis set, the difference between MR-CI and CASPT2 is about 0.2 kcal/mol, while SF-CCSD is within 0.4 kcal/mol of MR-CI. For the C-C bond breaking in ethane, the results of the SF-CCSD are within 1 kcal/mol of MR-CI for the entire curve and within 0.4 kcal/mol in the intermediate region. The corresponding NPEs for CASPT2 are 1.8 and 0.4 kcal/mol, respectively. Including the effect of triples by energy-additivity schemes is found to be insignificant for the intermediate region. For the entire range of nuclear separations, sufficiently large basis sets are required to avoid artifacts at small internuclear separations.     

Activation energies of pericyclic reactions: performance of DFT, MP2, and CBS-QB3 methods for the prediction of activation barriers and reaction energetics of 1,3-dipolar cycloadditions, and revised activation enthalpies for a standard set of hydrocarbon pericyclic reactions

Activation barriers and reaction energetics for the three main classes of 1,3-dipolar cycloadditions, including nine different reactions, were evaluated with the MPW1K and B3LYP density functional methods, MP2, and the multicomponent CBS-QB3 method. The CBS-QB3 values were used as standards for 1,3-dipolar cycloaddition activation barriers and reaction energetics, and the density functional theory (DFT) and MP2 methods were benchmarked against these values. The MPW1K/6-31G* method and basis set performs best for activation barriers, with a mean absolute deviation (MAD) value of 1.1 kcal/mol. The B3LYP/6-31G* method and basis set performs best for reaction enthalpies, with a MAD value of 2.4 kcal/mol, while the MPW1K method shows large errors for reaction energetics. The MP2 method gives the expected systematic underestimation of barriers. Concerted and nearly synchronous transition structures are predicted by all DFT and MP2 methods. Also reported are revised estimated 0 K experimental activation enthalpies for a standard set of hydrocarbon pericyclic reactions and updated comparisons to experiment for DFT, ab initio, and multicomponent methods. B3LYP and MPW1K methods with MAD values of 1.5 and 2.1 kcal/mol, respectively, fortuitously outperform the multicomponent CBS-QB3 method, which has a MAD value of 2.3. The MAD value of the O3LYP functional improves to 2.4 kcal/mol from the previously reported 3.0 kcal/mol.     

Polyfunctional methodology for improved DFT thermochemical predictions

Statistical error distributions for enthalpies of formation as predicted by 18 different density functionals have been analyzed using a test set of 675 molecules. Systematic errors, dependent on the number of valence electrons, have been identified for some functionals. A simple empirical correction makes a significant improvement in the prediction error for these single functionals. Linear combinations of enthalpy estimates from different density functionals are identified that exploit the error correlations among the functionals and allow for further improvements in the accuracy of thermodynamic predictions. A good compromise between accuracy and computational efforts is achieved by the BLUE (best linear unbiased estimator) combination of three functionals, B3LYP, BLYP, and VSXC (polyfunctional 3 or PF3). The PF3 method has a mean absolute deviation (MAD) from experiment of 2.4 kcal/mol on the G3 set of 271 molecules. This can be compared to the MAD of 4.9 kcal/mol for B3LYP and 1.2 kcal/mol for the more costly G3 method. On the larger set of 675 molecules, the MAD for PF3 is 3.0 kcal/mol. Opportunities for further improvements in the accuracy of this method are discussed.     

Validation of dispersion-corrected density functional theory approaches for ionic liquid systems

The performance of several general gradient approximation, meta general gradient approximation, and hybrid functionals is tested against M?ller-Plesset perturbation theory second-order for ionic liquid systems. Additionally, two dispersion-corrected approaches (addition of van der Waals forces by a 1/r(6) term and employing a dispersion-corrected atom-center dispersion pseudopotential) were studied. For the 1-butyl-3-methylimidazolium cation neglecting dispersion results in different trends for structural stabilities. The two applied correction schemes for density functional theory improve the results tremendously. Investigating several 1-butyl-3-methylimidazolium dicianamide ion pairs shows a mean absolute deviation from M?ller-Plesset perturbation theory of 35.7 kJ/mol for Hartree-Fock and up to 33.2 kJ/mol for the density functional theory methods. The dispersion-corrected methods reduce the mean absolute deviation to less than 10 kJ/mol. Comparing adducts of the 1-ethyl-3-methylimidazolium dicianamide ion pair with Diels-Alder educts (cyclopentadiene and methylacrylate) shows similar energetic differences as for the ion pairs. Furthermore large deviations in geometries for the intermolecular distances were found for the Hartree-Fock approach (mean absolute deviation: 190 pm) and density functional theory (mean absolute deviation up to 178 pm) while for the dispersion-corrected methods the mean absolute deviation is less than 50 pm.     

Testing the ONIOM G2R3 model against donor-acceptor dissociation energies of group 13-15 complexes: accuracy comparable to CCSD(T)/aug-CC-pVTZ at a fraction of the resource cost

The ability of the composite three-layer ONIOM G2R3 (OG2R3) method to match experimental dissociation energies for group 13-15 donor-acceptor complexes was examined for a database of 34 complexes. The composite approach provides energies that agree reasonably with experiment, performing nearly as well as both the CCSD(T)/aug-CC-pVTZ and CCSD(T)/6-311+G(2df, 2p) models for small molecules and nearly as well as the latter for slightly larger ones. Broadly, all three models exhibit average absolute errors of ～3 kcal mol(-1) , and root mean square errors of ～4 kcal mol(-1) . The average signed error suggest that the OG2R3 approach systematically underbinds by ～2.3 kcal mol(-1) ; if this is used as a general correction, the approach performs as well or better than the pure CCSD(T) models. However, the OG2R3 model can be applied to molecules too large to be studied by the other CCSD(T) methods, as it requires only a fraction of the time and computer resources.