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.     

Theoretical reference values for the AE6 and BH6 test sets from explicitly correlated coupled-cluster theory

We propose a new computational protocol to obtain highly accurate theoretical reference data. This protocol employs the explicitly correlated coupled-cluster method with iterative single and double excitations as well as perturbative triple excitations, CCSD(T)(F12), using quadruple-z\zeta basis sets. Higher excitations are accounted for by conventional CCSDT(Q) calculations using double-z\zeta basis sets, while core/core-valence correlation effects are estimated by conventional CCSD(T) calculations using quadruple-z\zeta basis sets. Finally, scalar-relativistic effects are accounted for by conventional CCSD(T) calculations using triple-z\zeta basis sets. In the present article, this protocol is applied to the popular test sets AE6 and BH6. An error analysis shows that the new reference values obtained by our computational protocol have an uncertainty of less than 1 kcal/mol (chemical accuracy). Furthermore, concerning the atomization energies, a cancellation of the basis set incompleteness error in the CCSD(T)(F12) perturbative triples contribution with the corresponding error in the contribution from higher excitations is observed. This error cancellation is diminished by the CCSD(T*)(F12) method. Thus, we recommend the use of the CCSD(T*)(F12) method only for small- and medium-sized basis sets, while the CCSD(T)(F12) approach is preferred for high-accuracy calculations in large basis sets.     

An explicitly correlated local coupled cluster method for calculations of large molecules close to the basis set limit

A new explicitly correlated local coupled-cluster method with single and double excitations and a perturbative treatment of triple excitations [DF-LCCSD(T0)-F12x (x = a,b)] is presented. By means of truncating the virtual orbital space to pair-specific local domains (domain approximation) and a simplified treatment of close, weak and distant pairs using LMP2-F12 (pair approximation) the scaling of the computational cost with molecular size is strongly reduced. The basis set incompleteness errors as well as the errors due to the domain approximation are largely eliminated by the explicitly correlated terms. All integrals are computed using efficient density fitting (DF) approximations. The accuracy of the method is investigated for 52 reactions involving medium size molecules. A comparison of DF-LCCSD(T0)-F12x reaction energies with canonical CCSD(T)-F12x calculations shows that the errors introduced by the domain approximation are indeed very small. Care must be taken to keep the errors due to the additional pair approximation equally small, and appropriate distance criteria are recommended. Using these parameters, the root mean square (RMS) deviations of DF-LCCSD(T0)-F12a calculations with triple-ζ basis sets from estimated CCSD(T) complete basis set (CBS) limits and experimental data amount to only 1.5 kJ mol(-1) and 2.9 kJ mol(-1), respectively. For comparison, the RMS deviation of the CCSD(T)/CBS values from the experimental values amounts to 3.0 kJ mol(-1). The potential of the method is demonstrated for five reactions of biochemical or pharmacological interest which include molecules with up to 61 atoms. These calculations show that molecules of this size can now be treated routinely and yield results that are close to the CCSD(T) complete basis set limits.     

Combustion chemistry: important features of the C3H5 potential energy surface, including allyl radical, propargyl + H2, allene + H, and eight transition states

The C(3)H(5) potential energy surface (PES) encompasses molecules of great significance to hydrocarbon combustion, including the resonantly stabilized free radicals propargyl (plus H(2)) and allyl. In this work, we investigate the interconversions that take place on this PES using high level coupled cluster methodology. Accurate geometries are obtained using coupled cluster theory with single, double, and perturbative triple excitations [CCSD(T)] combined with Dunning's correlation consistent quadruple-ζ basis set cc-pVQZ. The energies for these stationary points are then refined by a systematic series of computations, within the focal point scheme, using the cc-pVXZ (X = D, T, Q, 5, 6) basis sets and correlation treatments as extensive as coupled cluster with full single, double, and triple excitation and perturbative quadruple excitations [CCSDT(Q)]. Our benchmarks provide a zero-point vibrational energy (ZPVE) corrected barrier of 10.0 kcal mol(-1) for conversion of allene + H to propargyl + H(2). We also find that the barrier for H addition to a terminal carbon atom in allene leading to propenyl is 1.8 kcal mol(-1) lower than that for the addition to a central atom to form the allyl radical.     

New accurate reference energies for the G2/97 test set

A recently proposed computational protocol is employed to obtain highly accurate atomization energies for the full G2/97 test set, which consists of 148 diverse molecules. This computational protocol is based on the explicitly correlated coupled-cluster method with iterative single and double excitations as well as perturbative triple excitations, using quadruple-ζ basis sets. Corrections for higher excitations and core/core-valence correlation effects are accounted for in separate calculations. In this manner, suitable reference values are obtained with a mean deviation of -0.75 kJ/mol and a standard deviation of 1.06 kJ/mol with respect to the active thermochemical tables. Often, in the literature, new approximate methods (e.g., in the area of density functional theory) are compared to, or fitted to, experimental heats of formation of the G2/97 test set. We propose to use our atomization energies for this purpose because they are more accurate on average.     

Coupled-cluster methods including noniterative corrections for quadruple excitations

A new method is presented for treating the effects of quadruple excitations in coupled-cluster theory. In the approach, quadruple excitation contributions are computed from a formula based on a non-Hermitian perturbation theory analogous to that used previously to justify the usual noniterative triples correction used in the coupled cluster singles and doubles method with a perturbative treatment of the triple excitations (CCSD(T)). The method discussed in this paper plays a parallel role in improving energies obtained with the full coupled-cluster singles, doubles, and triples method (CCSDT) by adding a perturbative treatment of the quadruple excitations (CCSDT(Q)). The method is tested for an extensive set of examples, and is shown to provide total energies that compare favorably with those obtained with the full singles, doubles, triples, and quadruples (CCSDTQ) method.     

Ab initio structural and vibrational investigation of sulfuric acid monohydrate

We employ ab initio methods to find stable geometries and to calculate potential energy surfaces and vibrational wavenumbers for sulfuric acid monohydrate. Geometry optimizations are carried out with the explicitly correlated coupled-cluster approach that includes single, double, and perturbative triple excitations (CCSD(T)-F12a) with a valence double-ζ basis set (VDZ-F12). Four different stable geometries are found, and the two lowest are within 0.41 kJ mol(-1) (or 34 cm(-1)) of each other. Vibrational harmonic wavenumbers are calculated at both the density-fitted local spin component scaled second-order M?ller-Plesset perturbation theory (DF-SCS-LMP2) with the aug-cc-pV(T+d)Z basis set and the CCSD-F12/VDZ-F12 level. Water O-H stretching vibrations and two highly anharmonic large-amplitude motions connecting the three lowest potential energy minima are considered by limiting the dimensionality of the corresponding potential energy surfaces to small two- or three-dimensional subspaces that contain only strongly coupled vibrational degrees of freedom. In these anharmonic domains, the vibrational problem is solved variationally using potential energy surfaces calculated at the CCSD(T)-F12a/VDZ-F12 level.     

Accurate computational thermochemistry from explicitly correlated coupled-cluster theory

Explicitly correlated coupled-cluster theory has developed into a valuable computational tool for the calculation of electronic energies close to the limit of a complete basis set of atomic orbitals. In particular at the level of coupled-cluster theory with single and double excitations (CCSD), the space of double excitations is quickly extended towards a complete basis when Slater-type geminals are added to the wave function expansion. The purpose of the present article is to demonstrate the accuracy and efficiency that can be obtained in computational thermochemistry by a CCSD model that uses such Slater-type geminals. This model is denoted as CCSD(F12), where the acronym F12 highlights the fact that the Slater-type geminals are functions f(r ₁₂) of the interelectronic distances r ₁₂ in the system. The performance of explicitly correlated CCSD(F12) coupled-cluster theory is demonstrated by computing the atomization energies of 73 molecules (containing H, C, N, O, and F) with an estimated root-mean-square deviation from the values compiled in the Active Thermochemical Tables of σ = 0.10 kJ/mol per valence electron. To reach this accuracy, not only the frozen-core CCSD basis-set limit but also high-order excitations (connected triple and quadruple excitations), core–valence correlation effects, anharmonic vibrational zero-point energies, and scalar and spin–orbit relativistic effects must be taken into account.     

Basis set convergence of the coupled-cluster correction, δ(MP2)(CCSD(T)): best practices for benchmarking non-covalent interactions and the attendant revision of the S22, NBC10, HBC6, and HSG databases

In benchmark-quality studies of non-covalent interactions, it is common to estimate interaction energies at the complete basis set (CBS) coupled-cluster through perturbative triples [CCSD(T)] level of theory by adding to CBS second-order perturbation theory (MP2) a "coupled-cluster correction," δ(MP2)(CCSD(T)), evaluated in a modest basis set. This work illustrates that commonly used basis sets such as 6-31G*(0.25) can yield large, even wrongly signed, errors for δ(MP2)(CCSD(T)) that vary significantly by binding motif. Double-ζ basis sets show more reliable results when used with explicitly correlated methods to form a δ(MP2-F12)(CCSD(T(*))-F12) correction, yielding a mean absolute deviation of 0.11 kcal mol(-1) for the S22 test set. Examining the coupled-cluster correction for basis sets up to sextuple-ζ in quality reveals that δ(MP2)(CCSD(T)) converges monotonically only beyond a turning point at triple-ζ or quadruple-ζ quality. In consequence, CBS extrapolation of δ(MP2)(CCSD(T)) corrections before the turning point, generally CBS (aug-cc-pVDZ,aug-cc-pVTZ), are found to be unreliable and often inferior to aug-cc-pVTZ alone, especially for hydrogen-bonding systems. Using the findings of this paper, we revise some recent benchmarks for non-covalent interactions, namely the S22, NBC10, HBC6, and HSG test sets. The maximum differences in the revised benchmarks are 0.080, 0.060, 0.257, and 0.102 kcal mol(-1), respectively.     

The Beryllium tetramer: profiling an elusive molecule

The structure and energetics of Be(4) are investigated using state-of-the-art coupled-cluster methods. We compute the optimized bond length, dissociation energy, and anharmonic vibrational frequencies. A composite approach is employed, starting from coupled-cluster theory with single, double, and perturbative triple excitations extrapolated to the complete basis set (CBS) limit using Dunning's correlation consistent cc-pCVQZ and cc-pCV5Z basis sets. A correction for full triple and connected quadruple excitations in the smaller cc-pCVDZ basis set is then added, yielding an approximation to CCSDT(Q)/CBS denoted c～CCSDT(Q). Corrections are included for relativistic and non-Born-Oppenheimer effects. We obtain D(e) = 89.7 kcal mol(-1), D(0) = 84.9 kcal mol(-1), and r(e) = 2.043 A?. Second-order vibrational perturbation theory (VPT2) is applied to a full quartic force field computed at the c～CCSDT(Q) level of theory, yielding B(e) = 0.448 cm(-1) and fundamental frequencies of 666 (a(1)), 468 (e), and 571 (t(2)) cm(-1). Computations on the spectroscopically characterized Be(2) molecule are reported for the purpose of benchmarking our methods. Perturbative estimates of the effect of quadruple excitations are found to be essential to computing accurate parameters for Be(2); however, they seem to exert a much smaller influence on the structure and energetics of Be(4). Our extensive characterization of the Be(4) bonding potential energy surface should aid in the experimental identification of this thermodynamically viable but elusive molecule.     

Ab initio benchmark study for the oxidative addition of CH4 to Pd: importance of basis-set flexibility and polarization

To obtain a state-of-the-art benchmark potential energy surface (PES) for the archetypal oxidative addition of the methane C-H bond to the palladium atom, we have explored this PES using a hierarchical series of ab initio methods (Hartree-Fock, second-order M?ller-Plesset perturbation theory, fourth-order M?ller-Plesset perturbation theory with single, double and quadruple excitations, coupled cluster theory with single and double excitations (CCSD), and with triple excitations treated perturbatively [CCSD(T)]) and hybrid density functional theory using the B3LYP functional, in combination with a hierarchical series of ten Gaussian-type basis sets, up to g polarization. Relativistic effects are taken into account either through a relativistic effective core potential for palladium or through a full four-component all-electron approach. Counterpoise corrected relative energies of stationary points are converged to within 0.1-0.2 kcal/mol as a function of the basis-set size. Our best estimate of kinetic and thermodynamic parameters is -8.1 (-8.3) kcal/mol for the formation of the reactant complex, 5.8 (3.1) kcal/mol for the activation energy relative to the separate reactants, and 0.8 (-1.2) kcal/mol for the reaction energy (zero-point vibrational energy-corrected values in parentheses). This agrees well with available experimental data. Our work highlights the importance of sufficient higher angular momentum polarization functions, f and g, for correctly describing metal-d-electron correlation and, thus, for obtaining reliable relative energies. We show that standard basis sets, such as LANL2DZ+1f for palladium, are not sufficiently polarized for this purpose and lead to erroneous CCSD(T) results. B3LYP is associated with smaller basis set superposition errors and shows faster convergence with basis-set size but yields relative energies (in particular, a reaction barrier) that are ca. 3.5 kcal/mol higher than the corresponding CCSD(T) values.     

Calculation of the O-H stretching vibrational overtone spectrum of the water dimer

The O-H stretching vibrational overtone spectrum of the water dimer has been calculated with the dimer modeled as two individually vibrating monomer units. Vibrational term values and absorption intensities have been obtained variationally with a computed dipole moment surface and an internal coordinate Hamiltonian, which consists of exact kinetic energy operators within the Born-Oppenheimer approximation of the monomer units. Three-dimensional ab initio potential energy and dipole moment surfaces have been calculated using the internal coordinates of the monomer units using the coupled cluster method including single, double, and perturbative triple excitations [CCSD(T)] with the augmented correlation consistent valence triple zeta basis set (aug-cc-pVTZ). The augmented correlation consistent valence quadruple zeta basis set (aug-cc-pVQZ), counterpoise correction, basis set extrapolation to the complete basis set limit, relativistic corrections, and core and valence electron correlations effects have been included in one-dimensional potential energy surface cuts. The aim is both to investigate the level of ab initio and vibrational calculations necessary to produce accurate results when compared with experiment and to aid the detection of the water dimer under atmospheric conditions.     

Comment on "Revised electron affinity of SF6 from kinetic data" [J. Chem. Phys. 136, 121102 (2012)

The adiabatic electron affinity (AEA) of SF(6) has been calculated near the relativistic CCSDT(Q) basis set limit. Our best theoretical value (1.0340 ±?0.03 eV) is in excellent agreement with the recently revised experimental value of 1.03?±?0.05 eV reported by Troe et al. [J. Chem. Phys. 136, 121102 (2012)]. While our best nonrelativistic, clamped-nuclei, valence CCSD(T) basis set limit value of 0.9058 eV is in good accord with the previously reported CCSD(T)/CBS values, to obtain an accurate AEA, several additional contributions need to be taken into account. The most important one is scalar-relativistic effects (0.0839 eV), followed by inner-shell correlation (0.0216 eV) and post-CCSD(T) correlation effects (0.0248 eV), the latter almost entirely due to connected quadruple excitations. The diagonal Born-Oppenheimer correction is an order of magnitude less important at -0.0022 eV.     

Accurate pair interaction energies for helium from supermolecular Gaussian geminal calculations

Nonrelativistic clamped-nuclei pair interaction energy for ground-state helium atoms has been computed for 12 interatomic separations ranging from 3.0 to 9.0 bohr. The calculations applied the supermolecular approach. The major part of the interaction energy was obtained using the Gaussian geminal implementation of the coupled-cluster theory with double excitations (CCD). Relatively small contributions from single, triple, and quadruple excitations were subsequently included employing the conventional orbital coupled-cluster method with single, double, and noniterative triple excitations [CCSD(T)] and the full configuration interaction (FCI) method. For three distances, the single-excitation contribution was taken from literature Gaussian-geminal calculations at the CCSD level. The orbital CCSD(T) and FCI calculations used very large basis sets, up to doubly augmented septuple- and sextuple-zeta size, respectively, and were followed by extrapolations to the complete basis set limits. The accuracy of the total interaction energies has been estimated to be about 3 mK or 0.03% at the minimum of the potential well. For the attractive part of the well, the relative errors remain consistently smaller than 0.03%. In the repulsive part, the accuracy is even better, except, of course, for the region where the potential goes through zero. For interatomic separations smaller than 4.0 bohr, the relative errors do not exceed 0.01%. Such uncertainties are significantly smaller than the expected values of the relativistic and diagonal Born-Oppenheimer contributions to the potential.     

W3 theory: robust computational thermochemistry in the kJ/mol accuracy range

We are proposing a new computational thermochemistry protocol denoted W3 theory, as a successor to W1 and W2 theory proposed earlier [Martin and De Oliveira, J. Chem. Phys. 111, 1843 (1999)]. The new method is both more accurate overall (error statistics for total atomization energies approximately cut in half) and more robust (particularly towards systems exhibiting significant nondynamical correlation) than W2 theory. The cardinal improvement rests in an approximate account for post-CCSD(T) correlation effects. Iterative T3 (connected triple excitations) effects exhibit a basis set convergence behavior similar to the T3 contribution overall. They almost universally decrease molecular binding energies. Their inclusion in isolation yields less accurate results than CCSD(T) nearly across the board: It is only when T4 (connected quadruple excitations) effects are included that superior performance is achieved. T4 effects systematically increase molecular binding energies. Their basis set convergence is quite rapid, and even CCSDTQ/cc-pVDZ scaled by an empirical factor of 1.2532 will yield a quite passable quadruples contribution. The effect of still higher-order excitations was gauged for a subset of molecules (notably the eight-valence electron systems): T5 (connected quintuple excitations) contributions reach 0.3 kcal/mol for the pathologically multireference X 1Sigmag+ state of C2 but are quite small for other systems. A variety of avenues for achieving accuracy beyond that of W3 theory were explored, to no significant avail. W3 thus appears to represent a good compromise between accuracy and computational cost for those seeking a robust method for computational thermochemistry in the kJ/mol accuracy range on small systems.     

Relativistic effects in HgHe and HgXe CCSD(T) ground state potential curves. Low‐density viscosity simulations of Hg:Xe mixture

The comparison of coupled cluster with single and double excitations and with perturbative correction of triple excitations [CCSD(T)] ground state potential curves of mercury with rare gases (RG): HgHe and HgXe, at several levels of theory is presented. The scalar relativistic (REL) effects and spin‐orbit coupling effects in the ground state potential curves of these weakly bounded dimers are considered. The CCSD(T) ground state potential curves at the level of the Dirac‐Coulomb Hamiltonian (DCH) are compared with CCSD(T) curves at the level of 4‐component spin‐free modified DCH, the scalar 2nd order Douglas‐Kroll‐Hess (DKH2) and the nonrelativistic (NR‐LL) (Lévy‐Leblond) Hamiltonian. In addition, London‐Drude formula and SCF interaction energy curves are employed in the analysis of different contributions of REL effects in dissociation energies of HgRG and Hg₂ dimers. Moreover, the large anharmonicity of the HgHe ground state potential curve is highlighted. The computationally less demanding scalar DKH2 Hamiltonian is employed to calculate the HgXe, Hg₂, and Xe₂ all electron CCSD(T) ground state potential curves in highly augmented quadruple zeta basis sets. These potential curves are used to simulate the shear viscosity of mercury, xenon, and mercury‐xenon (Hg:Xe) mixture. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

Explicit correlation and basis set superposition error: the structure and energy of carbon dioxide dimer

We have investigated the slipped parallel and t-shaped structures of carbon dioxide dimer [(CO(2))(2)] using both conventional and explicitly correlated coupled cluster methods, inclusive and exclusive of counterpoise (CP) correction. We have determined the geometry of both structures with conventional coupled cluster singles doubles and perturbative triples theory [CCSD(T)] and explicitly correlated cluster singles doubles and perturbative triples theory [CCSD(T)-F12b] at the complete basis set (CBS) limits using custom optimization routines. Consistent with previous investigations, we find that the slipped parallel structure corresponds to the global minimum and is 1.09 kJ mol(-1) lower in energy. For a given cardinal number, the optimized geometries and interaction energies of (CO(2))(2) obtained with the explicitly correlated CCSD(T)-F12b method are closer to the CBS limit than the corresponding conventional CCSD(T) results. Furthermore, the magnitude of basis set superposition error (BSSE) in the CCSD(T)-F12b optimized geometries and interaction energies is appreciably smaller than the magnitude of BSSE in the conventional CCSD(T) results. We decompose the CCSD(T) and CCSD(T)-F12b interaction energies into the constituent HF or HF CABS, CCSD or CCSD-F12b, and (T) contributions. We find that the complementary auxiliary basis set (CABS) singles correction and the F12b approximation significantly reduce the magnitude of BSSE at the HF and CCSD levels of theory, respectively. For a given cardinal number, we find that non-CP corrected, unscaled triples CCSD(T)-F12b/VXZ-F12 interaction energies are in overall best agreement with the CBS limit.     

On the effectiveness of CCSD(T) complete basis set extrapolations for atomization energies

The leading cause of error in standard coupled cluster theory calculations of thermodynamic properties such as atomization energies and heats of formation originates with the truncation of the one-particle basis set expansion. Unfortunately, the use of finite basis sets is currently a computational necessity. Even with basis sets of quadruple zeta quality, errors can easily exceed 8 kcal/mol in small molecules, rendering the results of little practical use. Attempts to address this serious problem have led to a wide variety of proposals for simple complete basis set extrapolation formulas that exploit the regularity in the correlation consistent sequence of basis sets. This study explores the effectiveness of six formulas for reproducing the complete basis set limit. The W4 approach was also examined, although in lesser detail. Reference atomization energies were obtained from standard coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) calculations involving basis sets of 6ζ or better quality for a collection of 141 molecules. In addition, a subset of 51 atomization energies was treated with explicitly correlated CCSD(T)-F12b calculations and very large basis sets. Of the formulas considered, all proved reliable at reducing the one-particle expansion error. Even the least effective formulas cut the error in the raw values by more than half, a feat requiring a much larger basis set without the aid of extrapolation. The most effective formulas cut the mean absolute deviation by a further factor of two. Careful examination of the complete body of statistics failed to reveal a single choice that out performed the others for all basis set combinations and all classes of molecules.