Influence of diffuse and polarization functions on the second-order Møller–Plesset optimized dihedral angle of biphenyl

 Using 6-31G and 6-311G basis sets to which diffuse and polarization functions were added in a stepwise fashion (a total of 16 basis sets), Hartree–Fock (HF), MP2 and B3LYP geometry optimizations were performed on biphenyl. With the MP2 method, diffuse functions raise the dihedral angle φ, for example, from 46.3° for 6-31G to 54.1° for 6-311++G, while polarization functions lower it, for example, from 54.1° for 6-311++G to 42.1° for 6-311++G(2d,2p). For a single set of polarization functions, φ(MP2) lies close to or above φ(HF) (44–47°), but for a double set it is below φ(HF) and is close to B3LYP values (38–42°) which show little basis set dependence. The most reliable value for φ, 42.1° [MP2/6-311++G(2d,2p)], is expected to increase slightly by adding more diffuse functions. The corresponding best calculated energy barrier at 0° (coplanar conformation) is 2.83 kcal/mol, much higher than the experimental estimate (1.4 ± 0.5 kcal/mol). The barrier at 90° is 1.82 kcal/mol, in line with the experimental estimate (1.6 ± 0.5 kcal/mol) and with previous theoretical results. Received: 9 September 2002 / Accepted: 15 November 2002 / Published online: 1 April 2003 Correspondence to: Friedrich Grein e-mail: fritz@unb.ca Acknowledgement. The author would like to thank NSERC (Canada) for financial support.     

What are the most efficient basis set strategies for correlated wave function calculations of reaction energies and barrier heights?

As electronic structure methods are being used to obtain quantitatively accurate reaction energies and barrier heights for increasingly larger systems, the choice of an efficient basis set is becoming more critical. The optimum strategy for achieving basis set convergence can depend on the way that electron correlation is treated and can take advantage of flexibility in the order in which basis functions are added. Here we study several approaches for estimating accurate reaction energies and barrier heights from post-Hartree-Fock electronic structure calculations. First and second, we evaluate methods of estimating the basis set limit of second order Mo?ller-Plesset perturbation theory and of coupled cluster theory with single and double excitations and a quasiperturbative treatment of connected triple excitations by using explicitly correlated basis functions (in the F12a implementation) along with valence, polarization, and diffuse one-electron basis functions. Third, we test the scheme of adding a higher-order correction to MP2 results (sometimes called MP2∕CBS + ΔCCSD(T)). Finally, we evaluate the basis set requirements of these methods in light of comparisons to Weizmann-3.2, Weizmann-4, and CCSDT(2)(Q)∕CBS+CV+R results.     

The role of the basis set and the level of quantum mechanical theory in the prediction of the structure and reactivity of cisplatin

In this article, we conducted an extensive ab initio study on the importance of the level of theory and the basis set for theoretical predictions of the structure and reactivity of cisplatin [cis‐diamminedichloroplatinum(II) (cDDP)]. Initially, the role of the basis set for the Pt atom was assessed using 24 different basis sets, including three all‐electron basis sets (ABS). In addition, a modified all‐electron double zeta polarized basis set (mDZP) was proposed by adding a set of diffuse d functions onto the existing DZP basis set. The energy barrier and the rate constant for the first chloride/water exchange ligand process, namely, the aquation reaction, were taken as benchmarks for which reliable experimental data are available. At the B3LYP/mDZP/6‐31+G(d) level (the first basis set is for Pt and the last set is for all of the light atoms), the energy barrier was 22.8 kcal mol^?1, which is in agreement with the average experimental value, 22.9 ± 0.4 kcal mol^?1. For the other accessible ABS (DZP and ADZP), the corresponding values were 15.4 and 24.5 kcal mol^?1, respectively. The ADZP and mDZP are notably similar, raising the importance of diffuse d functions for the prediction of the kinetic properties of cDDP. In this article, we also analyze the ligand basis set and the level of theory effects by considering 36 basis sets at distinct levels of theory, namely, Hartree‐Fock, MP2, and several DFT functionals. From a survey of the data, we recommend the mPW1PW91/mDZP/6‐31+G(d) or B3PW91/mDZP/6‐31+G(d) levels to describe the structure and reactivity of cDDP and its small derivatives. Conversely, for large molecules containing a cisplatin motif (for example, the cDDP‐DNA complex), the lower levels B3LYP/LANL2DZ/6‐31+G(d) and B3LYP/SBKJC‐VDZ/6‐31+G(d) are suggested. At these levels of theory, the predicted energy barrier was 26.0 and 25.9 kcal mol^?1, respectively, which is only 13% higher than the actual value. © 2012 Wiley Periodicals, Inc.     

Counterpoise estimates of the BSSE in the evaluation of protonation energies

Counterpoise estimates of the BSSE in the evaluation of protonation energies have been calculated for basis sets ranging from minimal to split-valence plus polarization quality. Three-, five- and six-membered-ring heterocycles have been chosen as suitable model compounds for this study. Counterpoise corrections are significant, at the minimal basis set and 3–21G levels, when considering both, absolute and relative protonation energies and depend on the nature of the centre which undergoes protonation. In general, second- and third-order counterpoise corrections to the protonation energies are comparable to the corresponding SCF values. BSSE depend not only on the size of the basis sets but also on their quality. The presence in the basis of quite diffuse functions (either sp or d) leads to lower protonation energies and greater BSSE. Relative protonation energies are not substantially affected by BSSE or correlation effects.     

An ab initio benchmark study of the H + CO --> HCO reaction

The H + CO --> HCO reaction has been characterized with correlation consistent basis sets at five levels of theory in order to benchmark the sensitivities of the barrier height and reaction ergicity to the one-electron and n-electron expansions of the electronic wave function. Single and multireference methods are compared and contrasted. The coupled cluster method RCCSD(T) was found to be in very good agreement with Davidson-corrected internally-contracted multireference configuration interaction (MRCI+Q). Second-order Moller-Plesset perturbation theory (MP2) was also employed. The estimated complete basis set (CBS) limits for the barrier height (in kcal/mol) for the five methods, including harmonic zero-point energy corrections, are MP2, 4.66; RCCSD, 4.78; RCCSD(T), 4.15; MRCI, 5.10; and MRCI+Q, 4.07. Similarly, the estimated CBS limits for the ergicity of the reaction are: MP2, -17.99; RCCSD, -13.34; RCCSD(T), -13.79; MRCI, -11.46; and MRCI+Q, -13.70. Additional basis set explorations for the RCCSD(T) method demonstrate that aug-cc-pVTZ sets, even with some functions removed, are sufficient to reproduce the CBS limits to within 0.1-0.3 kcal/mol.     

Efficiency of numerical basis sets for predicting the binding energies of hydrogen bonded complexes: evidence of small basis set superposition error compared to Gaussian basis sets

Binding energies of selected hydrogen bonded complexes have been calculated within the framework of density functional theory (DFT) method to discuss the efficiency of numerical basis sets implemented in the DFT code DMol3 in comparison with Gaussian basis sets. The corrections of basis set superposition error (BSSE) are evaluated by means of counterpoise method. Two kinds of different numerical basis sets in size are examined; the size of the one is comparable to Gaussian double zeta plus polarization function basis set (DNP), and that of the other is comparable to triple zeta plus double polarization functions basis set (TNDP). We have confirmed that the magnitudes of BSSE in these numerical basis sets are comparative to or smaller than those in Gaussian basis sets whose sizes are much larger than the corresponding numerical basis sets; the BSSE corrections in DNP are less than those in the Gaussian 6-311+G(3df,2pd) basis set, and those in TNDP are comparable to those in the substantially large scale Gaussian basis set aug-cc-pVTZ. The differences in counterpoise corrected binding energies between calculated using DNP and calculated using aug-cc-pVTZ are less than 9 kJ/mol for all of the complexes studied in the present work. The present results have shown that the cost effectiveness in the numerical basis sets in DMol3 is superior to that in Gaussian basis sets in terms of accuracy per computational cost.     

Ab initio calculations on the barrier height for the hydrogen addition to ethylene and formaldehyde. The importance of spin projection

Heats of reaction and barrier heights have been computed for H + CH₂CH₂ → C₂H₅, H + CH₂O → CH₃O, and H + CH₂O → CH₂OH using unrestricted Hartree-Fock and Møller–Plesset perturbation theory up to fourth order (with and without spin annihilation), using single-reference configuration interaction, and using multiconfiguration self-consistent field methods with 3-21G, 6-31G(d), 6-31G(d,p), and 6-311G(d,p) basis sets. The barrier height in all three reactions appears to be relatively insensitive to the basis sets, but the heats of reaction are affected by p-type polarization functions on hydrogen. Computation of the harmonic vibrational frequencies and infrared intensities with two sets of polarization functions on heavy atoms [6-31G(2d)] improves the agreement with experiment. The experimental barrier height for H + C₂H₄ (2.04 ± 0.08 kcal/mol) is overestimated by 7?9 kcal/mol at the MP2, MP3, and MP4 levels. MCSCF and CISD calculations lower the barrier height by approximately 4 kcal/mol relative to the MP4 calculations but are still almost 4 kcal/mol too high compared to experiment. Annihilation of the largest spin contaminant lowers the MP4SDTQ computed barrier height by 8?9 kcal/mol. For the hydrogen addition to formaldehyde, the same trends are observed. The overestimation of the barrier height with Møller-Plesset perdicted barrier heights for H + C₂H₄ → C₂H₅, H + CH₂O → CH₃O, and H + CH₂O → CH₂OH at the MP4SDTQ /6-31G(d) after spin annihilation are respectively 1.8, 4.6, and 10.5 kcal/mol.     

Accurate calculation of the binding energy of the water dimer

The binding energy of the water dimer is calculated at the MP2 level using efficient basis sets augmented with bond functions. The intermolecular energy is determined by the supermolecular approach and the basis set superposition error is corrected by the counterpoise method. Bond functions are found useful and very effective in recovering the dispersion energy, which is traditionally achieved by polarization functions. The calculated binding energy of the water dimer is systematically converged to a value of 4.75 kcal mol⁻ as bond functions are gradually added to nucleus-centered basis sets.     

Bond functions,covalent potential curves,and the basis set superposition error

In the current practice of quantum chemistry, it is not clear whether corrections for basis set superposition errors should be applied to the calculation of potential energy curves, in order to improve agreement with experimental data. To examine this question, spectroscopic parameters derived from theoretical potential curves are reported for the homonuclear diatomics C₂, N₂, O₂, and F₂, using a configuration interaction method. Three different basis sets were used, including double zeta plus polarization, triple zeta plus double polarization, and double zeta polarization augmented by bond functions. The bond function basis sets, which were optimized in the preceding paper to obtain accurate dissociation energies, also gave the most accurate parameters. The potential curves were then corrected for basis set superposition error using the counterpoise correction, and the spectroscopic parameters were computed again. The BSSE-corrected curves showed worse agreement with experiment for all properties than the original (uncorrected) curves. The reasons for this finding are discussed. In addition to the numerical results, some problems in the application of the BSSE correction to basis sets containing bond functions are shown. In particular, there is an overcounting of the lowering due to the bond functions, regardless of which type of correction is applied. Also, genuine BSSE affects cannot be separated from energy-lowering effects due to basis set incompleteness, and we postulate that it is the latter which is strongly dominant in the calculation of covalent potential curves. Based on these arguments, two conclusions follow: (1) application of BSSE corrections to potential curves should not be routinely applied in situations where the bonding is strong, and (2) appropriate use of bond functions can lead to systematic improvement in the quality of potential curves.     

Can extrapolation to the basis set limit be an alternative to the counterpoise correction? A study on the helium dimer

Configuration interaction and coupled cluster calculations are reported for He₂ using various orbital basis sets of the d-aug-AVXZ type, with the results being extrapolated to the one electron basis set limit both with counterpoise and without counterpoise correction. A generalized uniform singlet- and triplet-pair extrapolation scheme has been utilized for such a purpose. Using appropriate corrections to mimic full configuration interaction, the energies were predicted in excellent agreement with the best available estimates. The results also suggest that extrapolation to the complete basis set limit may be a general alternative to the counterpoise correction that yields a more accurate potential energy while being more economical.     

Effects of counterpoise correction and basis set extrapolation on the MP2 geometries of hydrogen bonded dimers of ammonia, water, and hydrogen fluoride

We study the combined effects of counterpoise correction and basis set extrapolation on the second-order M?ller-Plesset (MP2) geometries of three hydrogen bonded dimers, namely (NH(3))(2), (H(2)O)(2) and (HF)(2). For (NH(3))(2), we study three characteristic structures on its potential energy surface. In addition, we look at the basis set convergence when diffuse functions on the hydrogen atoms are left out, as well as the errors introduced by including core correlation with valence-only correlation-consistent basis sets. Overall, the counterpoise-corrected and extrapolated geometries appear to be very reliable and are in convincing agreement with the geometries from explicitly correlated MP2-F12 calculations. Obtaining geometries with errors of less than 0.001 ?ngstrom and 0.5 degrees compared to the basis set limit is, however, even with these advanced methods a difficult task.     

Minimally augmented Karlsruhe basis sets

We propose an extension of the basis sets proposed by Ahlrichs and coworkers at Karlsruhe (these basis sets are designated as the second-generation default or “def2” basis sets in the Turbomole program). The Karlsruhe basis sets are very appealing because they constitute balanced and economical basis sets of graded quality from partially polarized double zeta to heavily polarized quadruple zeta for all elements up to radon (Z = 86). The extension consists of adding a minimal set of diffuse functions to a subset of the elements. This yields basis sets labeled minimally augmented or with “ma” as a prefix. We find that diffuse functions are not quite as important for the def2 basis sets as they are for Pople basis sets, but they are still necessary for good results on barrier heights and electron affinities. We provide assessments and validations of this extension for a variety of data sets and representative cases. We recommend the new ma-TZVP basis set for general-purpose applications of density functional theory.     

On basis set superposition error corrected stabilization energies for large n-body clusters

In this contribution, we propose an approximate basis set superposition error (BSSE) correction scheme for the site-site function counterpoise and for the Valiron-Mayer function counterpoise correction of second order to account for the basis set superposition error in clusters with a large number of subunits. The accuracy of the proposed scheme has been investigated for a water cluster series at the CCSD(T), CCSD, MP2, and self-consistent field levels of theory using Dunning's correlation consistent basis sets. The BSSE corrected stabilization energies for a series of water clusters are presented. A study regarding the possible savings with respect to computational resources has been carried out as well as a monitoring of the basis set dependence of the approximate BSSE corrections.     

An ab initio potential energy surface for the υ4 vibrational mode of hydrogen peroxide

In this paper, the levels and the torsional microstates of hydrogen peroxide are determined from fully optimized ab initio calculations using a nuclear model in one dimension. Calculations have been performed at the MP2 level with the 6-311 G(2df,2pd), 6-31 1+G(2df,2pd), cc-pVTZ and AUG-cc-pVTZ basis sets including polarization orbitals and diffuse functions. The most stable conformation, calculated with the MP2/AUG-cc-pVTZ approach, is a trans–gauche conformer lying at 67.5° from the trans structure. By using the same level of calculations, the heights of the trans and cis barriers have been determined to be 386.5 and 2643.8 cm⁻¹ in a good agreement with the experimental data. The variational torsional levels split into four components by the tunnelling effect of the barriers. The splitting of the fundamental level caused by the trans barrier has been found to be 11.8683 cm⁻¹, whereas the splitting caused by the cis barrier is insignificant under n=2. Current ab initio energies confirm the experimental assignments and verify the separability of the torsion from the rest of the vibrations. However, the experimental relation of dependence on the torsion of the rotational constants cannot be reproduced in one-dimension and depends on several additional vibrational effects.     

Acetamide, a challenge to theory and experiment? On the molecular structure, conformation, potential to internal rotation of the methyl group and force fields of free acetamide as studied by quantum chemical calculations

The molecular geometry has been optimized without any constraints using different basis sets and levels of theory as: Hartree-Fock with basis sets 6–31+G**, 6–311++G**, cc-pVTZ and aug-cc-pVTZ, MP2 with basis sets 6–311++G** and cc-pVTZ, MP3 with basis set 6–311++G**, and density functional theory with basis sets 6–311++G** and cc-pVTZ. Small basis sets up to 6-31G predict the syn conformation of the methyl group to be the most stable conformation. Larger basis sets predict an unsymmetrical conformation with one of the H atoms perpendicular to the amide skeleton or an anti-like conformation. Dunnings correlation consistent polarized valence triple zeta, cc-pVTZ, basis set including MP2 predict two conformations, one perpendicular and one anti to be the most stable. The DFT calculations predict anti-like conformations. The most accurate calculations predict anti-like conformations which have not been predicted previously. The vibrational frequencies have been calculated for several basis sets and compared to the observed frequencies. The wagging frequency of the NH₂ is very dependent on the basis sets and levels of theory. Most calculations predict a planar NH₂ group in agreement with experiment. A scaled molecular force field has been determined by fitting the calculated frequencies to the observed ones for the perpendicular conformation using MP2/cc-pVTZ. The barrier heights for the methyl group have been calculated. The rotational constants, I_A + I_B − I_C values and dipole moments are compared with experimental values.     

Is spin-component scaled second-order Møller-Plesset perturbation theory an appropriate method for the study of noncovalent interactions in molecules?

Testing of the spin-component scaled second-order M?ller-Plesset (SCS-MP2) method for the computation of noncovalent interaction energies is done with a database of 165 biologically relevant complexes. The effects of the spin-scaling procedure (i.e., MP2 vs SCS-MP2), the basis set size, and the corrections for basis set superposition error (BSSE) are systematically examined. When using two-point basis set extrapolations for the correlation energy, augmentation of the atomic orbital basis with computationally costly diffuse functions is found to be obsolete. In general, SCS-MP2 also improves results for noncovalent interactions statistically on MP2, and significant outliers are removed. Moreover, it is shown that effects of BSSE and one-particle basis set incompleteness almost cancel each other in the case of triple-zeta sets (SCS-MP2/TZVPP or SCS-MP2/cc-pVTZ without counterpoise correction), which opens a practical route to efficient computations for large systems. We recommend SCS-MP2 as the preferred quantum chemical wave function based method for the noncovalent interactions in large biologically relevant systems when reasonable coupled-cluster with single and double and perturbative triple excitations (CCSD(T)) calculations cannot be performed anymore. A comparison to MP2 and CCSD(T) interaction energies for n-alkane dimers, however, indicates (and this also holds to a lesser extent for hydrogen-bonded systems) limitations of SCS-MP2 when treating chemically "saturated" interactions. The different behavior of second-order perturbation theory for saturated and for stacked pi-systems is discussed.     

Thermochemistry of the chromium hydroxides Cr(OH)n, n = 2-6, and the oxyhydroxide CrO(OH)4: ab initio predictions

We here present a high-level ab initio study of the thermochemistry of the chromium hydroxides Cr(OH)n, n = 2-6, and of the oxyhydroxide CrO(OH)4. Optimum geometries and harmonic vibrational frequencies were determined at the B3LYP level of theory using basis sets of triple-zeta quality including polarization and diffuse functions. Heats of formation were obtained from isogyric reaction energies computed at the CCSD(T) level of theory using large basis sets and including corrections for core-valence correlation, scalar relativistic effects, and basis set incompleteness. Additionally, polynomial fits were performed for the heat capacity and the standard enthalpy and entropy over the 100-3000 K temperature range. While our computed heats of formation agree well with previously obtained experimental data for some of these species, our results suggest that revision of the experimental data for others may be appropriate.     

Approximations to complete basis set-extrapolated, highly correlated non-covalent interaction energies

The first-principles calculation of non-covalent (particularly dispersion) interactions between molecules is a considerable challenge. In this work we studied the binding energies for ten small non-covalently bonded dimers with several combinations of correlation methods (MP2, coupled-cluster single double, coupled-cluster single double (triple) (CCSD(T))), correlation-consistent basis sets (aug-cc-pVXZ, X = D, T, Q), two-point complete basis set energy extrapolations, and counterpoise corrections. For this work, complete basis set results were estimated from averaged counterpoise and non-counterpoise-corrected CCSD(T) binding energies obtained from extrapolations with aug-cc-pVQZ and aug-cc-pVTZ basis sets. It is demonstrated that, in almost all cases, binding energies converge more rapidly to the basis set limit by averaging the counterpoise and non-counterpoise corrected values than by using either counterpoise or non-counterpoise methods alone. Examination of the effect of basis set size and electron correlation shows that the triples contribution to the CCSD(T) binding energies is fairly constant with the basis set size, with a slight underestimation with CCSD(T)∕aug-cc-pVDZ compared to the value at the (estimated) complete basis set limit, and that contributions to the binding energies obtained by MP2 generally overestimate the analogous CCSD(T) contributions. Taking these factors together, we conclude that the binding energies for non-covalently bonded systems can be accurately determined using a composite method that combines CCSD(T)∕aug-cc-pVDZ with energy corrections obtained using basis set extrapolated MP2 (utilizing aug-cc-pVQZ and aug-cc-pVTZ basis sets), if all of the components are obtained by averaging the counterpoise and non-counterpoise energies. With such an approach, binding energies for the set of ten dimers are predicted with a mean absolute deviation of 0.02 kcal/mol, a maximum absolute deviation of 0.05 kcal/mol, and a mean percent absolute deviation of only 1.7%, relative to the (estimated) complete basis set CCSD(T) results. Use of this composite approach to an additional set of eight dimers gave binding energies to within 1% of previously published high-level data. It is also shown that binding within parallel and parallel-crossed conformations of naphthalene dimer is predicted by the composite approach to be 9% greater than that previously reported in the literature. The ability of some recently developed dispersion-corrected density-functional theory methods to predict the binding energies of the set of ten small dimers was also examined.     

Theoretical study of the pyrolysis of methyltrichlorosilane in the gas phase. 2. Reaction paths and transition states

The kinetics for the previously proposed 114-reaction mechanism for the chemical vapor deposition (CVD) process that leads from methyltrichlorosilane (MTS) to silicon carbide (SiC) are examined. Among the 114 reactions, 41 are predicted to proceed with no intervening barrier. For the remaining 73 reactions, transition states and their corresponding barrier heights have been explored using second-order perturbation theory (MP2) with the aug-cc-pVDZ basis set. Final energies for the reaction barriers were obtained using both MP2 with the aug-cc-pVTZ basis set and coupled cluster theory (CCSD(T)) with the aug-cc-pVDZ basis set. CCSD(T)/aug-cc-pVTZ energies were estimated by assuming additivity of basis set and correlation effects. Partition functions for the computation of thermodynamic properties of the transition states were calculated with MP2/aug-cc-pVDZ. Forward and reverse Gibbs free energy barriers were obtained at 11 temperatures ranging from 0 to 2000 K. Important reaction pathways are illustrated at 0 and 1400 K.