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Accurate benchmark calculations of gas-phase basicities of small molecules are presented and compared with available experimental results. The optimized geometries and thermochemical analyses were obtained from MP2/aug-cc-pVTZ calculations. Two different ab initio electron-correlated methods MP2 and CCSD(T) were employed for subsequent gas-phase basicity calculations and the single point energies were extrapolated to the complete basis set (CBS) limit. The overall accuracy for different ab initio methods are compared and the accuracy in descending order is CCSD(T)_CBS > CCSD(T)/aug-cc-pVDZ > (MP2/aug-cc-pVQZ ≈ MP2_CBS) > HF/ aug-cc-pVQZ. The best root-mean-squared-error obtained was 1.0 kcal mol−1 at the CCSD(T)_CBS//MP2/aug-cc-pVTZ level for a test set of 41 molecules. Clearly, accurate calculations for the electron correlation energy are important for the theoretical prediction of molecular gas-phase basicities. However, conformational effects were also found to be relevant in several instances when more complicated molecules were examined.
For continuum based condensed-phase molecular dynamics simulations, an accurate continuum solvation model is important in order to accurately simulate the motions of atoms in the aqueous phase.1 For many solvation models, a set of empirical parameters is finely tuned to reproduce experimental solvation free energies. In order to have a set of reliable experimental reference data, substantial effort has been devoted to compilations of solvation free energies.2-7 For neutral species, Truhlar and coworkers have concluded that the uncertainty in experimental solvation free energies is typically as low as 0.2 kcal mol−1.8 On the other hand, for the aqueous solvation free energies of ionic species, a typical experimental error of 4-5 kcal mol−1 was estimated because of the uncertainties in associated experimental quantities.8 Hence, the relatively large uncertainty of reference values for ionic solutes has hindered the critical assessment of current continuum solvation models.
The aqueous solvation free energies of an anion can be determined using the thermodynamic cycle shown in scheme 1.
and is defined as,9
where is the solvation free energy of the neutral species AH, is equal to 2.303RTpKa (AH) , (where pKa (AH) is the negative common logarithm of the aqueous-phase acid dissociation constant of AH). is the standard aqueous solvation free energy of the proton, is the gas-phase basicity of the anion A− defined as
Kelly et al. have reported the estimated uncertainties for the solvation free energy of anions using the root-sum-of-squares combinations of the experimentally measured quantities on the right side of the equation 1.3 The typical uncertainty of the solvation free energy of anions is 2-3 kcal mol−1. An average uncertainty of 0.2 kcal mol−1 for the solvation energy of neutral solutes was previously estimated.8 The experimental pKa (AH) within the range of 0-14 can be measured fairly precisely, therefore, the uncertainty of is negligible for the estimation the overall uncertainty of . For the aqueous solvation free energy of the proton, Kelly et al. assigned an uncertainty of 2 kcal mol−1,3 which has a large contribution to the overall uncertainty of . The gas-phase basicities of the anions were originally taken from the NIST standard reference database10. In this study, we took the values and their uncertainties from the data sets collected by Kelly et al.3. For several anions, there is more than one experimental measurement available, and a typical uncertainty of 2 kcal mol−1 is assigned for most of the anions.11-14 For some cases, the uncertainties of the gas-phase basicities are as large as 2.8 kcal mol−1, which significantly increases the overall uncertainties of the solvation free energies of anions.
During the past two decades, great progress has been made towards achieving the goal of predicting thermodynamic properties to “chemical” accuracy (1 kcal mol−1).15,16 High-level electron correlation theory, e.g. CCSD(T)17 incorporating high angular momentum basis functions has become the “gold standard” approach for obtaining thermochemical properties to chemical accuracy. Higher accuracy can be further attained by extrapolation of the energies to the complete basis set limit (CBS).18,19
Previous studies20-49 have been carried out to calculate the gas-phase basicities and acidities of molecules. Burk and co-workers,26,28 Koppel et al.21 have critically assessed the performance of density functional theory for prediction of gas-phase acidities and basicities. Burk et al. have concluded that the average absolute errors can fall below 2.5 kcal mol−1 for their test sets (49 acids and 32 bases) based on B3LYP/6-311+G(3df,3pd) calculations.26 Many-body perturbation theory (MBPT)50 and coupled-cluster theory (CC)51-55 in conjunction with G256, G357 and “multi-level” approaches (e.g. CBS-QB337,58, G3B359, G3MP2B359, MCCM/360 and SAC/360) have been proposed to obtain thermochemical data to chemical accuracy. In these procedures, a series of calculations are carried out at different levels of theory with different basis sets. Zero-point energy and high-level corrections were made based on the additivity approximation. For instance, the CBS-QB3 theory optimizes the geometries of molecules and calculates thermochemical data at the B3LYP/6-311G(2d,d,p) level, followed by a series of MP2, MP4 and CCSD(T) calculations using Pople type basis sets to obtain the electron correlation energy. Ervin and Deturi have found that CCSD(T)/aug-cc-pVTZ calculations give more accurate gas-phase acidities than CBS-QB3 theory for the molecules they tested,20 which indicates that large basis sets are required to obtain accurate electron correlation energies of molecules. However, CCSD(T) calculations using aug-cc-pVTZ are limited to small molecules due to the poor scaling properties (N7, where N is the number of basis functions) for CCSD(T) calculations. In addition, they did not extrapolate the CCSD(T) energies to the complete basis set limit.
Martin and co-workers have developed the W1 and W2 methods,32,34 where the CCSD and CCSD(T) energies are extrapolated to the infinite-basis limit. Moreover, contributions from inner-shell correlation, scalar relativity, atomic spin-orbit splitting and anharmonic zero-point energies were also included. One of the most sophisticated computations which have been done so far is by Allen and co-workers.27 They have performed all-electron coupled-cluster (AE-CC) calculations up to single, double, triple, quadruple and pentuple excitations with Dunning's augmented correlation-consistent, atom-centered Gaussian basis sets. They have also included the core electron correlation, scalar relativistic effects, diagonal Born-Oppenheimer corrections (DBOC)61-64 and anharmonic zero-point energies. However, such expensive calculations are currently limited to molecules with 2 heavy atoms and serve more as benchmark calculations rather than as an approach that can be applied generally.
It is well known that accurate calculation of the electron correlation energy requires a large atom-centered Gaussian basis set. In this work, we use Dunning's augmented correlation-consistent basis sets (aug-cc-pVnZ)65-67 (where n=D,T,Q) for benchmark MP2 and CCSD(T) calculations on gas-phase basicities and extrapolate the results to the complete basis set limit. Thereby, the errors arising from the incompleteness of the basis can be largely reduced.68 The goals of this study are (1) to benchmark the accuracy of different ab initio theories (HF, MP2 and CCSD(T)) for the theoretical estimation of the gas-phase basicities of molecules and (2) to identify an efficient approach which is able to achieve chemical accuracy for gas-phase basicity calculations on systems containing up to 10 heavy atoms. We can use the resultant approach as a useful computational protocol to validate experimental gas-phase basicities, when more than one experimental measurement is available, and to even make accurate theoretical estimates for the cases where experimental values are not available. In this study, we include some unusual molecules, such as hydroperoxides, in the test set of 41 molecules; furthermore, we have also examined the conformational effects for accurately theoretical prediction of gas-phase basicities.
We used the Gaussain03 package69 for all ab initio calculations. MP2/aug-cc-pVTZ calculations were carried out on all the molecules for geometry optimizations, vibrational frequencies and thermochemical analyses. The zero-point vibrational energies (ZPVEs) only include harmonic contributions. Subsequently, frozen-core MP2 and CCSD(T) single point energy calculations using augmented correlation-consistent basis sets (aug-cc-pVnZ) were employed on the optimized structures. The two point extrapolation scheme18
was used to obtain the complete basis set (CBS) extrapolated values of the MP2 correlation energies (EMP2_CBS) from energy calculations using two different basis sets, aug-cc-pVTZ and aug-cc-pVQZ, . The variable x in equation 3 represents their largest angular momentum of the basis set, i.e. x=3 for aug-cc-pVTZ and x=4 for aug-cc-pVQZ. The Hartree-Fock energies were not extrapolated and were simply taken from the results of the larger basis set (aug-cc-pVQZ) calculations. The CBS correlation energies for CCSD(T) were obtained using:
which is based on the observation that the difference between the MP2 and CCSD(T) correlation energies converges faster in basis set size than the correlation energies themselves70-72. The effectiveness of the computational approach shown in equation 4 is based on the propositions within the so-called focal-point analysis (FPA) scheme.27,73-75 The internal thermal energy corrections (translational Etrans, rotational Erot and vibrational Evib ) were made to the electronic energy,76
The Gibbs free energy G was calculated from
Where R is the gas constant, T is the temperature, H is the enthalpy and Stot = Strans + Srot+ Svib+ Selec (contributions from translational, rotational, vibrational and electronic motions, respectively). The gas-phase basicity of a species A− is defined in equation 2. The standard state was 298.15 K and 1 atm pressure.
First, to assess the accuracy of the complete basis set limit for MP2 and CCSD(T) calculations, we carried out full ab initio CCSD(T)/aug-cc-pVTZ and CCSD(T)/aug-cc-pVQZ calculations on five small molecules (H2O, H2S, HCN, C2H2, H2O2) for comparison. One can see from Table 1, for the same optimized geometries obtained from MP2/aug-cc-pVTZ calculations, HF/aug-cc-pVQZ has the largest RMSE of 5.6 kcal mol−1 compared to experimental values. MP2/aug-cc-pVQZ, MP2_CBS (MP2 with complete basis set estimate) and CCSD(T)/aug-cc-pVDZ results have smaller RMSEs between 2.0 kcal mol−1 and 2.6 kcal mol−1. CCSD(T)_CBS (CCSD(T) with complete basis set estimate) performs just as well as the significantly more expensive CCSD(T)/aug-cc-pVTZ and CCSD(T)/aug-cc-pVQZ levels. Note that the CCSD(T)_CBS results are extrapolated from MP2_CBS and CCSD(T)/aug-cc-pVDZ calculations with no additional computational cost. Due to the poor scaling of CCSD(T), it is not economical to calculate the Gibbs free energy for relatively larger molecules using large basis sets such as aug-cc-pVTZ and aug-cc-pVQZ, however, the extrapolation using equation 4 strikes a compromise between the computational expense incurred and the attained accuracy for our test on five representative small molecules.
Next, we applied the extrapolation approach using equation 4 for the remaining 36 molecules and the results are shown in Table 2. HF/aug-cc-pVQZ has the largest overall RMSE for this test set. MP2/aug-cc-pVQZ and MP2_CBS have similar performance with very close RMSEs of 3.0 kcal mol−1 and 3.2 kcal mol−1, respectively. CCSD(T)/aug-cc-pVDZ outperforms the MP2 results, with a RMSE of 2.2 kcal mol−1. Among all the approaches we tested, CCSD(T)_CBS has the lowest RMSE of 1.0 kcal mol−1. Only 6 gas-phase basicities (hydrogen cyanide, methanol, cyanamide, methyl hydroperoxide, acetic acid and 1,2-ethanediol) out of 41 obtained by CCSD(T)_CBS calculations fell outside the experimentally measured range. As the ab initio electron-correlation level increases from MP2 to CCSD(T), the accuracy gets better. From this comparison, we conclude, not unexpectedly, that accurate estimation of the electron correlation energy is important for theoretical gas-phase basicity predictions. Moreover, CCSD(T)_CBS calculations provide reliable gas-phase basicities of molecules at chemical accuracy at an affordable computational cost.
To further check the convergence of the extrapolation approach, we chose six molecules (hydrogen cyanide, methanol, cyanamide, methyl hydroperoxide, acetic acid and 1,2-ethanediol) whose calculated gas-phase basicities deviated from the experimental values for further analysis. As shown in equation 8, we computed the complete basis set limit for CCSD(T) by extrapolating the energies from CCSD(T)/aug-cc-pVTZ calculations instead of from the CCSD(T)/aug-cc-pVDZ level,
As shown in Table 3, the CCSD(T)_CBS extrapolated from CCSD(T)/aug-cc-pVDZ and CCSD(T)/aug-cc-pVTZ levels yield almost identical gas-phase basicities. In addition, we also obtained the CBS extrapolated values of the MP2 correlation energies (EMP2_CBS) from energy calculations using two larger basis sets, aug-cc-pVQZ and aug-cc-pV5Z using equation 3 (where x=4 for aug-cc-pVQZ and x=5 for aug-cc-pV5Z), and the Hartree-Fock energies were taken from the results of HF/aug-cc-pV5Z calculations. As shown in Table 3, using the MP2 CBS energies extrapolated from larger basis sets, the gas-phase basicities obtained from CCSD(T) CBS energies have very subtle changes. Therefore, the results are likely converged, or nearly converged, for these six molecules. It indicates that the CBS limit of CCSD(T) extrapolated from CCSD(T)/aug-cc-pVDZ level is, indeed, reliable for gas-phase basicity calculations.
Following the spirit of FPA approach,27,75 we further check the convergence of the HF, MP2 and CCSD(T) CBS limits using an extrapolation based on aug-cc-pV5Z and aug-cc-pV6Z for five representative molecules. For extrapolation of the Hartree-Fock energies, the two parameter exponential functions were used.77,78
The MP2 and CCSD(T) CBS energies were extrapolated using equation (3). As shown in Table 4, the gas-phase basicities calculated using MP2 energies extrapolated from smaller basis sets aug-cc-pVTZ and aug-cc-pVQZ are very close to those extrapolated gas-phase basicities using the much larger basis sets aug-cc-pV5Z and aug-cc-pV6Z. Among the five small molecules, the largest deviation of the MP2 extrapolated values is 0.39 kcal mol−1 for H2O. Meanwhile, the CCSD(T) computed gas-phase basicities using the extrapolation scheme of equation 4 are also very close to the CCSD(T) CBS limits. The largest deviation is also as low as 0.39 kcal mol−1 for C2H2 comparing the computed gas-phase basicities using equation 4 with the CCSD(T) CBS extrapolated values based on aug-cc-pV5Z and aug-cc-pV6Z basis sets. The observed deviations from the CBS limit calculations are well below our target accuracy (1 kcal mol−1). Overall, it is not currently routinely feasible to carry out MP2 and CCSD(T) calculations using aug-cc-pV5Z and aug-cc-pV6Z basis sets for molecules with more than 2 heavy atoms. Therefore, we conclude that the scheme proposed in this study provides an affordable approach for theoretical predictions of the gas-phase basicities of larger molecules within the accuracy of 1 kcal mol−1.
The fact that the computed results indicate that they are likely converged suggests that the experimental values may have larger associated errors than what have been estimated. This notion is bolstered by the fact that for 35 of the cases examined we obtained results well within experimental error, while for only six cases we found more significant differences between theory and experiment. For methyl hydroperoxide, whose predicted gas-phase basicity has the largest deviation from the experimental value, we have also examined the possible rearranged species CH2−-O-O-H and H-O-CH2-O− for the anion of methyl hydroperoxide, but the calculated gas-phase basicities for these two species are even poorer indicating that rearranged species are unlikely. Hence, at least for the case of methyl hydroperoxide, we suggest that it would be worthwhile reexamining the experimental value to validate that theory is failing. This is true in this case given that only one experimental measurement79 is cited in the NIST standard reference database10 for this compound. Further corrections examined previously, like relativistic, anharmonic effects or diagonal Born-Oppenheimer corrections are much smaller (~0.2 kcal mol−1)27 than the present computed error, but given the unusual nature of this molecule we cannot rule out theoretical shortcomings entirely.
We further check the role anharmonic effects play on the gas-phase basicities for the molecules which were found to have relatively larger deviations from experiment. One can see from Table 5, the anharmonic effect lowers the ZPVE by 0.1 kcal mol−1 to 1.0 kcal mol−1. Especially for the relatively floppy molecule 1,2-ethanediol, the anharmonic correction has the largest value of −0.95 kcal mol−1 among the six molecules we have examined in Table 5. However, the anharmonic correction is largely cancelled out when we calculate the gas-phase basicities by deducting the anharmonic correction of the molecule from its anion. As shown in Table 5, the anharmonic effects on the gas-phase basicities are less than or equal to 0.15 kcal mol−1 for all six molecules, which is much smaller than our target accuracy 1 kcal mol−1. Therefore, we conclude that the harmonic ZPVE is adequate for our theoretical prediction on the gas-phase basicities.
For a few flexible molecules in this test set, we performed geometry optimizations from different starting geometries. Different initial conformations are usually trapped at different local minima at the end of the geometry optimization. We took the structure with the lowest free energy for the gas-phase basicity calculation when the energy difference between the two conformers was larger than 2.0 kcal mol−1. Otherwise, we took the ensemble average of all low energy conformations (< 2.0 kcal mol−1 energy difference) based on the Maxwell-Boltzmann statistics,
where εi is the free energy of the i-th conformer and gi is the degeneracy of the energy level εi.
To illustrate this, we carried out a conformational study on 1,2-ethanediol. As shown in Figure 1a-1d, four different local minima (tTt, tGg', gGg' and g'Gg') were found for 1,2-ethanediol at the MP2/aug-cc-pVTZ level, which is consistent with previous studies.80-82 The conformer tGg' with a weak intramolecular hydrogen bond is 2.0 kcal mol−1 lower in total free energy than the conformer tTt without the intramolecular hydrogen bond. The other two conformers gGg' and g'Gg' are 0.5 kcal mol−1 and 0.3 kcal mol−1 higher than the conformer tGg', respectively. Previous study has shown that the conformer gGg' has a lower free energy than g'Gg' based on MP2/6-31G* calculations using the geometries optimized at the HF/6-31G* level80, while in this study, we find g'Gg' is more stable than gGg' at the MP2/aug-cc-pVTZ level. Moreover, for the anion of 1,2-ethanediol (CH2OHCH2O−), the conformer shown in Figure 1f has a stronger intramolecular hydrogen bonding interaction in terms of the donor-acceptor distance. Compared to the neutral 1,2-ethanediol at the tGg' configuration, the distance between hydrogen-donor and oxygen-acceptor is decreased from 2.32 Å to 1.63 Å, and the O-H-O angle is increased from 108.7° to 137.0°, and thus the total free energy of the conformer shown in Figure 1f is 12.2 kcal mol−1 lower than the conformer without the intramolecular hydrogen bond shown in Figure 1e. The gas-phase basicity calculations on 1,2-ethanediol further confirm that the structures with the intramolecular hydrogen bonds should be used for computing chemical properties. One can also see from Table 6, the calculated CCSD(T)_CBS gas-phase basicity of 1,2-ethanediol has a 2.5 kcal mol−1 deviation from experiment using the geometries with the lower energies (conformer f and ensemble average over b, c and d). On the other hand, the CCSD(T)_CBS predicted value derived from conformer e) and a) (see Figure 1) has a larger deviation of 7.7 kcal mol−1. This shows that conformational effects are relevant for theoretical predictions of the gas-phase basicities of molecules. Thus sampling represents yet another challenge associated with computing gas-phase basicities using extraordinarily sophisticated computational techniques.27 Further conformational studies for allyl alcohol, acrylic acid, propanoic acid, 2,2,2-trifluoroethanol, pyruvic acid and 2-butanol are presented in the supporting materials.
Through the theoretical study of the gas-phase basicities of 41 small molecules, chemical accuracy was achieved via CCSD(T) calculations with CBS extrapolation. For 35 of the cases studied theory and experiment were in excellent accord, while for six cases (hydrogen cyanide, methanol, cyanamide, methyl hydroperoxide, acetic acid and 1,2-ethanediol) theory predicted values outside of the experimental error bars. We suggested that a re-examination of the experimental value for methyl hydroperoxide will help us determine whether some aspect of the theoretical approach is less than optimal or if the experimental uncertainties are larger than currently believed. The electron correlation energy was found to be an important component in the theoretical estimation of gas-phase basicities. The least inexpensive ab initio electron correlation method MP2, which scales with the fifth power of molecular size, was not adequate for gas-phase basicity prediction. For cases, where experimental gas-phase basicities are not available, or large uncertainties (~3.0 kcal mol−1) are associated with the available values, the computational procedure proposed in this study provides a validated approach to accurately predict the gas-phase basicities of molecules with near chemical accuracy. Even though the computational expense scales with the seventh power of the molecular size for CCSD(T) calculations, modern parallel implementation of CCSD(T) calculations83-87 and low-order scaling local electron correlation methods88-91 have extended the power of coupled-cluster theory to systems beyond 10 heavy atoms.
We thank the NIH (GM066859 and GM44974) for financial support of this research. Computing support from the University of Florida High Performance Computing Center is gratefully acknowledged.
Supporting Information Available: Conformational studies on several medium-sized floppy molecules. This material is available free of charge via the Internet at http://pubs.acs.org.