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Benchmarking Compound Methods (CBS-QB3, CBS-APNO, G3, G4, W1BD) against the Active Thermochemical Tables: A Litmus Test for Cost-Effective Molecular Formation Enthalpies John M. Simmie* and Kieran P. Somers Combustion Chemistry Centre & School of Chemistry, National University of Ireland, Galway, Ireland S Supporting Information *

ABSTRACT: The theoretical atomization energies of some 45 CxHyOz molecules present in the Active Thermochemical Tables compilation and of particular interest to the combustion chemistry community have been computed using five composite model chemistries as titled. The species contain between 1−8 “heavy” atoms, and a few are conformationally diverse with up to nine conformers. The enthalpies of formation at 0 and 298.15 K are then derived via the atomization method and compared against the recommended values. In general, there is very good agreement between our averaged computed values and those in the ATcT; those for 1,3-cyclopentadiene exceptionally differ considerably, and we show from isodesmic reactions that the true value for 1,3cyclopentadiene is closer to 134 kJ mol−1 than the reported 101 kJ mol−1. If one is restricted to using a single method, statistical measures indicate that the best methods are in the rank order G3 ≈ G4 > W1BD > CBS-APNO > CBS-QB3. The CBS-x methods do on average predict ΔfH⊖(298.15 K) within ≈5 kJ mol−1 but are prone to occasional lapses. There are statistical advantages to be gained from using a number of methods in tandem, and all possible combinations have been tested. We find that the average formation enthalpy coming from using CBS-APNO/G4, CBS-APNO/G3, and G3/G4 show lower mean signed and mean unsigned errors, and lower standard and root-mean-squared deviations, than any of these methods in isolation. Combining these methods also leads to the added benefit of providing an uncertainty rooted in the chemical species under investigation. In general, CBS-APNO and W1BD tend to underestimate the formation enthalpies of target species, whereas CBS-QB3, G3, and G4 have a tendency to overestimate the same. Thus, combining CBS-APNO with a G3/G4 combination leads to an improvement in all statistical measures of accuracy and precision, predicting the ATcT values to within 0.14 ± 4.21 kJ mol−1, thus rivalling “chemical accuracy” (±4.184 kJ mol−1) without the excessive cost associated with higher-level methods such as W1BD.

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The accuracy of any ΔfH⊖ derived from quantum chemistry depends largely on the model chemistry applied and the size of the molecular system, with the theoretical thermodynamicist in constant pursuit of “chemical accuracy”  that is formation enthalpies computed within 1 kcal mol−1 (4.184 kJ mol−1) of the true formation enthalpy. The current “gold-standard” in model chemistries is the coupled-cluster method with perturbative quadruples, CCSDT(Q), with subsequent extrapolation of the computed energies to the basis-set limit; see for example, work on the C2H5 + O2 reaction.8 This is rarely achievable, and more practical standards are the CCSD(T) methods9 with perturbative inclusion of triples. The Wn,10−12 high-accuracy extrapolated ab initio thermochemistry (HEAT),13−15 focal-point extrapolation,16 and Feller-Peter-

INTRODUCTION

The determination of accurate formation enthalpies (ΔfH⊖) of stable, radical, and ionic species is central to the science of combustion modeling, serving as essential input information for a multitude of numerical simulations. Such is the importance of this fundamental property that great effort has been invested in their tabulation and refinement, with the JANAF,1 CODATA,2 and Third Millenium databases,3 and the Active Thermochemical Tables (ATcT),4−6 serving the combustion community well for the last number of decades. Although experimental determination (primarily calorimetry) is the ideal method to obtain accurate ΔfH⊖, it is both laborious and time-consuming. Add to this the number of species which may comprise a chemical kinetic mechanism7 (100s−1000s), many of which are radicals which may be experimentally inaccessible, it is clear that obtaining experimental ΔfH⊖ for all species of interest in combustion modeling is effectively impossible. However, it is now accepted that quantum chemistry, either in isolation or in tandem with experiment, can offer an effective means with which to obtain such information on feasible timescales. © 2015 American Chemical Society

Special Issue: 100 Years of Combustion Kinetics at Argonne: A Festschrift for Lawrence B. Harding, Joe V. Michael, and Albert F. Wagner Received: November 14, 2014 Revised: January 5, 2015 Published: January 12, 2015 7235

DOI: 10.1021/jp511403a J. Phys. Chem. A 2015, 119, 7235−7246

Article

The Journal of Physical Chemistry A son-Dixon17 methods have also shown accuracy to within ≈2 kJ mol−1 when compared with well-known experimental formation enthalpies. However, despite advances in computer software and hardware, these methods remain computationally expensive and are somewhat restricted in their application to molecules containing some five or so nonhydrogen atoms and, as a consequence, can rarely treat the full panoply of molecules which are to be encountered in detailed chemical kinetic models.7 Compound methods offer a cost-effective alternative to the above, albeit at the expense of accuracy. These methods consist of a series of less expensive geometry optimization, frequency, and single-point energy calculations with empirical corrections used to overcome deficiencies when the methods are benchmarked against formation enthalpies of well-known atomic and small-molecule standards. Numerous compound methods have entered widespread use within the combustion community in recent times; in particular, the CBS-x18,19 (x = QB3 and APNO) and Gaussian-x20−24 (x = 1, 2, 3, and 4) methods are regularly employed in thermodynamic and kinetics computations of relevance to combustion and atmospheric chemistry. Variants of the G3 method, such as G3B325 and G3MP226 are also popular, with the former frequently employed for ΔfH⊖ computations by Burcat et al. as part of the Third Millennium Database, with an uncertainty of ±8 kJ mol−1 typically quoted therein when they have applied this method. “Standard” methods have been tweaked on numerous occasions to remedy perceived difficulties and/or with the aim of improving the computational cost; for example, variants of G4,27,28 of W129,30 and W3,31,32 etc. It appears that divergent evolution is the order of the day. In order to overcome the large uncertainties in ΔfH⊖ when compound methods are employed, the isodesmic reaction method33 is frequently employed to reduce errors and uncertainties in the computation. However, a number of problems arise when employing this method. Well-known formation enthalpies are required from experiment or high-level theory to create these hypothetical working reactions; the computed ΔfH⊖ is very much dependent on the quality of the isodesmic reaction framed (given that there is no unique answer), and the computational cost is increased as numerous quantum chemical calculations must be carried out to determine the theoretical enthalpy of all the chaperone molecules. However, provided good working reactions can be framed the method can deliver excellent results; as an example, see previous work on cyclic esters.34 The empirical corrections employed in compound methods also reduce their predictive capability, as the databases against which they are benchmarked, and any empirical corrections thus derived tend to consist of a limited number of small molecule targets. Molecules of interest to combustion scientists can be large, conformationally complex, species extending well outside the benchmarked test-set in terms of size and the true uncertainty in any ΔfH⊖ computed from these methods is somewhat unknown. Thus, we arrive at the goals of this work: to benchmark a range of popular compound methods against a database of wellknown formation enthalpies, namely, the Active Thermochemical Tables of Ruscic.6 This database has been chosen as it is based on either direct experiment or experiment in combination with high-level theory and is undergoing constant development and refinement. A brief description of the

methodology behind ATcT and discussion of uncertainty quantification in general has been recently given by Ruscic.35 The isodesmic reaction method is not employed in this work as part of our benchmarking, owing to the problems discussed above, rather we use the atomization method alone to derive ΔfH⊖ throughout. Thus, the formation enthalpies derived herein are representative of the “pure” method, and are independent of experimental formation enthalpies required for isodesmic reactions. The results aim to provide an assessment of the performance of each method against a wide-range of stable and, in the future, radical species. Recently, Simmie et al.36 carried out a thermochemical study of furan derivatives, and they found that a ΔfH⊖ computed by taking an average ΔfH⊖ from CBS-QB3, CBS-APNO, and G3 atomization energies recreated those computed from wellframed isodesmic working reactions to within ≈4 kJ mol−1, despite the large uncertainties associated with the atomization calculations. The reasons for this interesting behavior were not investigated at the time, but Somers37 later ascribed it to a tendency of the G3 method to consistently overpredict ΔfH⊖, for the CBS-APNO method to consistently underpredict ΔfH⊖, and for the CBS-QB3 method to arrive at a similar answer to the isodesmic working reactions. A second goal of this work is therefore to assess whether a combination of compound methods can provide a more rigorous assessment of ΔfH⊖ than any of these methods in isolation, thus offering a cost-effective alternative to high-level methods. A primary reason for using two or more model chemistries, quite apart from the statistical gain, is that the final result is then less-dependent upon the actual geometrical structure if the optimization and frequency calculations, inbuilt into each composite method, differ. Of course if the two optimizations result in distinctly different geometries and/or electronic states then this approach would be compromised, but in our experience this has rarely happened for the species under consideration here.

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COMPUTATIONAL METHODOLOGY Five compound methods have been chosen for benchmarking as part of this work: CBS-QB3, CBS-APNO, G3, G4, and W1BD,38 as embedded within the application Gaussian.39 For the computation of 0 K enthalpies of formation, ΔfH0, we commence by calculating the theoretical atomization energy, TAE0, for the reaction: Cx HyOz → x 3C + y 2 H + z 3O

which is given by TAE0 = xH0(3C) + yH0(2 H) + zH0(3O) − H0(Cx HyOz ) (1)

where H0 is the 0 K enthalpy of an atom or molecule and is the sum of the electronic and zero-point energies. Zero-point energies are automatically computed, adjusted by a built-in scale factor and added to the 0 K electronic energy by each compound method as part of its predefined series of computations. The ΔfH0 of the molecule then follows knowing the theoretical atomization energy and the experimentally known formation enthalpies of the component atoms in their gaseous state: 7236


Article

The Journal of Physical Chemistry A

298.15 K. We display in Table 3, the results of computed formation enthalpies for all five model chemistries in comparison to ATcT values. Problem Molecules. Some molecules in the ATcT compendium are problematic in the sense that the composite methods under discussion here are unable to attain a satisfactory solution to their structure. Thus, for example, oxirene, a three-membered heterocyclic ring containing a CC bond resists optimization (or rather optimizes to a structure with an imaginary frequency) with the B3LYP functional, which is of course an essential component of CBS-QB3, G4, and W1BD. The basis sets used by these three methods are predefined at 6-311G(2d,d,p), 6-31G(2df,p), and cc-pVTZ+d; note that using a 6-311++G(d,p) basis set leads to a successful convergence but just not to a C2v structure. This molecule has a long and checkered history; in a telling phrase it has been described as “hovering on the edge of reality”, or it has been subtitled with a Shakespearean quotation “to be or not to be”, with some theoretical methods concluding that it is a saddle point and others a local minimum on the potential energy surface.42,43 Recent extremely expensive coupled-cluster interference-corrected explicitly correlated second-order perturbation theory results44 ascribe a total atomization energy of 1827.8 kJ mol−1 at 0 K, which implies an enthalpy of formation of 277.0 kJ mol−1, whereas equally expensive W4 calculations45 give 272.1 and 268.2 kJ mol−1 at 0 and 298.15 K, respectively. Viewed from this perspective, the CBS-APNO and G3 0 K results of 274.5 and 275.4 kJ mol−1, respectively, are surprisingly good. We should also take a moment to note if there are any considerable differences between any of our computations and those values recommended in the ATcT. The most obvious example is 1,3-cyclopentadiene where the ATcT value of ΔfH⊖(298.15K) = 101.30 ± 2.5 differs considerably from our result of 137.1 ± 6.7 kJ mol−1. In the compendium Thermochemical Data of Organic Compounds, Pedley et al. quote46 a value of 134.3 ± 1.5 kJ mol−1, whereas the NIST WebBook includes determinations47,48 of 139 and 133.4 kJ mol−1. In a thermochemical and kinetic analysis of addition reactions to 1,3-cyclopentadiene Zhong and Bozzelli49 use a value of 131 kJ mol−1. More recent photoacoustic spectroscopy and quantum chemical studies by Agapito et al.50 imply a heat of formation of 134 kJ mol−1. From a consideration of isodesmic reactions involving (i) furan, propane and dimethyl ether, (ii) ethyne, 2 molecules of ethene and methane as chaperones, it can be shown that the computed reaction enthalpies of −90.30 ± 1.58 and 275.53 ± 4.71 kJ mol−1 lead to values for ΔfH (298.15 K) of 135.21 ± 1.80 and 132.47 ± 4.76 kJ mol−1, respectively, from which a grand-weighted average of 134.9 ± 1.7 kJ mol−1 results. Thus, it appears that the ATcT value for 1,3-cyclopentadiene is incorrect. There are similar, but less severe, disparities for oxalic acid. We note that an isodesmic reaction:

Δf H0(Cx HyOz ) = [xΔf H0(3C) + yΔf H0(2 H) + zΔf H0(3O)] − TAE0

(2) ⊖

For formation enthalpies at 298.15 K, ΔfH , the same equations are employed, with thermal enthalpy corrections added to H0 in eq 1 to account for translational, rotational, and vibrational degrees of freedom of each molecule and atom. Standard state formation enthalpies of each gaseous atom, ΔfH⊖, also replace the 0 K counterparts in eq 2. Note that the influence of vibrational anharmonicities, rotational nonrigidities, and torsional degrees of freedom are not accounted for in the predefined routine of calculations in any of the compound methods employed, and therefore, in the final formation enthalpies reported herein. The 0 and 298.15 K formation enthalpies of atomic species used in the above equations are given in Table 1 and are adopted from the ATcT.6 Table 1. Atomic Formation Enthalpies (kJ mol−1) T (K)

C (3P)

H (2S1/2)

O (3P2)

0 298.15

711.38 716.87

216.034 217.998

246.844 249.229

In those cases where conformational diversity exists the Gibbs free energies, ΔG⊖ (298.15 K) were determined for each conformer and the Boltzmann distribution computed, taking due account of degeneracies, σ (eq 3). The contribution, xi, made by each conformer to the overall enthalpy of formation for species X is then calculated from n

xi = σi exp( −ΔGm⊖(i)/RT )/∑ [σi exp( −ΔGm⊖(i)/RT )] i=1

(3) n

Δf Hm⊖(X) =

∑ [xiΔf Hm⊖(i)] i=1

(4)

Thus, in the case of n-hexane nine conformers (out of a possible 12 as shown schematically by Morini et al.)40 with accompanying degeneracies or symmetry numbers were identified: ttt(1), gtt(4), tgt(2), tgg(4), gtg(2), g+t+g−(2), ggg(2), g+x−t+(4), and t+g+x−(4). These give rise to populations of 17.1, 45.0, 11.8, 13.6, 3.2, 5.9, 1.2, 1.4, and 0.8%, respectively, from G4 calculations. Each individual enthalpy of formation was then weighted with respect to the population to calculate a final value for that particular molecule (eq 4). In some cases, individual conformers are present in the ATcT such as syn- and anti-formic acid, and in these cases, no distributions are needed. In general, the literature was consulted for possible conformers of a particular molecule, but in any event, these were then checked by using the application Spartan and utilizing a modest basis set and B3LYP functional to generate the distributions.41 Only those which contributed to a significant extent were then retained for further computation.

(COOH)2 + C2H6 = 2CH3COOH

can be framed whose average reaction enthalpy is −43.86 ± 0.96 kJ mol−1, from which the predicted formation enthalpy of the dominant aa conformer of oxalic acid is −739.8 ± 1.2 kJ mol−1 where we have adopted the ATcT numbers for the chaperones ethane and cis acetic acid. This is in very good agreement with the atomization value of −735.5 ± 5.3 kJ mol−1 for the same conformer, thus indicating that the ATcT value for

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RESULTS AND DISCUSSION The results at 0 K are shown in Table 2; although interesting, the primary focus of our attention will be on the results at 298.15 K, which are much more important for chemical kinetic simulations and modeling purposes. 7237


Article

The Journal of Physical Chemistry A Table 2. Final Computed Formation Enthalpies of Compounds at 0 K/kJ mol−1 species

QB3

APNO

G3

G4

W1BD

ATcT

1,3-butadiyne 1,3-cyclopentadiene 1-butene (gauche) 1-butene (cis) 1-butyne 2,2,4-trimethylpentane gauche 2,3-butanedione 2-butyne 2-propanol (gauche) 2-propanol (trans) acetic acid (cis) acetone allene benzene carbon monoxide cis-2-butene cyclobutene cyclohexane cyclopropane cyclopropene dimethyl ether dioxirane ethanal ethane ethene ethylene glycol no. 1 ethylene glycol no. 2 ethylene glycol no. 3 ethyne ethynol formaldehyde formic acid (syn) formic acid (anti) glyoxal (trans) glyoxal (cis) isobutane isobutene ketene methane methanediol methanol methyl formate (cis Z) methyl hydroperoxide n-butane (trans) n-butane (gauche) n-hexane ttt gtt tgt tgg gtg g+t+g− ggg g+x−t+ t+g+x− o-benzyne oxalic acid (c/c) oxalic acid (t/t) oxalic acid (c/t) oxirane (1A1) oxirene (singlet)

468.41 161.39 28.49 29.32 188.37 −147.58 −160.09 −315.19 165.90 −249.47 −247.99 −422.67 −200.49 202.25 107.21 −115.57 21.79 186.71 −74.02 76.69 298.87 −171.42 0.42 −156.89 −66.45 64.11 −375.41 −373.05 −372.82 234.82 95.86 −110.24 −377.05 −360.12 −214.26 −194.85 −100.31 11.26 −45.49 −66.31 −385.18 −193.15 −354.08 −123.01 −92.35 −89.57 −119.90 −117.02 −117.10 −114.34 −114.07 −113.52 −112.14 −107.68 −107.19 477.32 −716.78 −734.03 −723.06 −43.62 270.38

462.06 148.19 15.49 16.30 178.03 −178.69 −191.51 −319.76 155.80 −257.53 −256.49 −422.63 −−207.47 196.48 91.80 −113.78 8.40 175.51 −97.33 65.64 292.28 −173.81 5.71 −158.95 −74.99 59.75 −374.64 −372.87 −376.46 233.54 97.37 −107.06 −372.59 −356.14 −209.65 −190.70 −116.91 −1.83 −47.19 −70.91 −381.72 −193.46 −351.59 −117.57 −109.03 −106.14 −144.30 −141.50 −141.41 −139.78 −138.68 −138.25 −138.09 −132.51 −132.15 463.97 −710.04 −726.40 −715.92 −44.31 274.54

462.03 155.65 22.23 23.07 181.67 −159.19 −171.70 −313.36 160.26 −249.86 −248.78 −418.61 −200.81 196.62 103.83 −114.71 16.17 183.12 −82.55 73.82 295.82 −167.66 9.70 −155.71 −69.15 60.55 −370.04 −367.84 −366.17 230.90 95.93 −107.15 −371.39 −354.84 −209.62 −190.25 −105.72 5.36 −47.14 −67.78 −378.43 −190.06 −348.58 −112.96 −97.89 −95.06 −128.06 −125.39 −125.38 −123.75 −122.71 −122.20 −122.10 −116.30 −116.18 470.99 −707.19 −724.48 −713.53 −39.64 275.44

458.61 153.43 23.05 24.11 181.33 −156.69 −169.26 −311.53 160.51 −247.65 −246.28 −416.90 −199.61 197.07 102.26 −117.59 16.91 180.29 −80.66 71.92 293.58 −166.80 6.86 −155.49 −67.35 60.84 −368.17 −365.49 −364.65 229.18 95.82 −108.04 −371.35 −354.93 −210.61 −192.15 −103.79 5.60 −−45.51 −66.61 −378.13 −189.87 −347.45 −114.09 −95.63 −92.69 −125.53 −122.61 −122.65 −120.02 −119.69 −119.07 −117.97 −113.51 −112.80 467.35 −704.49 −719.94 −709.83 −41.35 269.63

458.08 144.16 15.39 15.85 176.13 −168.22 −181.87 −318.11 154.99 −257.09 −255.95 −423.43 −206.14 194.26 92.65 −114.07 8.30 172.02 −93.44 65.95 288.91 −171.77 8.66 −158.85 −73.12 58.11 −378.03 −376.18 −375.53 228.61 92.19 −107.36 −374.83 −358.01 −210.77 −192.13 −113.44 −1.81 −47.47 −68.60 −385.35 −194.75 −350.69 −117.97 −106.21 −103.17 −140.24 −137.14 −137.18 −134.17 −134.04 −133.46 −131.84 −127.90 −127.20 464.34 −711.63 −728.90 −717.78 −44.17 270.16

458.39 118.20 20.30 20.93 178.97

7238

−171.20 −310.25 158.72

−419.59 −199.39 197.63 100.70 −113.81 13.94 173.90 −82.52 70.76 292.64 −166.51 9.00 −154.98 −68.13 61.08

228.89 94.80 −105.33 −371.62 −355.28 −206.85 −188.42 −106.76 3.46 −45.47 −66.56 −379.21 −189.82 −344.54 −114.99 −98.54 −98.54

469.60

−40.17 276.60


Article

The Journal of Physical Chemistry A Table 2. continued species propane propene propyne succinic acid (aAa) sGs sGa gAs toluene trans-2-butene vinyl alcohol (anti) vinyl alcohol (syn)

QB3 −78.96 40.32 198.64 −806.20 −810.18 −804.18 −801.13 80.45 16.56 −108.62 −112.65

APNO −91.57 31.34 191.56 −805.87 −809.03 −802.68 −800.04 60.99 3.38 −110.86 −114.81

oxalic acid of −721.4 ± 2.1 is probably somewhat on the high side. In a similar manner, an isodesmic reaction targeting cyclobutene can be constructed with ethane, cyclobutane, and ethene as chaperons. This reaction is, not unexpectedly, very nearly thermoneutral at 1.74 ± 1.16 kJ mol−1 and based on a formation enthalpy for cyclobutane,46 which is absent from the ATcT, of 28.4 ± 0.6 kJ mol−1; this results in ΔfH⊖(298.15 K) = 163.0 ± 1.3 kJ mol−1, which is somewhat higher than the ATcT value of 156.9 ± 1.6 but in excellent agreement with our computed average of 163.9 ± 5.9 kJ mol−1. Statistical analyses in subsequent sections therefore do not include the molecules 1,3-cyclopentadiene and oxalic acid but cyclobutene is included in our later calculations. Conformers. Some of the values in ATcT v1.110 are composites, for example 1-butene is quoted at −0.03 ± 0.48 kJ mol−1, but in reality this molecule is comprised of two conformers, namely cis and gauche, with CCCC dihedral angles of ≈0° and ≈120°, respectively. The degenerate gauche conformer is dominant at 78.5% ± 1.5%, but the differences in enthalpy are slight, so the overall enthalpy of formation at 298.15 K is +0.78 ± 5.59, in very good agreement with the listed ATcT value of −0.03 ± 0.48 kJ mol−1. The results for 1butene and other systems are tabulated in Table 4. In general, most of the conformer distributions are consistent across all five composite methods, but the treatment of those compounds whose geometries are very sensitive to slight changes in dihedral angles, such as the gylcols and dibasic acids, can be much less successful. For oxalic acid, (COOH)2, three conformers prevail with the OC−CO always anti and the H−OC−C angles either anti or syn (Figure 1). In the case of the higher acid succinic, (CH2COOH)2, 4 principal conformers were identified in which three dihedral angles differ. These are classified in the order shown, Figure 2, as syn s, anti a, or gauche g, with the middle dihedral capitalized. Thus, the all trans configuration is denoted as aAa, etc. These lowest energy conformers all exhibit syn H−O−C O dihedrals. In this, we follow the notation and conclusions of a recent study by Vogt et al.51 The optimization procedures inbuilt in CBS-QB3 and G4 converge to a structure with an imaginary frequency if a C2h symmetry is imposed; this is a problem with the functional and basis sets used for these two cases. The incorporation of one (or more) diffuse terms in the 6-311G(2d,d,p) basis set, which is the default option in the model chemistry CBS-QB3, so that it now becomes CBS-QB3(+)52, does yield a nonimaginary solution with C2h symmetry.

G3

G4

W1BD

ATcT

−82.99 35.52 193.73 −799.30 −802.93 −796.99 −793.68 74.99 10.73 −109.00 −112.72

−80.87 35.95 192.74 −793.72 −798.32 −792.09 −789.33 73.56 11.40 −108.14 −111.93

−89.26 30.35 189.92 −806.69 −808.41 −802.00 −800.58 63.80 3.36 −113.89 −118.27

−82.13 35.36 192.86

73.69 9.39 −112.45

The conformer distribution outlined in Table 4 are consistent except for the W1BD results which give less prominence to the sGs conformer than to the aAa. These were among the most challenging calculations undertaken (Table 5), and perhaps a relaxation, in the case of the sGs conformer, of the convergence-related options for the Berny algorithm is responsible for this anomaly. Computational Cost. A crude example of the computational cost in terms of job times is shown in Table 5 from which it can be seen that G3 emerges as a relatively cheap yet, to be described below, effective method. Comparison of Methods with ATcT. We benchmark each theoretical method against the formation enthalpies recommended in the ATcT via descriptive statistics, and the results are shown in Tables 6 and 7. In order to assess how well a combination of different compound methods may perform, unweighted average formation enthalpies for various combinations of the CBS-QB3, CBS-APNO, G3, and G4 methods have also been computed. Note that combinations involving the W1BD method have generally been excluded, owing to its expense, but it is included in a combination of “all methods” as shown in subsequent figures and tables. For each method, or combination of methods, the difference between the ATcT recommendation and our computed formation enthalpy for species, i, is denoted as Φi = ΔfH⊖ ATcT − ΔfH⊖ i , and the mean signed errors (MSE) and mean unsigned errors (MUE) are computed as follows where n is the number of species in the benchmark set: MSE =

∑ (Φi)/n

(5)

MUE =

∑ |Φ|i /n

(6)

In terms of uncertainties, the maximum absolute deviation (|Φmax|) from the ATcT is determined from eq 6 and is essentially an indicator of “worst-case scenario” performance for each method; it is rarely used in thermochemical work as a true indicator of statistical accuracy or precision. In line with the recommendations of Ruscic, uncertainties are considered for 95% confidence intervals via computation of the population standard deviation (σMSE) and root-mean-square deviations (RMSD): σMSE = 2 ×

RMSD = 2 × 7239

∑ (Φi − MSE)2 /n

∑ (Φi)2 /n

(7)

(8) DOI: 10.1021/jp511403a J. Phys. Chem. A 2015, 119, 7235−7246

Article

The Journal of Physical Chemistry A Table 3. Final Computed Formation Enthalpies of Compounds at 298.15 K/kJ mol−1 species

QB3

APNO

G3

G4

W1BD

ATcT

±

1,3-butadiyne 1,3-cyclopentadiene 1-butene 1-butyne 2,2,4-trimethylpentane 2,3-butanedione 2-butyne 2-propanol acetic acid acetone allene benzene carbon monoxide cis-2-butene cyclobutene cyclohexane cyclopropane cyclopropene dimethyl ether dioxirane ethanal ethane ethene ethylene glycol ethyne ethynol formaldehyde formic acid glyoxal isobutane isobutene ketene singlet methane methanediol (sc,sc) methanol methyl formate (cis Z) methyl hydroperoxide n-butane n-hexane o-benzyne oxalic acid oxirane (1A1) oxirene (singlet) propane propene propyne succinic acid toluene trans-2-butene vinyl alcohol

471.07 145.90 8.28 175.81 −210.74 −329.50 155.35 −272.13 −434.23 −214.95 195.66 91.65 −111.44 2.10 171.05 −111.55 60.53 291.06 −188.22 −5.95 −166.36 −81.61 56.24 −393.53 234.81 94.53 −113.24 −383.10 −218.25 −127.920 −8.92 −47.50 −73.95 −398.12 −203.44 −365.61 −134.33 −118.72 −155.63 470.39 −742.52 −55.01 267.82 −100.52 26.05 192.48 −829.62 60.59 −3.18 −121.93

464.24 132.59 −4.76 165.16 −242.31 −334.43 144.95 −280.26 −434.25 −224.33 189.77 76.16 −109.62 −11.22 159.78 −134.71 49.45 284.32 −190.50 −0.72 −168.38 −90.15 51.84 −394.68 233.21 95.57 −110.05 −378.60 −213.70 −144.48 −22.01 −49.37 −78.54 −394.59 −203.71 −363.24 −129.09 −135.75 −180.08 457.41 −734.26 −55.77 272.45 −113.11 17.07 185.07 −827.47 41.08 −16.34 −123.86

463.89 140.30 2.24 169.07 −221.92 −327.74 149.65 −272.30 −430.05 −215.26 190.04 88.47 −110.56 −3.24 167.55 −119.53 57.73 287.98 −184.27 3.30 −165.00 −84.21 52.70 −386.99 230.69 94.34 −110.13 −377.30 −213.57 −133.05 −14.57 −49.19 −75.41 −391.18 −200.26 −360.10 −124.40 −124.35 −163.49 464.76 −732.26 −51.02 273.50 −104.35 21.40 187.42 −820.98 55.48 −8.74 −121.56

461.47 138.00 2.96 168.97 −219.66 −325.78 150.01 −270.09 −428.31 −214.24 190.47 86.81 −113.43 −2.78 164.67 −117.87 55.82 285.72 −183.56 0.55 −164.90 −82.48 53.00 −384.86 229.44 94.75 −111.01 −377.30 −214.51 −131.24 −14.56 −47.46 −74.22 −390.87 −200.09 −358.88 −125.60 −121.86 −161.05 460.69 −728.21 −52.70 267.18 −102.36 21.71 186.75 −816.80 53.85 −8.31 −121.16

460.74 128.72 −4.83 163.64 −232.28 −332.37 144.48 −279.53 −434.87 −220.67 187.70 77.14 −109.91 −11.32 156.42 −130.86 49.84 281.14 −188.43 2.36 −168.26 −88.23 50.26 −394.21 228.62 90.96 −110.33 −380.80 −214.65 −140.92 −21.95 −49.38 −76.21 −398.03 −204.88 −362.14 −129.37 −132.42 −175.67 457.45 −737.19 −55.51 267.60 −110.74 16.12 183.79 −826.02 44.02 −16.33 −127.54

460.11 101.30 −0.03 165.39 −223.70 −326.81 145.76 −272.81 −433.71 −216.09 190.15 83.18 −110.53 −7.33 156.90 −122.08 53.61 283.91 −184.02 1.30 −165.46 −83.79 52.56 −389.42 228.33 92.70 −109.17 −378.94 −212.48 −135.36 −17.60 −48.58 −74.53 −392.61 −200.71 −357.80 −127.73 −125.85 −166.94 460.70 −721.40 −52.72 275.90 −104.41 20.35 185.80 −817.75 50.41 −11.18 −123.76

0.87 2.50 0.48 0.85 1.50 0.98 0.79 0.37 0.49 0.37 0.37 0.26 0.03 0.53 1.60 0.68 0.53 0.59 0.44 1.20 0.32 0.17 0.15 0.49 0.15 1.40 0.11 0.27 0.59 0.40 0.53 0.15 0.06 0.96 0.18 0.59 0.91 0.38 0.48 1.40 2.10 0.44 3.10 0.29 0.33 0.38 0.61 0.37 0.51 0.91

We will first consider the performance of individual compound methods against the ATcT benchmark. As a general trend, the G3, G4, and CBS-QB3 methods tend to overpredict the formation enthalpies of the compounds studied, thus their negative MSE. Conversely, the W1BD and CBS-APNO methods underpredict ΔfH⊖ leading to a positive MSE. The computed MSEs and MUEs imply the following trend with respect to accuracy: G3 ≈ G4 > W1BD > CBS-APNO > CBSQB3 (Figure 3). These results are somewhat surprising, with

Note first the 2-fold multiplier in the above formulas which provides us with 95% confidence, and second that there is a subtle difference in our two measures of dispersion, σMSE and the RMSD. The former assumes the average deviation from the ATcT (MSE) is the central value for the distribution, whereas the latter assumes the central line is zero deviation from the ATcT. Both measures ultimately highlight the same trends in the performance of each method. In the text, uncertainties typically refer to the MSE ± σMSE. 7240


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The Journal of Physical Chemistry A Table 4. Composition-Averaged Enthalpies of Formation at 298.15 K/kJ mol−1 composition gauche cis final

a

QB3 0.788 0.212 8.28

APNO 0.781 0.219 −4.76

gauche transa final

0.9981 0.0019 −210.74

0.9981 0.0019 −242.31

gauche trans final

0.787 0.213 −272.13

0.755 0.245 −280.26

no. 1 no. 2 no. 3 final

0.609 0.206 0.185 −393.53

0.316 0.144 0.540 −394.68

trans gauche final

0.593 0.407 −118.72

0.756 0.244 −135.75

ttt gtt tgt tgg gtg g+t+gggg g+x-t+ t+g+xfinal

0.174 0.460 0.122 0.124 0.031 0.059 0.010 0.012 0.008 −155.63

0.161 0.423 0.102 0.179 0.033 0.066 0.018 0.010 0.009 −180.08

syn/syn anti/anti syn/anti final

0.003 0.966 0.031 −742.52

0.007 0.919 0.074 −734.26

aAa sGs sGa gAs final

0.166 0.733 0.059 0.042 −829.62

0.128 0.765 0.056 0.052 −827.47

anti syn final

0.195 0.805 −121.93

0.229 0.772 −123.86

G3 1-Butene 0.786 0.214 2.24 2,2,4-Trimethylpentane 0.9978 0.0022 −221.90 2-Propanol 0.755 0.246 −272.30 Ethylene Glycol 0.547 0.211 0.242 −386.99 n-Butane 0.750 0.250 −124.35 n-Hexane 0.146 0.423 0.105 0.185 0.035 0.068 0.019 0.010 0.009 −163.49 Oxalic Acid 0.001 0.936 0.063 −732.26 Succinic Acid 0.108 0.778 0.065 0.049 −820.98 Vinyl Alcohol 0.255 0.745 −121.56

G4

W1BD

0.805 0.195 2.96

0.763 0.237 −4.83

ATcT

−0.03 ± 0.48

0.9980 0.0020 −219.66

0.9980 0.0022 −232.28

−223.7 ± 1.5

0.776 0.224 −270.09

0.755 0.245 −279.53

−272.81 ± 0.37

0.492 0.144 0.364 −384.86

0.298 0.127 0.575 −394.21

−389.42 ± 0.49

0.614 0.386 −121.86

0.625 0.375 −132.42

−125.85 ± 0.38

0.171 0.450 0.118 0.136 0.032 0.059 0.012 0.014 0.008 −161.05

0.191 0.462 0.120 0.111 0.030 0.055 0.009 0.014 0.008 −175.67

−166.94 ± 0.48

0.007 0.950 0.044 −728.21

0.005 0.966 0.030 −737.19

−721.4 ± 2.1

0.136 0.782 0.042 0.040 −816.80

0.480 0.433 0.033 0.053 −826.02

−817.8 ± 0.61

0.209 0.791 −121.16

0.169 0.832 −127.54

−123.76 ± 0.91

One imaginary frequency of 26 cm−1.

Figure 2. Key dihedrals of succinic acid. Figure 1. Conformers of oxalic acid.

both Gx methods outperforming the considerably more expensive W1BD. 7241


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species, this result manifesting itself as a positive MSE. As a result, combining CBS-APNO with either G3/G4, CBS-QB3/ G3, or CBS-QB3/G4 leads to improvements in the computed MSEs when the results are compared with individual methods. In fact, adding the CBS-APNO to the G3/G4 combination leads to improvements in all statistical indicators of accuracy and precision when compared to a G3/G4 combination. As the CBS-QB3, G3, and G4 methods all tend to overpredict the formation enthalpies of the ATcT (negative MSE), combining these methods offers no improvement on a G3/G4 combination. Also shown in Table 7 are results from a combination of all methods, both including and excluding W1BD computations. Adding the CBS-QB3 method to a CBS-APNO/G3/G4 combination offers nothing in terms of reducing absolute deviations and uncertainties. In turn, incorporating our W1BD results into the average reduces the MUE, RMSD, and standard deviation, but only marginally, and perhaps not significantly enough to warrant the added expense. Surprisingly, the W1BD method shows poorer performance than many so-called “cheaper” methods against this test-set of formation enthalpies. On the basis of the present results, the following methods, or combination of methods, could be argued to offer more accurate and precise results than W1BD at significant cost savings: G3, G4, G3/G4, CBS-APNO/G3, CBS-APNO/G4, CBS-APNO/G3/G4, CBS-QB3/CBSAPNO/G4, CBS-QB3/CBS-APNO/G4, CBS-QB3/CBSAPNO/G3/G4. The added advantages of using multiple methods which were alluded to earlier would also apply. The question also arises as to whether the accuracy of any given method deteriorates with the size of the system under study. Plots of MSE versus molecular weight (MW) (Figure 5) show that over the admittedly narrow range explored here, increasing MW leads to increasing MSEs for both CBS-APNO and W1BD. Conversely, a slight, perhaps negligible, decreasing MSE is observed for CBS-QB3 computations as f(MW), and almost no change for G3 and G4. Figure 5 also illustrates the general trend of increasing variation in ΔfH⊖ across methods as MW increases.

Table 5. Crude Timings (Minutes) of Calculations Steps for sGs Succinic Acid; 8 Heavy Atoms, with Total Computational Time no. 1 2 3 4 5 6 7 8 9

step Opt Freq Opt SPE SPE SPE SPE SPE SPE hours relative

QB3

APNO

G3

19 17 − 127 6 24 − − − Total 3.2 1.0

7 3 4 1 1579 9 1907 73 158 53 143 362 317 193 − − − − Simulation Times 68.6 11.6 21.3 3.6

G4

W1BD

81 45 − 79 53 359 277 355 3480

142 86 − 435 14760 56340 23280 39240 −

78.8 24.5

2238 695.8

Rayne and Forrest have shown in a study of a number of compounds, not just of type CxHyOz but also containing nitrogen, sulfur, fluorine, and chlorine, with up to 4 “heavy” atoms that W1BD ΔfH⊖(298.15 K) are systematically lower than comparable G4 ones.53 Some of these results have been questioned,54,55 but for all the 18 species in common between our work and Rayne’s and Forrest’s, there are only very small differences in the reported formation enthalpies. The CBS-APNO and CBS-QB3 methods tend to perform worst of all, and while on average they tend to predict the ΔfH⊖ to within ≈5 kJ mol−1, in some instances these methods deviate absolutely from the ATcT by >10 kJ mol−1, with respective |Φmax| of 18.61 and 14.10 kJ mol−1 computed. |Φmax| in the cases of G3, G4, and W1BD are computed as 10.60, 8.70, and 8.82 kJ mol−1, respectively. These trends are visible in Figure 4, where statistical measures of variation (2σ about the MSE, and twice the RMSD) show the following trends in terms of precision: G3 ≈ G4 > W1BD > CBS-APNO > CBS-QB3. The CBS-x methods are therefore more prone to producing an extremely erroneous result, a certain warning flag should one attempt to use either of these methods in isolation for thermochemical work. Thus, Figures 3 and 4 also delineate the performance of combinations of two or more different compound methods in terms of accuracy and precision. For combinations of two compound methods, six combination are possible using the CBS-QB3, CBS-APNO, G3, and G4 methods. An average formation enthalpy from any combination of the CBS-APNO, G3, and G4 methods shows the lowest MSE and MUE, and RMSD and 2σ deviations, of all individual and combined methods. Thus, combining any two of these methods may offer an improvement over their individual implementation. Likewise, the CBS-QB3 and CBS-APNO methods are shown to perform quite poorly in isolation, but a combination of these methods outperforms either method in isolation in terms of accuracy and precision. In terms of computational time, Table 5, a combination of CBS-QB3 and the CBS-APNO methods is of a similar expense to a combination of the G3 and G4 methods, so a combination of the latter two methods offers increased accuracy and precision for roughly the same cost. Combining CBS-APNO with G4 does lead to improvements from the CBS-APNO method in isolation, but combining CBS-APNO with G3, or G3 with G4, would provide a more cost-effective approach. Excluding W1BD, CBS-APNO was the only method which tended to under-predict the formation enthalpy of the target

■

CONCLUSION AND RECOMMENDATIONS In identifying the combination CBS-APNO/G3, CBS-APNO/ G4, and/or G3/G4 as the best combinations of methods for computing atomization energies, we are conscious of the fact that the different approaches used by these two methods are poles apart. Thus, for example, G3 utilizes HF/6-31G(d) for geometry optimization and frequency determination, whereas G4 employs B3LYP/6-31G(2df,p) for that purpose. Subsequent single-point energy calculations are also considerably different as is the fact that G3 reoptimizes the structure at MP2(Full)/6-31G(d) before embarking on the single-point corrections. Thus, the results from this combination are to some extent insulated from errors arising from specific contributions, and very good data can be obtained. The fact that the model chemistry CBS-APNO can only be applied to first row atoms does militate against its more widespread application, although a CBS-APNO/G3/G4 combination offers formation enthalpies which rival chemical accuracy (0.14 ± 4.21 kJ mol−1). These uncertainties are expected to be lower should one employ this trio of methods in combination with well-framed isodesmic reactions. For a discussion of the statistical advantages and the relevant equations, applied in that case to a series of isodesmic reactions 7242


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Table 6. Calculated Differences (Φi, kJ mol−1) for the Five Compound Methods Tested with ATcT Recommendations and Uncertaintiesa

a

molecule

QB3

APNO

G3

G4

W1BD

ATcT

±

1,3-butadiyne 1,3-cyclopentadienea 1-butene 1-butyne 2,2,4-trimethylpentane 2,3-butanedione 2-butyne 2-propanol acetic acid acetone allene benzene carbon monoxide cis-2-butene cyclobutene cyclohexane cyclopropane cyclopropene dimethyl ether dioxirane ethanal ethane ethene ethylene glycol ethyne ethynol formaldehyde formic acid glyoxal isobutane isobutene ketene singlet methane methanediol (sc,sc) methanol methyl formate (cis Z) methyl hydroperoxide n-butane n-hexane o-benzyne oxalic acida oxirane (1A1) oxirene (singlet) propane propene propyne succinic acid toluene trans-2-butene vinyl alcohol

−10.99 −44.60 −8.33 −10.41 −12.96 2.69 −9.54 −0.71 0.49 −1.09 −5.55 −8.42 0.88 −9.43 −14.10 −10.48 −6.89 −7.19 4.18 7.20 0.94 −2.19 −3.64 4.08 −6.47 −1.80 4.03 4.16 5.72 −7.46 −8.70 −1.08 −0.53 5.49 2.69 7.80 6.57 −7.15 −11.34 −9.70 21.10 2.28 8.10 −3.91 −5.65 −6.70 11.87 −10.19 −7.98 −1.86

−4.09 −31.30 4.77 0.19 18.61 7.59 0.76 7.49 0.49 8.21 0.35 6.98 −0.93 3.87 −2.90 12.62 4.11 −0.39 6.48 2.00 2.94 6.31 0.76 5.28 −4.87 −2.90 0.93 −0.34 1.22 9.14 4.40 0.82 3.97 1.99 2.99 5.40 1.37 9.85 13.16 3.30 12.90 3.08 3.40 8.69 3.25 0.70 9.75 9.31 5.12 0.14

−3.79 −39.00 −2.23 −3.71 −1.80 0.89 −3.84 −0.51 −3.61 −0.79 0.15 −5.32 0.07 −4.13 −10.60 −2.58 −4.09 −4.09 0.28 −2.00 −0.46 0.41 −0.14 −2.42 −2.37 −1.60 0.93 −1.64 1.12 −2.36 −3.00 0.62 0.87 −1.41 −0.41 2.30 −3.33 −1.55 −3.44 −4.10 10.90 −1.72 2.40 −0.01 −1.05 −1.60 3.25 −5.09 −2.48 −2.16

−1.39 −36.70 −3.03 −3.61 −4.04 −1.01 −4.24 −2.71 −5.41 −1.89 −0.35 −3.62 2.88 −4.53 −7.80 −4.18 −2.19 −1.79 −0.42 0.80 −0.56 −1.29 −0.44 −4.52 −1.07 −2.10 1.83 −1.64 2.02 −4.16 −3.00 −1.08 −0.33 −1.71 −0.61 1.10 −2.13 −3.95 −5.84 0.00 6.80 −0.02 8.70 −2.01 −1.35 −0.90 −0.95 −3.39 −2.88 −2.56

−0.59 −27.40 4.77 1.79 8.58 5.59 1.26 6.69 1.19 4.61 2.45 6.08 −0.63 3.97 0.50 8.82 3.81 2.81 4.38 −1.10 2.84 4.41 2.26 4.78 −0.27 1.70 1.13 1.86 2.12 5.54 4.40 0.82 1.67 5.39 4.19 4.30 1.67 6.55 8.76 3.30 15.80 2.78 8.30 6.29 4.25 2.00 8.25 6.41 5.12 3.74

460.11 101.30 −0.03 165.39 −223.70 −326.81 145.76 −272.81 −433.71 −216.09 190.15 83.18 −110.53 −7.33 156.90 −122.08 53.61 283.91 −184.02 1.30 −165.46 −83.79 52.56 −389.42 228.33 92.70 −109.17 −378.94 −212.48 −135.36 −17.60 −48.58 −74.53 −392.61 −200.71 −357.80 −127.73 −125.85 −166.94 460.70 −721.40 −52.72 275.90 −104.41 20.35 185.80 −817.75 50.41 −11.18 −123.76

0.87 2.50 0.48 0.85 1.50 0.98 0.79 0.37 0.49 0.37 0.37 0.26 0.03 0.53 1.60 0.68 0.53 0.59 0.44 1.20 0.32 0.17 0.15 0.49 0.15 1.40 0.11 0.27 0.59 0.40 0.53 0.15 0.06 0.96 0.18 0.59 0.91 0.38 0.48 1.40 2.10 0.44 3.10 0.29 0.33 0.38 0.61 0.37 0.51 0.91

Species are omitted from statistical analyses.

A model chemistry which combines the best functional and basis set for optimization and frequency calculations and not too expensive higher-order corrections is badly needed for usage by inexpert computational chemists. It should not have two geometry optimization routines, as in G3 or CBS-APNO for example, because occasional failures can occur in the second optimization which is perforce starting from the previous

but capable of much wider interpretation, see earlier work on alkyl hydroperoxides by Simmie et al.56 Usually transferable uncertainties are used by theoreticians developing new methodologies; that is, they benchmark their method against an existing database and then transfer that uncertainty to new species. But there is no evidence that such an approach has any grounding in reality. 7243


Article

The Journal of Physical Chemistry A Table 7. Summary Statistics (kJ mol−1) Based on Results shown in Table 6 model chemistry G3 G4 W1BD APNO QB3 APNO/G4 APNO/G3 G3/G4 QB3/APNO QB3/G3 QB3/G4 APNO/G3/G4 QB3/APNO/G4 QB3/APNO/G3 QB3/G3/G4 QB3/APNO/G3/G4 all methods

MUE

MSE

−1.71 −1.74 3.74 3.86 −2.78 2 Methods 2.05 1.06 2.24 1.08 2.27 −1.72 3.06 0.54 3.85 −2.24 3.98 −2.26 3 Methods 1.58 0.14 2.37 −0.22 2.45 −0.21 3.23 −2.07 4+ Methods 2.17 −0.59 1.80 0.28 2.27 2.46 3.85 4.55 6.08

|Φmax|

±2σ

2RMSD

10.60 8.70 8.82 18.61 14.10

4.76 5.15 5.13 9.21 13.02

5.86 6.21 8.82 12.02 14.16

7.29 8.41 9.20 10.81 12.35 10.95

4.71 5.51 4.56 7.50 8.33 8.33

5.17 5.91 5.72 7.58 9.46 9.47

7.10 8.27 9.20 10.83

4.21 5.87 6.21 6.83

4.22 5.89 6.22 7.99

8.85 6.98

5.40 4.75

5.53 4.79

Figure 4. Measures of dispersion (±2σ from MSE, 2 times root-meansquared deviation) for individual compound methods, and combinations thereof, upon comparison with the ATcT.

Figure 5. Signed error in standard state formation enthalpy versus molecular mass for compounds studied. Lines are fitted via leastsquares regression. Consult online version of manuscript for interpretation of color in figure.

Figure 3. Mean signed errors (MSE), mean unsigned errors (MUE), and maximum absolute deviations (|Φmax|) for individual compound methods, and combinations thereof, upon comparison with the ATcT.

geometry. This is particularly the case in the location of transition states where energy minimization failures are rather more common. Some progress in this direction has been made by da Silva who has combined the M06-2X functional within a G3-type framework to create G3X-K theory.57 Three out of the five model chemistries surveyed here employ the B3LYP functional, which although not as popular now as it was at the time of a 2007 review by Sousa et al.,58 is still very common. A range of better functionals are available, see for example recent comprehensive reviews,59,60 but have not yet come into widespread use. The treatment of vibrational and rotational partition functions is crucial because from an end-user perspective thermochemistry at 0 K is of a much lesser value than the same data at 298.15 K. The harmonic oscillator approximation is

commonly used to determine vibrational frequencies, but this necessitates scale factors to correct the resultant zero-point energies and vibrational partition functions. Unfortunately, separate factors seem to be required to scale the frequencies (“low” frequencies being treated differently than “high” frequencies)61 to compute zero-point energies, entropies and enthalpies, and worse still, these latter are functions of temperature.62 An effort is needed here to simplify without losing essential details perhaps by way of universal scale factors.63 The incorporation of anharmonic effects has not been seriously attempted for moderately sized molecules for reasons discussed by Irikura.64 In order to reach the wider computational chemistry community, the composite method should be embedded or 7244


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(9) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A 5th-Order Perturbation Comparison of Electron Correlation Theories. Chem. Phys. Lett. 1989, 157 (6), 479−483. (10) Martin, J. M. L.; de Oliveira, G. Towards Standard Methods for Benchmark Quality Ab Initio Thermochemistry: W1 and W2 Theory. J. Chem. Phys. 1999, 111 (5), 1843−1856. (11) Boese, A. D.; Oren, M.; Atasoylu, O.; Martin, J. M. L.; Kallay, M.; Gauss, J. W3 Theory: Robust Computational Thermochemistry in the kJ/mol Accuracy Range. J. Chem. Phys. 2004, 120 (9), 4129−4141. (12) Karton, A.; Rabinovich, E.; Martin, J. M. L.; Ruscic, B. W4 Theory for Computational Thermochemistry: In Pursuit of Confident Sub-kJ/mol Predictions. J. Chem. Phys. 2006, 125 (14), 144108. (13) Tajti, A.; Szalay, P. G.; Császár, A. G.; Kallay, M.; Gauss, J.; Valeev, E. F.; Flowers, B. A.; Vazquez, J.; Stanton, J. F. HEAT: High Accuracy Extrapolated Ab Initio Thermochemistry. J. Chem. Phys. 2004, 121 (23), 11599−11613. (14) Bomble, Y. J.; Vazquez, J.; Kallay, M.; Michauk, C.; Szalay, P. G.; Császár, A. G.; Gauss, J.; Stanton, J. F. High-Accuracy Extrapolated Ab Initio Thermochemistry. II. Minor Improvements to the Protocol and a Vital Simplification. J. Chem. Phys. 2006, 125 (6), 064108. (15) Harding, M. E.; Vazquez, J.; Ruscic, B.; Wilson, A. K.; Gauss, J.; Stanton, J. F. High-Accuracy Extrapolated Ab Initio Thermochemistry. III. Additional Improvements and Overview. J. Chem. Phys. 2008, 128 (11), 114111. (16) Császár, A. G.; Allen, W. D.; Schaefer, H. F. In Pursuit of the Ab Initio Limit for Conformational Energy Prototypes. J. Chem. Phys. 1998, 108 (23), 9751−9764. (17) Feller, D.; Peterson, K. A.; Dixon, D. A. Further Benchmarks of a Composite, Convergent, Statistically Calibrated Coupled-ClusterBased Approach for Thermochemical and Spectroscopic Studies. Mol. Phys. 2012, 110 (19−20), 2381−2399. (18) Montgomery, J. A.; Frisch, M. J.; Ochterski, J. W.; Petersson, G. A. A Complete Basis Set Model Chemistry. VII. Use of the Minimum Population Localization Method. J. Chem. Phys. 2000, 112 (15), 6532−6542. (19) Ochterski, J. W.; Petersson, G. A.; Montgomery, J. A. A Complete Basis Set Model Chemistry. V. Extensions to Six or More Heavy Atoms. J. Chem. Phys. 1996, 104 (7), 2598−2619. (20) Pople, J. A.; Head-Gordon, M.; Fox, D. J.; Raghavachari, K.; Curtiss, L. A. Gaussian-1 Theory: A General Procedure for Prediction of Molecular Energies. J. Chem. Phys. 1989, 90, 5622. (21) Curtiss, L. A.; Jones, C.; Trucks, G. W.; Raghavachari, K.; Pople, J. A. Gaussian-1 Theory of Molecular-Energies for 2nd-Row Compounds. J. Chem. Phys. 1990, 93 (4), 2537−2545. (22) Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. Gaussian-2 Theory for Molecular-Energies of 1st-Row and 2nd-Row Compounds. J. Chem. Phys. 1991, 94 (11), 7221−7230. (23) Curtiss, L. A.; Raghavachari, K.; Redfern, P. C.; Rassolov, V.; Pople, J. A. Gaussian-3 (G3) Theory for Molecules Containing First and Second-Row Atoms. J. Chem. Phys. 1998, 109 (18), 7764−7776. (24) Curtiss, L. A.; Redfern, P. C.; Raghavachari, K. Gaussian-4 Theory. J. Chem. Phys. 2007, 126, 084108. (25) Baboul, A. G.; Curtiss, L. A.; Redfern, P. C.; Raghavachari, K. Gaussian-3 Theory Using Density Functional Geometries and ZeroPoint Energies. J. Chem. Phys. 1999, 110 (7), 7650−7657. (26) Curtiss, L. A.; Redfern, P. C.; Raghavachari, K.; Rassolov, V.; Pople, J. A. Gaussian-3 Theory Using Reduced Möller-Plesset Order. J. Chem. Phys. 1999, 110 (10), 4703−4709. (27) Chan, B.; Coote, M. L.; Radom, L. G4-SP, G4(MP2)-SP, G4SC, and G4(MP2)-SC: Modifications to G4 and G4(MP2) for the Treatment of Medium-Sized Radicals. J. Chem. Theory Comput. 2010, 6 (9), 2647−2653. (28) Chan, B.; Deng, J.; Radom, L. G4(MP2)-6x: A Cost-Effective Improvement to G4(MP2). J. Chem. Theory Comput. 2011, 7 (1), 112−120. (29) Chan, B.; Radom, L. W1x-1 and W1x-2: W1-Quality Accuracy with an Order of Magnitude Reduction in Computational Cost. J. Chem. Theory Comput. 2012, 8 (11), 4259−4269.

easily interfaced with all popular computational codes including those which carry a licensing fee. Although the ATcT probably represents the current state of the art for thermochemical tabulations, and the results presented herein reinforce the large majority of recommendations in this database, there remains a certain lack of transparency in how some formation enthalpies reported therein have been derived, 1,3-cyclopentadiene, oxalic and succinic acids, and cyclobutene being exemplary. A minor but important improvement to benefit auditability of the compendium would be direct referencing of any relevant experiment or theory which were primarily used to derive a given value, as done in the Third Millennium Database.3

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ASSOCIATED CONTENT

S Supporting Information *

0 K Zero-point energy-corrected electronic energies of all compounds studied and 298.15 K zero-point and thermal energy-corrected electronic energies of all compounds studied. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Computational resources were provided by the Irish Centre for High-End Computing, ICHEC. We are grateful to Alin Elena (ICHEC), C. Franklin Goldsmith (Brown), Amir Karton (Western Australia), Wim Klopper (Karlsruhe), Errol Lewars (Trent), Jan Martin (Weizmann Institute), and Andrew Yeung (Texas A & M) for their considerable assistance. We thank the reviewers for their useful comments.

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REFERENCES

(1) NIST-JANAF Thermochemical Tables, 4th ed.; Chase, M. W. Jr., Ed.; J. Phys. Chem. Ref. Data Monogr. 9; American Institute of Physics and American Chemical Society: New York, 1998. (2) Cox, J. D.; Wagman, D. D.; Medvedev, V. A. CODATA Key Values for Thermochemistry; Hemisphere, New York, 1989. (3) Goos, E.; Burcat, A.; Ruscic, B. Ideal Gas Thermochemical Database with Updates from Active Thermochemical Tables. http:// garfield.chem.elte.hu/Burcat/burcat.html (accessed January 20, 2015). (4) Ruscic, B.; Pinzon, R. E.; Morton, M. L.; von Laszevski, G.; Bittner, S. J.; Nijsure, S. G.; Amin, K. A.; Minkoff, M.; Wagner, A. F. Introduction to Active Thermochemical Tables: Several “Key” Enthalpies of Formation Revisited. J. Phys. Chem. A 2004, 108 (45), 9979−9997. (5) Ruscic, B.; Pinzon, R. E.; von Laszewski, G.; Kodeboyina, D.; Burcat, A.; Leahy, D.; Montoya, D.; Wagner, A. F. Active Thermochemical Tables: Thermochemistry for the 21st Century. J. Phys. Conf. Ser. 2005, 16, 561−570. (6) Ruscic, B. Active Thermochemical Tables. ATcT.anl.gov (updated Active Thermochemical Tables (ATcT) values based on v 1.110 of the Thermochemical Network (2012); last update Nov 30, 2013.) (7) Simmie, J. M. Detailed Chemical Kinetic Models for the Combustion of Hydrocarbon Fuels. Prog. Energy Combust. Sci. 2003, 29 (6), 599−634. (8) Wilke, J. J.; Allen, W. D.; Schaefer, H. F. Establishment of the C2H5+O2 Reaction Mechanism: A Combustion Archetype. J. Chem. Phys. 2008, 128 (7), 074308. 7245


Article

The Journal of Physical Chemistry A (30) Karton, A.; Martin, J. M. L. Explicitly Correlated Wn Theory: W1-F12 and W2-F12. J. Chem. Phys. 2012, 136 (12), 124114. (31) Chan, B.; Radom, L. W3x: A Cost-Effective Post-CCSD(T) Composite Procedure. J. Chem. Theory Comput. 2013, 9 (11), 4769− 4778. (32) Karton, A.; Gruzman, D.; Martin, J. M. L. Benchmark Thermochemistry of the CnH2n+2 Alkane Isomers (N = 2−8) and Performance of DFT and Composite Ab Initio Methods for Dispersion-Driven Isomeric Equilibria. J. Phys. Chem. A 2009, 113 (29), 8434−8447. (33) Wheeler, S. E.; Houk, K. N.; Schleyer, P. V. R.; Allen, W. D. A Hierarchy of Homodesmotic Reactions for Thermochemistry. J. Am. Chem. Soc. 2009, 131 (7), 2547−2560. (34) Würmel, J.; Simmie, J. M. Thermochemistry and Kinetics of Angelica and Cognate Lactones. J. Phys. Chem. A 2014, 118, 4172− 4183. (35) Ruscic, B. Uncertainty Quantification in Thermochemistry, Benchmarking Electronic Structure Computations, and Active Thermochemical Tables. Int. J. Quantum Chem. 2014, 114 (17), 1097−1101. (36) Simmie, J. M.; Somers, K. P.; Metcalfe, W. K.; Curran, H. J. Substituent Effects in the Thermochemistry of Furans: A Theoretical (CBS-QB3, CBS-APNO and G3) Study. J. Chem. Thermodyn. 2013, 58, 117−128. (37) Somers, K. P. On the Pyrolysis and Combustion of Furans: Quantum Chemical, Statistical Rate Theory and Chemical Kinetic Modelling Studies. Ph.D. Thesis, National University of Ireland, Galway, 2014. (38) Barnes, E. C.; Petersson, G. A.; Montgomery, J. A.; Frisch, M. J.; Martin, J. M. L. Unrestricted Coupled Cluster and Brueckner Doubles Variations of W1 Theory. J. Chem. Theory Comput. 2009, 5 (10), 2687−2693. (39) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; et al. Gaussian 09, revision D.01; Gaussian, Inc.: Wallingford, CT, 2009. (40) Morini, F.; Knippenberg, S.; Deleuze, M. S.; Hajgato, B. Quantum Chemical Study of Conformational Fingerprints in the Photoelectron Spectra and (e, 2e) Electron Momentum Distributions of n-Hexane. J. Phys. Chem. A 2010, 114 (12), 4400−4417. (41) Spartan ‘08, v 1.0.0; Wavefunction Inc., Irvine, CA, 2008. (42) Vacek, G.; Galbraith, J. M.; Yamaguchi, Y.; Schaefer, H. F.; Nobes, R. H.; Scott, A. P.; Radom, L. Oxirene: To Be or Not to Be. J. Phys. Chem. 1994, 98 (35), 8660−8665. (43) Lewars, E. Modeling Marvels, Chapter 3; Springer: Amsterdam, 2008. (44) Vogiatzis, K. D.; Haunschild, R.; Klopper, W. Accurate Atomization Energies from Combining Coupled-Cluster Computations with Interference-Corrected Explicitly Correlated Second-Order Perturbation Theory. Theor. Chem. Acc. 2014, 133 (3), 1−12. (45) Karton, A.; Daon, S.; Martin, J. M. L. W4−11: A HighConfidence Benchmark Dataset for Computational Thermochemistry Derived from First-Principles W4 Data. Chem. Phys. Lett. 2011, 510 (4−6), 165−178. (46) Pedley, J. B.; Naylor, R. D.; Kirby, S. P. Thermochemical Data of Organic Compounds, 2nd ed.; Chapman & Hall: London, 1986. (47) Roth, W. R.; Adamczak, O.; Breuckmann, R.; Lennartz, H. W.; Boese, R. Resonance Energy Calculation: The MM2ERW Force-Field. Chem. Ber. 1991, 124 (11), 2499−2521. (48) Furuyama, S.; Golden, D. M.; Benson, S. W. Thermochemistry of Cyclopentene and Cyclopentadiene from Studies of Gas-Phase Equilibria. J. Chem. Thermodyn. 1970, 2 (2), 161−169. (49) Zhong, X.; Bozzelli, J. W. Thermochemical and Kinetic Analysis of the H, OH, HO2, O, and O2 Association Reaction with Cyclopentadienyl Radical. J. Phys. Chem. A 1998, 102, 3537−3555. (50) Agapito, F.; Nunes, P. A.; Cabral, B. J. C.; dos Santos, R. M. B.; Simoes, J. A. M. Energetic Differences between the Five- and SixMembered Ring Hydrocarbons: Strain Energies in the Parent and Radical Molecules. J. Org. Chem. 2008, 73 (16), 6213−6223.

(51) Vogt, N.; Abaev, M. A.; Rykov, A. N.; Shishkov, I. F. Determination of Molecular Structure of Succinic Acid in a Very Complex Conformational Landscape: Gas-Phase Electron Diffraction (GED) and Ab Initio Studies. J. Mol. Struct. 2011, 996 (1−3), 120− 127. (52) Parthiban, S.; de Oliveira, G.; Martin, J. M. L. Benchmark Ab Initio Energy Profiles for the Gas-Phase SN2 Reactions Y− + CH3X → CH3Y + X− (X,Y = F,Cl,Br). Validation of Hybrid DFT Methods. J. Phys. Chem. A 2001, 105 (5), 895−904. (53) Rayne, S.; Forest, K. Estimated Gas-Phase Standard State Enthalpies of Formation for Organic Compounds Using the Gaussian4 (G4) and W1BD Theoretical Methods. J. Chem. Eng. Data 2010, 55 (11), 5359−5364. (54) Dorofeeva, O. V. Comments on “Estimated Gas-Phase Standard State Enthalpies of Formation for Organic Compounds Using the Gaussian-4 (G4) and W1BD Theoretical Methods” (Rayne, S.; Forest, K. J. Chem. Eng. Data 2010, 55, 5359−5364). J. Chem. Eng. Data 2011, 56 (3), 682−683. (55) Rayne, S.; Forest, K. Reply to Comments by O. V. Dorofeeva on J. Chem. Eng. Data 2010, 55, 5359−5364. J. Chem. Eng. Data 2011, 56 (3), 684−685. (56) Simmie, J. M.; Black, G.; Curran, H. J.; Hinde, J. P. Enthalpies of Formation and Bond Dissociation Energies of Lower Alkyl Hydroperoxides and Related Hydroperoxy and Alkoxy Radicals. J. Phys. Chem. A 2008, 112 (22), 5010−5016. (57) da Silva, G. G3X-K Theory: A Composite Theoretical Method for Thermochemical Kinetics. Chem. Phys. Lett. 2013, 558, 109−113. (58) Sousa, S. F.; Fernandes, P. A.; Ramos, M. J. General Performance of Density Functionals. J. Phys. Chem. A 2007, 111 (42), 10439−10452. (59) Peverati, R.; Truhlar, D. G. Quest for a Universal Density Functional: The Accuracy of Density Functionals across a Broad Spectrum of Databases in Chemistry and Physics. Philos. Trans. R. Soc., A 2014, 372, 20120476. (60) Zhang, I. Y.; Xu, X. Doubly Hybrid Density Functional for Accurate Description of Thermochemistry, Thermochemical Kinetics and Nonbonded Interactions. Int. Rev. Phys. Chem. 2011, 30 (1), 115− 160. (61) Laury, M. L.; Boesch, S. E.; Haken, I.; Sinha, P.; Wheeler, R. A.; Wilson, A. K. Harmonic Vibrational Frequencies: Scale Factors for Pure, Hybrid, Hybrid Meta, and Double-Hybrid Functionals in Conjunction with Correlation Consistent Basis Sets. J. Comput. Chem. 2011, 32 (11), 2339−2347. (62) Merrick, J. P.; Moran, D.; Radom, L. An Evaluation of Harmonic Vibrational Frequency Scale Factors. J. Phys. Chem. A 2007, 111 (45), 11683−11700. (63) Alecu, I. M.; Zheng, J. J.; Zhao, Y.; Truhlar, D. G. Computational Thermochemistry: Scale Factor Databases and Scale Factors for Vibrational Frequencies Obtained from Electronic Model Chemistries. J. Chem. Theory Comput. 2010, 6 (9), 2872−2887. (64) Irikura, K. K. Anharmonic Partition Functions for Polyatomic Thermochemistry. J. Chem. Thermodyn. 2014, 73, 183−189.

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