Enthalpy of Formation of C2H2O4 (Oxalic Acid) from High-Level

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Enthalpy of Formation of CHO (Oxalic Acid) from High-Level Calculations and the Active Thermochemical Tables Approach David F. Feller, David H. Bross, and Branko Ruscic J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b12329 • Publication Date (Web): 22 Mar 2019 Downloaded from http://pubs.acs.org on March 22, 2019

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The Journal of Physical Chemistry

Enthalpy of Formation of C2H2O4 (Oxalic Acid) from High-Level Calculations and the Active Thermochemical Tables Approach David Feller,*,† David H. Bross,*‡ and Branko Ruscic*,‡↨ †Department

‡Chemical

of Chemistry, Washington State University, Pullman, WA 99164-4630

Sciences and Engineering Division, Argonne National Laboratory, Argonne, IL 60439

↨Consortium

for Advanced Science and Engineering, The University of Chicago, Chicago, IL 60637

The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.

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Abstract High-level coupled cluster calculations obtained with the Feller-Peterson-Dixon (FPD) approach and new data from the most recent version of the Active Thermochemical Tables (ATcT) are used to reassess the enthalpy of formation of gas-phase C2H2O4 (oxalic acid). The theoretical value was further calibrated by comparing FPD and ATcT gas-phase enthalpies of formation for H2CO (formaldehyde) and the two low-lying conformations of C2H4O2 (syn and anti acetic acid). The FPD approach produces a theoretical enthalpy of formation of gas-phase oxalic acid of -732.2 ± 4.0 kJ/mol at 298.15 K (-721.8 ± 4.0 kJ/mol at 0 K). An independently obtained ATcT value, based on reassessing the existent experimental determinations and expanding the resulting thermochemical network with select mid-level composite theoretical results, disagrees with several earlier recommendations that were based solely on experimental determinations, but is in excellent accord with the current FPD value. The inclusion of the latter in the most recent ATcT thermochemical network produces a further refined value for the gas phase enthalpy of formation, -731.6 ± 1.2 kJ/mol at 298.15 K (-721.0 ± 1.2 kJ/mol at 0 K). The condensed phase ATcT enthalpy of formation of oxalic acid is -829.7 ± 0.5 kJ/mol, and the resulting sublimation enthalpy is 98.1 ± 1.3 kJ/mol, both at 298.15 K. I. Introduction We recently re-examined the enthalpy of formation of N2H4 (hydrazine)1 in light of questions about the values provided by the Active Thermochemical Tables (ATcT)2,3 versions 1.110 through 1.118, 4,5,6 with the latter version reporting values of fH°(0 K) = 109.71 ± 0.19 kJ/mol and fH°(298.15 K) = 95.55 ± 0.19 kJ/mol, based primarily on existing experimental measurements.

Articles by Karton et al.,7 Simmie8 and Dorofeeva et al.9 recommended

theoretical values that were larger than the ATcT values by 2-6 kJ/mol: fH°(0 K) = 111.77 kJ/mol (Karton et al.), fH°(0 K) = 115.4 ± 2.0 kJ/mol (Simmie) and fH°(298.15 K) = 97.0 ± 3.0 kJ/mol (Dorofeeva et al.). Karton et al.7 based their estimate on the W410 composite theoretical method, the Simmie8 value relied on an average of four mid-level theoretical methods (CBS-QB3,11,12 CBS-APNO,13 G314 and G415) and Dorofeeva et al.9 used 75 isogyric reactions at the G4 level of theory involving 50 reference species. The descriptor “mid-level” used here with regard to the levels of theory used by Simmie and by Dorofeeva et al. implies that their accuracy in thermochemical calculations lies somewhere between uncorrected density functional theory (DFT) methods and much more time-consuming high-level approaches based on coupled cluster theory or configuration interaction calculations. 2

For hydrazine, our analysis of the salient


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experimental data used by ATcT indicated that an overly optimistic interpretation of the vaporization enthalpy - adopted directly from the JANAF Tables16 - was the likely source of the anomalous enthalpy of formation, causing ATcT to rely exclusively on what initially appeared to be reliable experimental data.

After correcting for that effect and balancing all available

information, the revised ATcT enthalpy of formation, 111.57 ± 0.47 kJ/mol at 0 K (97.42 ± 0.47 kJ/mol at 298.15 K) was found to be in excellent agreement with the Feller-Peterson-Dixon (FPD)17-21 approach theoretical fH°(0 K) of 111.64 ± 1.73 kJ/mol (97.49 ± 1.73 kJ/mol at 298.15 K). The situation with C2H2O4 (oxalic acid) bears interesting similarities to that encountered previously with hydrazine. ATcT ver. 1.1186 (the main focus of which was to derive enthalpies for diatomics and a few triatomics22 as well as to provide the initial benchmarks used during the development of high-accuracy ANLn composite electronic structure methods that are primarily focused on combustion-related species23) and ATcT ver. 1.12224 (which was used to derive the sequential bond dissociation energies of methane, ethane, and methanol,25 and has provided thermochemical values during advanced stages of development and benchmarking of ANLn methods23) contain experimental thermochemical determinations involving oxalic acid. Its gas phase enthalpy of formation at 298.15 K, -721.4 ± 2.1 kJ/mol, reported in these two versions, relies on a small set of existing experimental data, essentially identical to that considered by Pedley et al.26 Not surprisingly, the value recommended in the latter compilation, -723.7 ± 4.9 kJ/mol, is rather similar to the ATcT value, differing by no more than 2.3 kJ/mol. In their derivation, Pedley et al.26 considered the sublimation enthalpy of oxalic acid of 23.40 ± 0.50 kcal/mol (97.9 ± 2.1 kJ/mol) of Bradley and Cleasby,27 combined with two measurements of combustion calorimetry of solid oxalic acid: a 1926 determination by Verkade et al.28 and a 1964 determination by Wilhoit and Shiao.29 The 298.15 K combustion enthalpies resulting from these two measurements are listed in Pedley as -60.59 ± 0.11 kcal or -253.5 ± 0.5 kJ/mol (from Verkade et al.,28 implying fH°(298.15 K) = -819.4 ± 0.5 kJ/mol for the solid) and -58.06 ± 0.22 kcal or -242.9 ± 0.9 kJ/mol (from Wilhoit and Shiao,29 implying fH°(298.15 K) = -829.9 ± 0.9 kJ/mol for the solid). Clearly, the two calorimetric measurements are not mutually consistent, given that their difference (> 10 kJ/mol) egregiously exceeds their assigned (sub-kJ/mol) uncertainties by more than order of magnitude. This implies that at least one of the two is incorrect (i.e. has an overly optimistic uncertainty). Therefore, they cannot be legitimately averaged. Nevertheless, Pedley et al.26 appears to have taken a weighted average of the two calorimetric results (increasing - albeit in retrospect insufficiently - the resulting uncertainty in 3



an attempt to bridge the difference), and recommended a 298.15 K enthalpy of formation for the condensed phase of -821.7 ± 4.4 kJ/mol, which translates to fH°(298.15 K) = -723.7 ± 4.9 kJ/mol for the gas-phase once the sublimation enthalpy is added. The actual values for the two calorimetric determinations and the sublimation enthalpy of oxalic acid that are listed in Pedley et al. were adopted directly from the predecessor compilation by Cox and Pilcher,30 who performed a critical examination and reanalysis of the three published determinations. Curiously, the 298.15 K enthalpy of formation of the condensed phase recommended by Cox and Pilcher, 198.36 ± 0.23 kcal/mol (-829.9 ± 1.0 kJ/mol), and the resulting value for the gas phase, fH° = 174.96 ± 0.59 kcal/mol (-732.0 ± 2.5 kJ/mol), are both lower (more negative) than the values recommended by Pedley et al.26 by more than 8 kJ/mol, though both compilations used exactly the same experimental data. The reason is that instead of averaging the two calorimetries, as was done by Pedley et al.,26 Cox and Pilcher30 tacitly discarded the older calorimetry and used the newer (though nominally less accurate) determination to derive their recommended values for the solid and the gas phases. ATcT Thermochemical Network (TN) ver. 1.1186 (as well as ver. 1.12224 and several subsequent versions) contains the reaction enthalpies (as reanalyzed by Cox and Pilcher30) of all three experimental determinations discussed above.

However, in contrast to the averaging

approach of Pedley et al.,26 ATcT attempts to resolve internal inconsistencies by dynamically reweighting the individual determinations via a statistical analysis that precedes the final solution of the TN and iteratively augments the initially assigned uncertainties of determinations that appear statistically less likely to be correct. Thus, if the TN contains two (or more) inconsistent competing determinations of the same quantity, the ATcT analysis attempts to arbitrate between them, based on information that is available from additional thermochemical cycles in the TN. These additional thermochemical cycles that enable successful arbitration are normally created as the TN is further expanded by adding chemical reactions involving the species in question (using tailored theoretical computations and/or new experimental measurements). Which one of the competing measurements will prevail during the arbitration process depends on how much the individual values of these determinations are reinforced by the cumulative statistical weight of the additional thermochemical cycles. However, in TN vers. 1.118 and 1.122, the data regarding oxalic acid was augmented only in a token manner, by inserting a single mid-level theoretical atomization energy of oxalic acid obtained at the G3B331 level of theory, and further expansion of the TN connections related to oxalic acid, based on more accurate atomization energies and/or isodesmic reactions, were deferred to future TN sub-versions. In the absence of 4


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additional thermochemical cycles that might enable an earnest arbitration between the two calorimetric determinations, the ATcT analysis of TN vers. 1.118 and 1.122 defaults to preferring the older (nominally more accurate) experimental calorimetry, ultimately producing a fH°(298.15 K) of -819.3 ± 0.5 kJ/mol for the solid phase and -721.4 ± 2.1 kJ/mol for the gas phase. The ATcT value discussed above (as well as the values in Cox and Pilcher30 and Pedley et al.26), appeared to differ quite significantly from the mid-level gas phase theoretical values In particular, Simmie and Somers32 recently

that subsequently appeared in the literature.

reported two oxalic acid enthalpies of formation that both fall well outside the error bars of the results obtainable directly from experiment. The first of Simmie and Somers’ values (-739.8 ± 1.2 kJ/mol) was based on the unweighted average of five composite theoretical methods applied to the following isodesmic reaction: C2H2O4 (oxalic acid) + C2H6 (ethane)  2(C2H4O2) (acetic acid) These authors demonstrated that for a collection of 50 enthalpies of formation, taking an average produced smaller errors than any single method taken alone due to compensating errors. The specific levels of theory were CBS-QB3,11,12 CBS-APNO,13 G3,14 G4,15 and W1BD.33 A second value (-735.5 ± 5.3 kJ/mol) was based on the atomization energy of oxalic acid averaged over the same five methods. Both enthalpies were measured with respect to the lowest energy conformation. Thus, there is a 14 – 18 kJ/mol difference between the Simmie and Somers32 enthalpies and the ATcT ver. 1.122 value.24 An even more negative theoretical value appeared in the work of Cioslowski et al.,34 who found fH°(298.15 K) = -820.5 kJ/mol (-196.1 kcal/mol) using the B3LYP/6-311++G** bond density functional scheme. Duan et al.35 applied B3LYP DFT with a 6-parameter empiricallybased linear regression correction to obtain a much higher (less negative) fH°(298.15 K) = 689.9 kJ/mol with the small 6-31G(d) basis set and a more modestly higher value of -709.6 kJ/mol with the 6-311+G(d,p) basis set. Values without the linear regression correction were not reported. Cohen36 reported a fH°(298.15 K) group additivity (GA) value of -730.5 kJ/mol, where the group values were based on a large number of experimental enthalpies of formation of C-HO compounds (mostly based on the compilation of Pedley et al.26). Although the general accuracy of GA values is rather limited, Cohen’s value36 for oxalic acid curiously appeared to be much closer to the value of Cox and Pilcher30 than that given by Pedley et al.,26 further 5



suggesting that the former recommendation may be better. Considering all of the theoretical and experimental values and the virtual absence of overlapping error bars, it was of interest to revisit the enthalpy of formation for oxalic acid from the perspective of ATcT and in light of new high level calculations based on the FPD approach. As discussed in our previous study,1 the Simmie8 enthalpy of formation for hydrazine differed by only ~6 kJ/mol from the ATcT ver. 1.122 value24 and just ~4 kJ/mol from the final ATcT and FPD results. Thus, the discrepancy between mid-level theory and the ATcT value is much larger for oxalic acid than it was for hydrazine. When compared to previous compilations of thermochemical data, such as the NISTJANAF Tables,16 the tables of Gurvich et al.,37 or the organic compound tables of Pedley et al.,26 the Active Thermochemical Tables (ATcT)2,3 are in general able to achieve much higher accuracy due to the fact that the values are based on a thermochemical network covering nearly 2000 chemical species connected via a set of constraints which have to be simultaneously satisfied, to the extent possible, by the resulting enthalpies of formation. The TN contains both experimental and theoretical determinations of thermochemical properties, such as bond dissociation energies, reaction enthalpies, ionization potentials and electron affinities.38,39 The TN is statistically analyzed, iteratively corrected for inconsistencies, and then simultaneously solved for the enthalpies of formation of all included chemical species.2,25,3,40-43 In qualitative terms, the FPD approach consists of a combined flexible, systematic sequence of coupled cluster (or configuration interaction in the case of strongly multiconfiguration systems) component calculations used for the prediction of thermochemical and spectroscopic properties. Because of the convergent nature of the FPD approach, crude estimates of the molecule-and-property specific uncertainty can be attached to the final predictions. FPD results stored in the Computational Results Database (CRDB)44 provide a calibrated, statistical measure of the performance of this level of theory when compared to reliable experimental data.

The current version of the CRDB contains theoretical and

experimental information on 580 chemical species. The root mean square (RMS) error across 237 enthalpies of formation was 1.84 kJ/mol (0.44 kcal/mol), and the corresponding mean absolute deviation (MAD) was 0.96 kJ/mol (0.23 kcal/mol). Some of the experimental values involved in the aforementioned RMS calculation possessed uncertainties exceeding 4 kJ/mol. In general, the uncertainty of a method, as determined by benchmarking, udetermined, roughly corresponds to a sum in quadrature of the intrinsic uncertainty of the method, uintrinsic, and the average uncertainty of the external benchmark values, ubenchmark:45 6


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udetermined = (u2intrinsic + u2benchmark)1/2 Clearly, unless ubenchmark is much smaller than uintrinsic, it will have a tendency to increase udetermined. Indeed, when the statistical analysis of the CRDB entries used in the statistical comparison is restricted to experimental values with uncertainties less than 0.8 kJ/mol, the RMS deviation drops to 0.54 kJ/mol (0.13 kcal/mol) across 104 comparisons. In light of these findings, we expect the FPD approach to be adequate for distinguishing among the various enthalpies of formation of oxalic acid. To gain further insight into the likely level of accuracy to be expected for oxalic acid, we have included FPD calibration studies of formaldehyde and acetic acid in the current work. II. Theoretical Approach Because the details of the FPD approach have been discussed previously,20,21 they will only be summarized here. This approach consists of a sequence of (up to) 13 steps. The initial step is geometry optimization with frozen core (FC) coupled cluster theory through single and double excitations, combined with a noniterative, quasiperturbative estimate of the effect of triple excitations, CCSD(T).46-49

Valence basis sets are chosen from the diffuse function

augmented correlation consistent basis set family of Dunning, Peterson and co-workers.50-70 These basis sets are commonly denoted aug-cc-pVnZ, n = D, T, Q,…10, although for the sake of brevity we will shorten the names to aVnZ. CCSD(T) calculations with up through  = 6 (i-functions) were performed with MOLPRO.71 Calculations involving k- and l-functions were performed with Gaussian 09,72 because the current version of MOLPRO does not support these angular momentum functions. In certain cases, such as with H2CO and the aV8Z basis and C2H4O2 with the aV7Z basis set, a single energy evaluation required several days. Consequently, geometry optimization proved to be prohibitively time consuming. In instances where geometry optimization wasn’t feasible, such as with oxalic acid calculations with the aV6Z basis set in Cs symmetry, geometries were estimated with an exponential extrapolation of the internal coordinates using optimized values from the prior three smaller basis sets. Calculations with large basis sets, such as aV7Z, can be plagued by near linear dependency problems. This often leads to convergence difficulties in the solution of the CCSD equations. For example, applying the aV7Z basis set to acetic acid produces a minimum overlap eigenvalue of ~4  10-8, indicative of near linear dependency. This undesirable situation can be partially ameliorated by increasing the accuracy of the 2-electron integrals to 64 bit machine 7



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tolerance and by performing a rectangular transformation to eliminate the eigenvectors with the very smallest overlap eigenvalues from the function space in cases where the software allows it. The current version of MOLPRO does not support such a transformation, but Gaussian 09 does. Undesirable basis set truncation errors were reduced through a combination of large basis sets and extrapolation of energies to the complete basis set (CBS) limit. Past studies have shown that no single extrapolation formula from the large number in the literature is superior for all ranges of basis sets and all classes of molecules.65,73,74 An expression involving the inverse power of max, the highest angular momentum present in the basis set, has been found to be a reasonable compromise for basis sets with max values between 5 and 8:74-78 E(max) = ECBS + A/(max + ½)4

,

(1)

The spread among four (five for aVQ56Z) different formulas will be taken as a conservative, albeit crude, indicator of the residual uncertainty in the extrapolation. Other components of the FPD approach represent smaller corrections to CCSD(T)(FC) and display varying sign. Thus, a certain amount of fortuitous cancellation of error can be expected. Moreover, these smaller corrections are assumed to be additive. This assumption has been partially tested and found to be adequate for all but the very highest accuracy studies.17 Core/valence (CV) effects due to the inclusion of the outermost core orbitals (1s2 in the case of carbon and oxygen) are recovered from CCSD(T)(CV) calculations with the weighted core/valence correlation consistent basis sets, cc-pwCVnZ, n = D, T, Q, 5.54 Scalar relativistic (SR) second order Douglas-Kroll-Hess (DKH) CCSD(T)(FC)79,80 calculations were performed with the aug-cc-pVQZ-DK correlation consistent basis sets specifically designed for the DKH level of theory.81 For first row elements the SR correction is insensitive to the level of theory and choice of basis set. As was the case with CCSD(T)(FC), CV and DKH calculations were performed with optimized geometries. For chemical systems with wave functions dominated by the Hartree-Fock configuration, which includes all systems included in this study, CCSD(T) provides an excellent baseline method. Higher order (HO) correlation recovery beyond CCSD(T) was accomplished with CCSDT, CCSDT(Q), CCSDTQ and CCSDTQ5 calculations performed with the MRCC program of Kállay and co-workers interfaced to MOLPRO.82 The impact of including iterative triple excitations is often opposite in sign to the effect of quadruple (or quintuple) excitations and of similar magnitude. Consequently, care must be taken to balance the uncertainty associated with these two effects. Whenever it was possible to perform CCSDT calculations with the VTZ and VQZ basis sets and simultaneously carry out CCSDTQ with VDZ and VTZ, CBS extrapolations 8


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were performed for both triples and quadruples corrections. In other cases, where quadruple excitations could only be treated with the small VDZ basis set, no extrapolation was attempted for the triples for fear of biasing the results. With our current software and computer hardware we were limited to fully iterative methods, e.g. CCSDT or CCSDTQ, involving no more than roughly 5  109 determinants. The same HO methods are used to describe the impact of higher order correlation on the core/valence contribution to the atomization energy. Such calculations can be extremely time consuming, even for systems as small as acetic acid where the number of quadruple excitation determinants with the wCVDZ basis set reached 1.1  1011. The HO CV correction for HartreeFock (HF) dominant systems of the size examined here are expected to fall into the range of 0.2 – 0.5 kJ/mol. For systems with more multiconfiguration character, such as C2 (1g+),20 this correction can easily exceed 1 kJ/mol. An estimate of the remaining higher order correlation energy beyond what was explicitly recovered with coupled cluster quadruple (or quintuple) excitations was obtained from the continued fraction (cf) full configuration interaction (FCI) approximant originally suggested by Goodson, who based his approach on HF, CCSD and CCSD(T) energies.83 Our benchmarks indicated that Goodson’s original formulation was insufficiently accurate for reliably improving upon CCSDTQ results.84 Fortunately, when the original 3-energy sequence was replaced by the CCSD/CCSDT/CCSDTQ or CCSDT/CCSDTQ/CCSDTQ5 energy sequences the results more consistently replicated FCI. Strictly speaking, according to the analysis of Schröder et al.,85 the cf CCSD/CCSDT/CCSDTQ approximant is limited to recovering the contribution of 5-fold excitations, i.e. one order higher than the highest considered excitation level. When the wave function is dominated by the HF configuration, as it is in the systems examined in the current study, the distinction between CCSDTQ5 and FCI is of little practical significance for thermochemical properties. FCI represents the exact solution of the stationary state, nonrelativistic Schrödinger equation, but in a straightforward application of the method it is too expensive for most systems due to the enormous number of slightly interacting configurations.

Fortunately, when the

continued fraction approximant correction is used in a composite approach, like FPD, a basis set of double  or augmented double  quality is normally sufficient to account for the majority of effect. Our current hardware and software limits us to a maximum of slightly more than 5  109 determinants in an FCI calculation, the same number of determinants as can be accommodated in an iterative coupled cluster calculation. Perhaps surprisingly, despite the expense of explicit FCI 9



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the CRDB includes almost 300 instances in which it was possible to include this correction for a thermochemical property. Techniques for partially circumventing the problem of the explosive growth in the number of determinants involved in FCI are an active area of research. We have investigated several alternatives, but have not found them to be reliable. At this point the various alternatives to the continued fraction approximant for estimating the FCI correction to CCSDTQ (or CCSDTQ5) do not appear to provide sufficient accuracy for studies already including CCSDTQ or CCSDTQ5. Open shell CCSD(T) calculations for the constituent atoms were carried out with the R/UCCSD(T) method, which begins with restricted open-shell Hartree-Fock (ROHF) orbitals, but allows a small amount of spin contamination in the solution of the CCSD equations.86,87 Orbital symmetry equivalence constraints were imposed on the isolated atoms.

Because

Gaussian 09 does not support the R/UCCSD(T) method it was necessary to estimate the R/U energies using the scheme described in an earlier work.65 That approach assumes that the difference between UCCSD(T) and R/UCCSD(T) energies for basis sets beyond 6 can be determined to an accuracy ≤ 10-6 Eh using results up through 6. Open shell CCSDT(Q) calculations were based on unrestricted Hartree-Fock orbitals. Harmonic zero point vibrational energies (ZPEs) were obtained at the CCSD(T)(FC) level of theory. Anharmonic corrections were determined at the MP2(FC) level of theory with Gaussian 09. Since the anharmonic correction to the ZPE is relatively insensitive to the quality of the basis set, we combined harmonic CCSD(T)/aVnZ with anharmonic MP2/aV(n-1)Z. Studies indicate that this approach should provide ZPEs with sufficient accuracy for our purpose.88 CV and HO corrections to stretching frequencies are often of opposite signs and of roughly comparable magnitudes.

Therefore, we avoid including the less expensive CV

correction when the HO correction can’t also be included due to its cost. Computing the second derivative with a finite difference algorithm, such as the one implemented in MOLPRO, requires a potentially large number of displacement geometries in C1 symmetry. In the case of H2CO it proved possible to include the effects of CV, HO, and SR corrections. For acetic acid and oxalic acid the CV and HO corrections to the harmonic ZPE were ignored. Atomic spin-orbit (SO) coupling effects decrease the energies of the isolated atoms relative to the spin multiplet average values obtained from standard CCSD(T). It is necessary to include the atomic SO contribution for accurate atomization energies. For this purpose, we have chosen to use the tabulated values of Moore.89 10


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A correction to account for the breakdown in the Born-Oppenheimer approximation, which can become significant for systems with hydrogen atoms, was made to the computed atomization energies. In the present work a first-order, adiabatic correction, known as the diagonal Born-Oppenheimer correction (DBOC), was obtained from CCSD(FC)/aVTZ calculations performed with the CFOUR program90 using CCSD(T)(FC)/aVTZ geometries. The open shell atomic energies used in computing the DBOC were based on unrestricted HartreeFock wave functions and UCCSD. FPD is a flexible, composite coupled cluster-based theoretical approach to the prediction of thermochemical and spectroscopic properties. The literature contains at least 187 named, fixed-recipe composite approaches. Although the details differ amongst approaches, many of them, including FPD, use some of the same basic components. Examples include Weizmann-n (Wn),10,

91-93

High-accuracy Extrapolated Ab Initio Thermochemistry (HEAT),94-96 Correlation

Consistent Composite Approach (ccCA),97-99 and the Focal-Point Analysis (FPA) scheme.100,101 Like FPD, the last of these does not involve a fixed recipe. III. Theoretical Results and Discussion H2CO Formaldehyde (1A1) A complete breakdown of the individual FPD components for H2CO is given in Table S1 in the Supporting Information. The best composite atomization energy is 1495.36 ± 0.80 kJ/mol (including zero point vibrational energy) and the corresponding f is -105.05 ± 0.80 (0 K) and -108.89 ± 0.80 kJ/mol (298.15 K). Due to the small size of formaldehyde molecule, it was possible to use basis sets up through aV8Z in the valence CCSD(T) piece. Despite the large size of the basis set, the CCSD(T)(FC)/CBS component of the electronic, vibrationless atomization energy (De) still contributes the largest uncertainty (± 0.44 kJ/mol) to the final result. While it might be argued that the admittedly crude uncertainty estimate is overly pessimistic, even if the current value were cut in half it would still be among the largest potential sources of error. The raw aV7Z and aV8Z CCSD(T)(FC) atomization energies are 1559.98 and 1560.64 kJ/mol. Thus, the observed aV7Z  aV8Z increase in the atomization energy (0.66 kJ/mol) is larger than the proposed uncertainty in the CCSD(T)(FC)/CBS limit and lends credence to an uncertainty on the order of ± 0.4 kJ/mol for the valence CCSD(T) limit. Significantly decreasing the overall uncertainty of ± 0.80 kJ/mol is very challenging with the software and hardware currently at our disposal. Among the numerous smaller corrections, ΔR/UCCSD(T)(CV) was found to be the 11



largest, as is often the case for molecules composed of hydrogen and first row elements. It increases the atomization energy, while the rest of the smaller corrections are of varying sign and therefore partially cancel each other. The final 0 K and 298.15 K FPD enthalpies of formation for formaldehyde are in good agreement with the recent ATcT values. The theoretical error bars, ± 0.80 kJ/mol, encompass the ATcT value25 and the value reported by Pedley et al.,26 and indicate that the often-quoted JANAF value16 is much too negative, as suggested earlier.25 In the FPD approach, the overall theoretical uncertainty is taken as the simple sum of the individual component uncertainties, i.e. it corresponds to a worst case scenario. This choice assumes that the errors for all components are inherently highly correlated. One factor that is not included in either the best composite f or the overall uncertainty is the impact of the additivity approximation.

In an earlier study we estimated the FC + CV + SR additivity

approximation correction to atomization energies for first row diatomics to fall in the range of 0.08 – 0.21 kJ/mol.20 We were unable to estimate the uncertainty in these corrections. More accurately accounting for the intrinsic error associated with the additivity approximation would require large basis set, higher order all-electron, scalar relativistic calculations that are currently beyond our ability to perform. The impact of including the expensive CV + HO + SR corrections on the harmonic zero point vibrational energy was small (0.12 kJ/mol) due to partial cancellation of the effects, as mentioned above. The CV corrections were uniformly positive across all modes (max = 5.6 cm-1), while the CCSDTQ corrections were uniformly negative (max = -6.6 cm-1). The final composite harmonic frequencies (in cm-1) for formaldehyde are: a1 = 2939.7, a1 = 1778.0, a1 = 1539.1, b1 = 3002.8, b1 = 1281.3, and b2 = 1191.5, from which the following fundamentals can be obtained: a1 = 2793.2 (2782.5), a1 = 1745.9 (1746.1), a1 = 1506.9 (1500.1), b1 = 2818.0 (2843.1), b1 = 1260.9 (1249.1), and b2 = 1174.6 (1167.3), where the experimental values102 are given in parentheses. In light of the approximate nature of the CBS extrapolation and the alternating signs of the various smaller corrections, it is clearly possible to obtain fortuitously closer agreement with experiment using lower levels of theory. Formaldehyde has been the subject of a very large number of previous studies. A comprehensive review of the literature is beyond the scope of the present work. In 1999 Martin and Oliveira91 published a W2 zero point inclusive atomization energy (D0) of 1495.6 kJ/mol, compared to a concurrent ATcT value of 1494.7 kJ/mol. The largest W2 error in the 28 member test set described in their paper was 2.6 kJ/mol. Seven years 12


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later Karton et al.10 computed identical W4 and W4.2 D0 values of 1495.8 kJ/mol. In that same year DeYonker et al.98 listed a correlation consistent composite approach (ccCA-CBS2+SO+SR) f(298.15K) value of -111.7 kJ/mol, roughly 2.5 kJ/mol more negative than the ATcT value. A 2008 application of a slightly lower level FPD approach73 found f(298.15K) = -109.2 ± 1.2 kJ/mol, in essentially exact agreement with the then current ATcT value.4 The Gaussian-4 enthalpy of formation was found to be 3.5 kJ/mol smaller than experiment.15 In a 2009 investigation that combined coupled cluster theory and explicitly correlated second order perturbation theory, Klopper et al.103 reported a zero point inclusive atomization energy of 1496.6 kJ/mol compared to an ATcT value of 1495.8 ± 0.2 kJ/mol. The FPD composite structure is in good agreement with an equilibrium semiexperimental structure reported by Pawlowski et al. 104: rCO = 1.2044 Å (1.2046 Å), rCH = 1.1003 Å (1.007 Å) and HCO = 121.7° (121.6°), where the semi-experimental values are given in parentheses. C2H4O2 Acetic Acid The second calibration system is acetic acid, C2H4O2. A schematic representation of the acetic acid HOCC torsional angle rotational potential energy surface is shown in Figure 1. The surface contains two minima, with the 1A' anti conformation lying 21.55 kJ/mol above the 1A' syn conformation without inclusion of ZPE corrections. The two minima are connected by a C1 symmetry transition state, which is 51.91 kJ/mol (CCSD(T)(FC)/aVDZ, ignoring ZPE corrections) higher in energy than the syn conformation. Figure 2 shows the corresponding lowlying, triply degenerate rotational transition states associated with the methyl group rotation. At the CCSD(T)(FC)/aVDZ level of theory the syn methyl group transition state lies 1.85 kJ/mol above the syn minimum and the anti transition state lies 3.16 kJ/mol above the anti conformation. All transition states were verified to possess a single CCSD(T) imaginary frequency 546.0i syn/anti torsional T.S.; 66.0i syn methyl rotation T.S.; 102.0i anti methyl group rotation T.S. Compared to the theoretical procedure followed for formaldehyde, some components were necessarily obtained with smaller basis sets due to the increased size of the molecule. For example, with H2CO the largest valence CCSD(T) calculation involved the aV8Z basis set. The same aV8Z basis set in the case of acetic acid would have required 2,536 basis functions while the exploitable symmetry would have simultaneously been reduced from C2v to Cs. Such a calculation would have been prohibitively expensive with our current hardware and software. In 13



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addition, the problem of near linear dependency would have been exacerbated in a hypothetical aV8Z calculation on acetic acid. With the aV7Z basis set the smallest overlap matrix eigenvalue for acetic acid (syn) was already at 3  10-8, which is close to the smallest practical value. As a consequence of the reduction in size of the largest basis set from aV8Z to aV7Z and the doubling in the magnitude of the correlation energy, the uncertainty associated with the CCSD(T)(FC)/CBS component increased by a factor of 3 (H2CO vs. C2H4O2). Formaldehyde atomization

energies

determined

at

the

estimated

CCSD(T)(FC)/CBS(aV78Z)

and

CCSD(T)(FC)/CBS(aV67Z) levels of theory differed by only 0.05 kJ/mol (0.01 kcal/mol), suggesting that expansion of the acetic acid basis set to aV8Z would not likely change the CBS(aV67Z) value significantly relative to our 1.57 kJ/mol overall uncertainty. The T1 CCSD diagnostic,105 which is often used as a crude indicator of the degree of multireference character in the wave function, is 0.014 for acetic acid and 0.015 for formaldehyde. Both values are typical of wave functions strongly dominated by the HartreeFock configuration. This conclusion is further supported by the relatively small magnitude of the total higher order valence correlation corrections (1.47 kJ/mol for acetic acid, 0.08 kJ/mol for formaldehyde). Based on our experience across hundreds of first and second row compounds, we anticipated that the FPD approach would be capable of accurately predicting the enthalpies of formation for both conformations of acetic acid when applied with the current basis sets. This was indeed found to be the case, as shown in Table S2 in the Supporting Information. The best composite acetic acid syn enthalpies of formation are: -418.28 ± 1.57 kJ/mol (0 K) and -432.40 ± 1.57 kJ/mol (298.15 K), where the latter value compares to -432.50 ± 0.5 kJ/mol (ATcT ver. 1.122). The best composite anti enthalpies of formation are: -397.58 ± 1.63 kJ/mol (0 K) and -411.38 ± 1.63 kJ/mol (298.15 K), with the latter value compared to -411.1 ± 1.2 kJ/mol (ATcT ver. 1.122). Consequently, due to the close agreement with the corresponding ATcT values found for formaldehyde and acetic acid, we expect similarly good agreement between the FPD enthalpy for oxalic acid and the ATcT value, barring unforeseen complications. The best composite FPD equilibrium structures for the syn and anti conformations of acetic acid are shown in Table S3 in the Supporting Information. As mentioned previously, the CCSD(T)(FC)/aV7Z structures were estimated by performing an exponential extrapolation of the aVQ56Z sequence of optimized internal coordinates. The theoretical re structure for the syn conformation is in reasonable agreement with the microwave and electron diffraction experimental structures. To the best of our knowledge, no experimental structure has been reported for the anti conformation. 14


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C2H2O4 Oxalic Acid The oxalic acid potential energy surface, schematically displayed in Figure 3, is considerably more complex than the acetic acid surface. Our CCSD(T)(FC) calculations reveal several minima interconnected by multiple first order transition states. The most prominent of the torsional stationary points are indicated in Figure 3. The largest affordable basis set used in the valence calculations was aV6Z. An aV7Z calculation would have involved 1986 basis functions and was judged to be prohibitively expensive with current hardware and  functioncapable software (Gaussian 09). Our best composite theoretical equilibrium structures for the stationary points on the oxalic acid potential surface are given in Table S4 in the Supporting Information. These structures combine the largest basis set CCSD(T)(FC) structures corrected by the largest basis set CV and DK changes in geometry. A variety of naming conventions have been used in previous studies for the conformers of oxalic acid. Oxalic acid has three torsional modes, one corresponding to the relative rotation of the two carboxyl groups along the C-C bond, and two corresponding to the rotation of the OH groups along the single C-O bond in each of the carboxyl groups.

The lowest energy

conformation, in which the two carboxyl groups are trans to each other and both OH groups are cis with respect to the C-C bond, referred to simply as trans-1 in this work and cTc elsewhere, possesses a 1Ag electronic configuration (C2h symmetry). Next in terms of increasing energy is the 1A' trans-2 conformation (designated cTt elsewhere), of Cs symmetry, with one of the two OH groups trans with respect to the C-C bond. It lies at approximately 11.7 kJ/mol (11.0 kJ/mol after ZPE inclusion) above trans-1. At a slightly higher energy is the 1Ag trans-3 conformation (C2h, both OH groups are trans with respect to the C-C bond) at 18.6 kJ/mol (16.5 kJ/mol after ZPE inclusion). The C-C cis analog of trans-2 conformation, labeled here as cis-2 conformation (1A' Cs), is found as a minimum at 25.6 kJ/mol (23.7 kJ/mol with ZPE inclusion). The C-C cis analog of trans-1 conformation, termed here as cis-3 (1A1, C2v), corresponds to a transition state at 79.5 kJ/mol (72.7 kJ/mol after ZPE inclusion), with two enantiomeric gauche conformers nearby (inflection points or very shallow minima, depending on the method/basis set combination employed), termed here gauche-3 (1A, C2) and located roughly 25 kJ/mol lower than the cis-3 transition state. Though the nature of the C-C cis analog of trans-3 conformation, termed here as cis-1 (1A1, C2v), is also somewhat method/basis set dependent, it generally corresponds to a transition state (about 2.4 kJ/mol higher than the trans-3 minimum), with two adjacent shallow (about a tenth of a kJ/mol lower than the cis-1 TS) enantiomeric gauche 15



minima (1A, C2), further separated from the trans-3 conformer by a very modest forward barrier (less than 1 kJ/mol). Table 1 shows the relatively energy differences for all stationary points examined in this study. In general, improving the basis set increases the energy differences between the studied conformers, but beyond the aVTZ the effect is small. At that point the relative energies are converged to approximately 0.2 kJ/mol.

Transition state calculations were limited to the

CCSD(T)(FC)/aVDZ level of theory and all such states were verified to possess a single imaginary frequency. IV. ATcT Results and Discussion The FPD enthalpy of formation for oxalic acid is almost 8 kJ/mol higher than the preferred result of Simmie and Somers,32 but is still nearly 10 kJ/mol lower than the gas phase ATcT results from TN ver. 1.118 and ver. 1.122.6,24 As discussed in the Introduction, the information relating to oxalic acid included in these two versions of ATcT TN was restricted to three experimental measurements27-29 that were originally re-analyzed by Cox and Pilcher30 and later reused in the evaluation of Pedley et al.26 Based on this rather restricted set of experimental data, the resulting 298.15 K ATcT enthalpy of formation24 was -819.3 ± 0.5 kJ/mol for the condensed phase, and -721.4 ± 2.1 kJ/mol for the gas phase (or -710.8 ± 2.1 kJ/mol at 0 K for the latter). The variance decomposition analysis, which provides an account of the provenance of each individual ATcT enthalpy of formation,22 indicates that in these two versions the value for the condensed phase is heavily dominated (99 %) by one of the two calorimetric determination (Verkade et al.,28 as reanalyzed by Cox and Pilcher30), while the value for the gas phase value is dominated by the resulting condensed phase value and the sublimation enthalpy of Bradley and Cleasby.27 In contrast to traditional sequential thermochemistry, where the provenances of the derived enthalpies of formation are rather straightforward, the provenances of ATcT enthalpies of formation for most species are distributed over a significant number of determinations. This salient feature of the ATcT approach, which is a consequence of the underlying TN approach, normally imparts both an enhanced robustness and an increased accuracy to the resulting enthalpies of formation.22 Given its rather contained provenance, this is clearly not the case for the enthalpy of formation of oxalic acid in ATcT TN vers. 1.118 and 1.122.6,24 The essential reason is the paucity of relevant data and the resulting relatively weak integration with the reminder of the TN. Consequently, the quality of the result for oxalic acid from the discussed 16


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versions of TN cannot be expected to be significantly better than what would have been obtained from the same measurements by the traditional sequential approach. The progression of the ATcT TN from one version to the next is achieved by an expansion of its knowledge content, both by inserting additional experimental and/or theoretical determinations that improve the definition of already existing chemical species, as well as by expanding the coverage by introducing additional species. Thus, following ver. 1.122, the TN was expanded several times as part of various projects, creating new sub-versions such as 1.122b,23,106,107 1.122d,108 1.122e,109 1.122h,110 1.122m,111 1.122o,112,113 1.122p,1 and 1.122q.114 The interim sub-version of the TN of relevance for the present discussion is ver. 1.122i (the immediate predecessor of ver. 1.122h), because the related expansion of data content involved, inter alia, the addition of several computational results on oxalic acid. These additions followed the pattern that is typically an integral part of the process that introduces a new gas phase species in the TN, and involves the creation of an initial bare-bones network scaffolding by using a select set of mid-level composite methods, including G3X,115 G4,15 CBS-QB3,11,12 CBS-APNO,13,116 and W1,91,117 in order to complement the experimental determinations found in the existing literature. In subsequent stages of the growth of the TN, this information is replaced or superseded by results from more elaborate computations. Thus, in addition to the three experimental results that were already in the TN, the expansion in ver. 1.122i added mid-level theoretical reaction enthalpies at 0 K, computed using the five composite methods mentioned above, corresponding to the C-C bond dissociation energy (C(O)OH)2  2 HOCO and the following two isodesmic reactions that correspond to the difference in the carbon-carbon bond dissociation energy in oxalic acid and in glyoxal or ethane (C(O)OH)2 + 2 HCO  (CHO)2 + 2 HOCO (C(O)OH)2 + 2 CH3  2 HOCO + C2H6 Purely for documentation purposes, the total atomization enthalpies of oxalic acid obtained with mid-level composite methods were also added to the TN, though these were prevented from contributing directly to the result by giving them zero weight. The additions and changes incorporated in TN ver. 1.122i result in 298.15 K enthalpies of formation of oxalic acid of -731.9 ± 1.6 kJ/mol for the gas phase and -819.3 ± 0.5 kJ/mol for the condensed phase. One immediately notices that with respect to the enthalpy of formation of the condensed phase, the additions in TN ver. 1.122i did not cause a change compared to TN ver. 1.122. Indeed, 17



a variance decomposition analysis shows that the provenance of the condensed phase value in that sub-version continues to be entirely restricted to the combustion calorimetry of Verkade et al.28 However, the provenance of the gas phase enthalpy of formation has in this version changed and has became quite distributed: the top 90% of the provenance involves no less than 63 different determinations extant in the TN, 16 of which directly involve oxalic acid, with the rest playing vital roles in defining other related species, such as HOCO or HCO. The top ten contributors are all based on the newly inserted mid-level theoretical results, and correspond to the differences in the C-C bond dissociation energies in oxalic acid and ethane computed at the W1 (6.8%), CBS-APNO (5.5%), G4 (5.5%), and CBS-QB3 (3.3%) composite levels, and in oxalic acid and glyoxal at the W1 (6.0%), CBS-APNO and G3X (5.4% each), G4 and CBS-QB3 (4.3% each) levels, as well as the decomposition of HOCO to H2, O2, and gas phase C atom118 obtained at the HEAT345-(Q) level of theory (6.3%). Importantly, the new gas phase value, which now evidently relies heavily on mid-level theoretical results, is 10.5 kJ/mol lower than the earlier ATcT result from TN ver. 1.12224 (-721.4 ± 2.1 kJ/mol at 298.15 K) that relied entirely on experimental data, and 8.2 kJ/mol lower than the experimental value recommended by Pedley et al.26 (-723.7 ± 4.9 kJ/mol). It fortuitously becomes nearly identical to the experimental value recommended by Cox and Pilcher,30 -732.0 ± 2.5 kJ/mol. The 0 K total atomization energy implied by the revised ATcT result, 3563.5 ± 1.6 kJ/mol, is in excellent accord with the current FPD computational result, 3564.0 ± 4.0 kJ/mol (see Table 2). Thus, the revised ATcT result for the gas phase and the high-level computational result, since they are independent, essentially validate each other. However, as a consequence of the fact that the introduction of mid-level theoretical results in the TN significantly changes the gas-phase enthalpy of formation and produces an ATcT value that is in excellent agreement with the high-level FPD value, while that of the condensed phase value remains unchanged, their difference, 87.4 ± 1.7 kJ/mol at 298.15 K, which corresponds to the sublimation enthalpy, becomes intrinsically incongruent with the experimental values for this quantity, such as that of Bradley and Cleasby27 (97.9 ± 2.1 kJ/mol), suggesting unresolved problems with the experimental data included in the TN. Since ver. 1.122i, the ATcT TN has undergone further additions and expansions (not related to oxalic acid), eventually leading to TN ver. 1.122q.114 In preparation for the inclusion of the FPD computations detailed above, and in view of the uncovered incongruences with the experimental determinations, the experimental determinations involving oxalic acid were independently reevaluated based on the original reports, rather than relying on their earlier 18


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reanalysis by Cox and Pilcher.30 A critical scrutiny of the calorimetric determination by Wilhoit and Shiao29 did not uncover any discernible problems, and, based on their reported constantvolume heat of combustion of -654.7 ± 2.5 cal/g, produced combustH°(298.15 K) = -58.06 ± 0.23 kcal/mol or -242.9 ± 1.0 kJ/mol - a value that is essentially identical to that derived by Cox and Pilcher30 (-58.06 ± 0.22 kcal/mol).

However, a similar reexamination of the calorimetric

enthalpy of combustion of Verkade et al.28 suggested that both the value and the ± 0.11 kcal/mol uncertainty assigned to this measurement by Cox and Pilcher30 are less than optimal. Verkade et al.28 reported the constant volume heat released by combustion of oxalic acid as -678.0 cal15/g at 19.5 °C. Assuming that the cal15 unit used by those authors corresponds to 1.0011 modern (thermal) calories42,119,120 and converting that measurement to a molar constant-pressure enthalpy using a molecular weight of 90.0349, leads to a 298 K combustion enthalpy of -60.20 kcal/mol, somewhat smaller than the value of -60.59 kcal/mol obtained by Cox and Pilcher.30 Even more importantly, the overall accuracy of 0.2% or better, as claimed by Verkade et al.28 (which is the origin of the ± 0.11 kcal/mol uncertainty assigned by Cox and Pilcher30), turned out to be veritably optimistic.

Namely, the standard deviation of the mean of the four calorimetric

measurements reported by Verkade et al.28 is 0.6 cal/g. The corresponding 95% confidence interval (which is the required standard in thermochemistry45) for such a small sample is 3.2 cal/g, or ± 0.29 kcal/mol, significantly exceeding the uncertainty assigned by Verkade et al.28 and Cox and Pilcher.30

Moreover, this uncertainty reflects just the random error in the

calorimetric determinations of Verkade et al.,28 and would need to be further augmented by an estimate of potential sources of systematic errors, which is frustrated by lack of sufficient details in their paper. Systematic error could easily augment the overall uncertainty to as much as one or even two kcal/mol.121

Consequently, the determination of Verkade et al.28 has to be

considered as less than reliable, and was given zero weight in the updated version of the TN - a decision ultimately not very different from that made by Cox and Pilcher,30 who tacitly ignored this determination when deriving the recommended value for oxalic acid. While this would nominally reduce the amount of viable determinations involving solid oxalic acid to just one, a thorough review of literature on the enthalpy of formation of solid oxalic acid produced one more calorimetric determination,122 reported in the Ph.D. thesis of Brown,123 which, with the thermochemistry prevalent in the late 1960s,124 corresponds to combustH°(298.15 K) = -58.13 ± 0.11 kcal or -243.2 ± 0.5 kJ/mol, in apparently very good accord with the calorimetric determination of Wilhoit and Shiao.29 19



With respect to the sublimation enthalpy, a literature search uncovered several relevant determinations: 23.47 (± 1.0) kcal/mol or 98.2 (± 4.2) kJ/mol at 316.5 K by Bradley and Cotson,125 (a further refinement of the prior determination by Bradley and Cleasby27 that was used by Cox and Pilcher), 98.5 (± 3) kJ/mol at 322 K by de Wit et al.126 (from combined torsion and mass loss effusion measurements), and 100 ± 4 kJ/mol at 298 K by Bilde et al.127 (from a combined analysis of data available at the time, such as Noyes and Wobbe,128 de Wit et al.,126 Booth et al.,129 and Bradley and Cotson125). In addition, Ribeiro da Silva et al.130 obtained 101.7 ± 1.1 kJ/mol at 298.15 K by a reanalysis of the fit of partial pressures of Bradley and Cotson,125 Noyes and Wobbe,128 obtained 21.65 ± 0.50 kcal/mol (90.6 ± 2.1 kJ/mol) at 348 K by a fit of their measurements of equilibrium pressure, and Booth et al.129 reported 75 ± 19 kJ/mol at 298 K from Knudsen effusion mass spectrometry. The latter three measurements were also inserted in the TN, but, together with the measurement originally used by Cox and Pilcher,30 given zero weight on account of either being superseded by subsequent measurements from the same group (Bradley and Cleasby27), suspected problems (such as the measurement of Noyes and Wobbe,128 which was subsequently criticized125 as pertaining to the less stable monoclinic β-oxalic acid, rather than the orthorhombic α-oxalic acid), insufficient accuracy for the present purpose (Booth et al.129), or simply lacking the necessary independence from measurements already included in the ATcT TN (Ribeiro da Silva et al.130). At this point we need to briefly examine the fact that the sublimation enthalpies that were reported in the literature appear to be devoid of corrections for non-ideality. Namely, in order to represent a pure substance, the thermochemical properties of gas-phase species should correspond to ideal-gas behavior,131 the underlying reason being that real gases are complex equilibrated mixtures containing monomers, dimers, etc., with a composition that is a function of both temperature and pressure. Essentially, each oligomer is a distinct thermochemical species, contributing its own partial pressure to the measured total vapor pressure of a real gas above the condensed phase.42 The real-to-ideal correction to the experimentally determined sublimation or vaporization enthalpy is normally a positive quantity that extricates the process of evaporation of the gas phase monomers from that of a real gas.42,131 Given their strong propensity to form gas phase dimers and possibly higher oligomers, the corrections to vaporization or sublimation enthalpies for organic acids can be substantial. Prima facie, the known corrections for vaporization enthalpies of simple aliphatic monoacids (+26.2 kJ/mol for formic acid, +28.2 kJ/mol for acetic acid, +22.9 kJ/mol for propanoic acid, +17.6 kJ/mol for butanoic acid)132 would suggest that this correction cannot be simply ignored. However, besides the propensity to 20


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associate and form oligomers, a relevant component that governs the importance of the real-toideal correction is the equilibrium pressure of the gas phase above the condensed phase. Thus, at room temperature, the equilibrium pressure of oxalic acid (12-14 mPa125-130) is many orders of magnitude smaller than that133 of formic (5.7 kPa), acetic (2.1 kPa), propanoic (0.5 kPa), and even butanoic (0.1 kPa) acid. Using an estimated second virial coefficient for oxalic acid,133,134 one can infer that the partial pressure of the oxalic acid dimer at room temperature is likely to be nine or ten orders of magnitude lower than that of the monomer, implying that for oxalic acid the real-to-ideal correction can be safely neglected, and the literature values for the sublimation enthalpy can be used in the TN without further modification. Amending the experimental entries of TN ver. 1.122q as described above, and retaining the mid-level theoretical entries related to oxalic acid that were described earlier - though at this point still not utilizing the current FPD result in the TN - produces 298.15 K enthalpies of formation of oxalic acid of -731.5 ± 1.3 kJ/mol for the gas phase and -829.7 ± 0.5 kJ/mol for the condensed phase. Compared to TN ver. 1.122i, the change in the value for the gas phase enthalpy of formation is minimal (less than 0.5 kJ/mol, accompanied by a modest improvement in its uncertainty), but the change in the condensed phase enthalpy of formation is substantial, which is not entirely surprising: The variance decomposition indicates that the provenance of the latter, though still rather restricted (given the paucity of relevant experimental measurements), now relies on the concordant combustion calorimetry determinations of Brown123 (78.7 %) and Wilhoit and Shiao.29 (18.0 %), rather than Verkade et al.28 The provenance of the gas phase enthalpy of formation continues to be quite distributed, except that the contributions from midlevel theoretical reaction enthalpies (discussed in conjunction with the provenance in TN ver. 1.122i) are now preceded by three sublimation enthalpies: de Wit et al.126 (18.2 %), Bilde et al.127 (10.2 %), and Bradley and Cotson125 (9.4 %), which, within their individual uncertainties, are now found to be entirely consistent with the difference in the enthalpies of formation of the condensed and gas phases of -98.2 ± 1.3 kJ/mol. The final step, which uses all the available knowledge to derive the best currently possible ATcT value for the enthalpy of formation of oxalic acid, involves the addition of the current FPD total atomization energy to the TN. This further refines the resulting gas phase enthalpy of formation of oxalic acid, producing fH°(298.15 K) = -731.6 ± 1.2 kJ/mol and fH°(0 K) = -721.0 ± 1.2 kJ/mol. Notably, the current ATcT value is significantly different than the experimental gasphase value recommended by Pedley et al.26 (-723.7 ± 4.9 kJ/mol at 298 K), as well as the ATcT 21



value obtained earlier from TN vers. 1.118 and 1.122,6,24 both of which were derived entirely from experimental data. Curiously, the value recommended in the older compilation of Cox and Pilcher30 (-732.0 ± 2.5 kJ/mol), which relied on manually selecting one of the two available calorimetric determination and combining the resulting enthalpy of formation of the solid phase with a particular sublimation enthalpy, appears to be in excellent agreement with the current ATcT value. The current gas phase result sustains the notion that the DFT-based theoretical results proposed earlier in the literature are rather inaccurate, ranging from being 22 to 42 kJ/mol too high (Duan et al.35) to as much as 88 kJ/mol too low (Cioslowski et al.34). While one of the two theoretical values proposed by Simmie and Sommers32 (-735.5 ± 5.3 kJ/mol at 298 K, which was based on averaging mid-level theoretical atomization energies) agrees within its error bar with the current ATcT enthalpy of formation of gas-phase oxalic acid, their preferred value (739.8 ± 1.2 kJ/mol at 298 K, obtained by averaging the isodesmic reaction enthalpies obtained from composite methods), somewhat surprisingly, appears discordant (being lower by 7.6 kJ/mol, with the difference substantially exceeding the combined uncertainties). As could have been expected, the insertion of the high-level FPD atomization energy did not substantially affect the ATcT condensed phase enthalpy, which continues to be -829.7 ± 0.5 kJ/mol.

The provenance analysis indicates that its value still relies on the combustion

calorimetric determinations of Brown123 and of Wilhoit and Shiao.29 The difference between the enthalpies of formation of the gas phase and the condensed phase produces a sublimation enthalpy fH°(298.15 K) = 98.1 ± 1.3 kJ/mol, in excellent agreement with the (less accurate) determinations of Bradley and Cotson125 (corresponding to 98.6 ± 4.2 kJ/mol at 298.15 K), de Wit et al.126 (corresponding to 99.0 ± 3.0 kJ/mol at 298.15 K), and Bilde et al.127 (100.0 ± 4.0 kJ/mol at 298.15 K). V. Conclusion High-level coupled cluster computations using the FPD approach, calibrated by comparing the computed and ATcT enthalpies of formation for formaldehyde and syn- and antiacetic acid, produce a gas-phase enthalpy of formation of oxalic acid of -721.8 ± 4.0 kJ/mol at 0 K or -732.2 ± 4.0 kJ/mol at 298.15 K. The high level theoretical results confirmed the suspicion raised recently by Simmie and Somers32 that the earlier ATcT result, which was obtained from a TN that for oxalic acid contained only experimental data (two combustion calorimetries on condensed phase oxalic acid and one sublimation enthalpy), was probably too high. The ATcT analysis progressed in several steps. Initially, the section of the ATcT TN 22


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involving oxalic acid was simply expanded by adding the theoretical results from a set of select mid-level composite approaches. This shifted the provenance of the enthalpy of formation for gas phase oxalic acid from purely experimental data to theoretical data, producing a significantly more negative value, which showed very good agreement with the FPD theoretical result, but given that the condensed phase enthalpy of formation remained unchanged - it implied a sublimation enthalpy that was incongruent with the experimental determinations of this quantity. In a second step of ATcT analysis, the experimental determinations relevant to oxalic acid that were included in the earlier versions of ATcT were critically reanalyzed. This process suggested that one of the two literature calorimetric determinations (Verkade et al.28) is not entirely reliable, and uncovered a third calorimetric determination.

These changes and

enhancements of the ATcT TN resulted in a significantly revised enthalpy of formation for the condensed phase, and a further enhancement of the enthalpy of formation for the gas phase. The revised condensed phase value of -829.7 ± 0.5 kJ/mol at 298.15 K is in good agreement with the older evaluation of Cox and Pilcher,30 but differs substantially from the value subsequently recommended by Pedley et al.,26 as well as from earlier ATcT TNs that did not include theoretical results for oxalic acid. At the same time, the enhanced gas phase value, -720.9 ± 1.3 kJ/mol at 0 K or -731.5 ± 1.3 kJ/mol at 298.15 K, is in excellent agreement with the FPD theoretical result. Since the high-level theoretical value and the ATcT result are independent, they tend to provide mutual validation. In the final step of the ATcT analysis, the FPD high-level total atomization energy was included in the TN. This produced a further enhancement of the ATcT gas phase enthalpy of formation of oxalic acid, resulting in -731.6 ± 1.2 kJ/mol at 298.15 K, and -721.0 ± 1.2 kJ/mol at 0 K. The difference of 98.1 ± 1.3 kJ/mol between the final value for the condensed phase enthalpy of formation, -829.7 ± 0.5 kJ/mol, and that for the gas phase, is in excellent agreement with several experimental determinations of the sublimation enthalpy, further validating the current results. Supporting Information The Supporting Information, which contains the decomposition of the FPD atomization energies and enthalpies of formation for formaldehyde (H2CO), acetic acid (C2H4O2), and best composite structures for acetic acid and oxalic acid, is available free of charge on the ACS Publications website.

23



Author Information *E-mail: [email protected]; [email protected]; [email protected]. Acknowledgments The work at Argonne National Laboratory (BR and DHB) was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences, under Contract No. DE-AC02-06CH11357, through the Gas-Phase Chemical Physics Program (BR) and the Computational Chemical Sciences Program (DHB).

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Figure Captions 1. Schematic representation of the zero-point-exclusive potential energy surface of C2H4O2 (acetic acid). Not shown are the transition states associated with methyl rotations. 2. Transition states associated with methyl rotation in the syn and anti conformations of acetic acid. 3. Schematic representation of a portion of the potential energy surface of C2H2O4 (oxalic acid).

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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Table 1. CCSD(T)(FC) Energies for Oxalic Acid Conformations Relative to the Lowest Energy (trans-1) Form (kJ/mol). Basis aVDZ aVTZ aVQZ aV5Z aV6Z Est. CBSa Basis aVDZ aVTZ aVQZ aV5Z aV6Z Est. CBSa a b

trans-2 11.487 11.501 11.720 11.741 11.735 11.733 trans-2 10.795 10.745 10.962 10.982 10.979 10.970

trans-3 16.239 18.093 18.594 18.649 18.631 18.616

Relative Energies Excluding ZPEs cis-2 cis-1 T.S. cis-3 T.S. trans-1-2 T.S. 23.409 18.543 79.535 58.213 25.113 25.490 25.594 25.585 25.581

trans-3 14.677 16.008 16.510 16.564 16.548 16.531

Relative Energies Including ZPEsb cis-2 cis-1 T.S. cis-3 T.S. trans-1-2 T.S. 21.522 16.769 72.651 53.359 23.221 23.598 23.702 23.694 23.690

CBS estimate based on 1/(max +0.5)4. Zero point energies were obtained from CCSD(T)(FC)/aVDZ and aVTZ calculations. The aVTZ ZPE corrections were applied to aVnZ, n = T,Q,5,6 electronic energy differences.

26


trans-2-3 T.S 60.672

trans-2-3 T.S 54.392

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Table 2. Atomization Energy/Enthalpy Components for C2H2O4 Oxalic Acid (kJ/mol). trans-1 (1Ag) Conformation Component De/D0/corrections a 3685.22 ± 2.84 De R/UCCSD(T)(FC)/CBS(aV56Z) ΔR/UCCSD(T)(CV)/CBS(wCVTQZ)a 14.15 ± 0.31 ΔR/UCCSD(T)-DKH/aVQZ-DK -5.04 ± 0.02 ΔCCSDT(FC)/VTZ -6.10 ΔCCSDTQ(FC)/VDZ 7.43 Δcf CCSDTQ(5)(FC)/VDZ 0.80 ΔCCSDT(CV)/wCVDZ(no p on H) 0.07 ΔUCCSD(FC)-DBOC/aVTZ 0.81 ΔAtomic S.O. -4.43 3692.92 ± 3.17 Best Composite De Estimate Anharm. ZPE/aVTZb -128.88 ± 0.84 3564.04 ± 4.01 Best Composite D0 Estimate Best Composite fc ATcT v1.122d Burcat and Ruscice Cox and Pilcherf Pedley et al.g Dorofeeva et al.h trans-2 (1A') Conformation Component De R/UCCSD(T)(FC)/CBS(aV56Z)a ΔR/UCCSD(T)(CV)/CBS(wCVQ5Z)a ΔR/UCCSD(T)-DKH/aVQZ-DK ΔCCSDT(FC)/VTZ(no d on H) ΔCCSDTQ(FC)/VDZ Δcf CCSDTQ5(FC)/VDZ ΔCCSDT(CV)/wCVDZ(no p on H) ΔUCCSD(FC)-DBOC/aVTZ ΔAtomic S.O. Best Composite De Estimate Anharm. ZPE/aVTZb Best Composite D0 Estimate Best Composite f

d

0K -721.83 ± 4.01 -710.8 ± 2.1 -721.2 ± 2.0 -721.2 ± 2.0

298.15 K -732.24 ± 4.01 -721.4 ± 2.1 -731.8 ± 2.0 -732.2 ± 2.5 -723.7 ± 4.9 -731.8 ± 2.0

De/D0/corrections 3673.47 ± 2.82 14.01 ± 0.31 -4.99 ± 0.01 -5.57 7.39 0.81 0.08 0.78 -4.43 3681.55 ± 3.14 -128.12 ± 0.74 3553.43 ± 3.88 0K -711.20 ± 3.88

27



trans-3 (1Ag) Conformation Component De R/UCCSD(T)(FC)/CBS(aV56Z)a ΔR/UCCSD(T)(CV)/CBS(wCVTQZ)a ΔR/UCCSD(T)-DKH/aVQZ-DK ΔCCSDT(FC)/VTZ ΔCCSDTQ(FC)/VDZ Δcf CCSDTQ(5)(FC)/VDZ ΔCCSDT(CV)/wCVDZ(no p on H) ΔUCCSD(FC)-DBOC/aVTZ ΔAtomic S.O. Best Composite De Estimate Anharm. ZPE/aVTZb Best Composite D0 Estimate d

Best Composite f

cis-2 (1A’) Conformation Component ∑De R/UCCSD(T)(FC)/CBS(aV56Z)a ΔR/UCCSD(T)(CV)/CBS(wCVQ5Z)a ΔR/UCCSD(T)-DKH/aVQZ-DK ΔCCSDT(FC)/VTZ(no d on H) ΔCCSDT(Q)(FC)/VDZ Δcf CCSDTQ5(FC)/VDZ ΔCCSDT(CV)/wCVDZ(no p on H) ΔUCCSD(FC)-DBOC/aVTZ ΔAtomic S.O. Best Composite De Estimate Anharm. ZPE/aVTZb Best Composite D0 Estimate d

Best Composite f

De/D0/corrections 3666.59 ± 2.83 13.94 ± 0.32 -4.95 ± 0.02 -6.20 7.41 0.82 0.08 0.76 -4.43 3674.02 ± 3.17 -126.79 ± 0.84 3547.23 ± 4.01 0K -704.90 ± 4.01 De/D0/corrections 3659.62 ± 2.83 13.99 ± 0.32 -4.99 ± 0.02 -5.44 7.38 0.80 0.08 0.77 -4.43 3667.78 ± 3.17 -126.98 ± 0.84 3540.80 ± 4.01 0K -698.59 ± 4.01

a

CBS estimate based on 1/(max +0.5)4. At the CCSD(T) level of theory, atoms are described with R/UCCSD(T) and atomic symmetry is imposed. At the CCSDT and CCSDT(Q) levels of theory atomic symmetry was not imposed. b Anharmonic zero point vibrational energy is based on CCSD(T)(FC)/aug-cc-pVTZ harmonic frequencies plus an MP2(FC)/aug-cc-pVDZ anharmonic correction. c Based on ATcT 0 K atomic enthalpies of formation24,25 (kJ/mol): H 216.034 ± 0.000; C 711.401 ± 0.050; O 246.844 ± 0.002.

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Active Thermochemical Tables v 1.122.24 e Burcat and Ruscic.135 f Cox and Pilcher.30 g Pedley et al.26 h Dorofeeva et al.136 d

29



Figure 1.

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Figure 2.

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Figure 3.

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TOC Graphic FPD theory

 fH°(C2 H2 O4 )

experiment

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Li, H.; Feng, H.; Sun, W.; Zhang, Y.; Fan, Q.; Peterson, K. A.; Xie, Y.; Schaefer, H. F., III The Alkaline Earth Dimer Cations (Be2+, Mg2+, Ca2+, Sr2+, and Ba2+). Coupled Cluster and Full Configuration Interaction Studies. Mol. Phys. 2013, 111, 2292-2298.

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Porterfield, J. P.; Bross, D. H.; Ruscic, B.; Thorpe, J. H.; Nguyen, T. L.; Baraban, J. H.; Stanton, J. F.; Daily, J. W.; Ellison, G. B. Thermal Decomposition of Potential Ester Biofuels, Part I: Methyl Acetate and Methyl Butanoate. 2017, 121, 4658-4677.

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120

The now obsolete cal15 was defined as the amount of heat needed to increase the temperature of 1 g of liquid water from 14.5 °C to 15.5 °C. Using the best currently available partition function for liquid water (Ref. 42, which is based on the equation of state of water from Ref. 119), or, alternatively, using directly the equation of state of water from Ref. 119, one finds that the difference between the enthalpy increments at 287.65 and 288.65 K is 4.18846 J/g or 1.00107 cal/g.

121

There are numerous potential sources of error that are hard to estimate because of insufficient details in Ref. 25, such as the actual definition of the cal15 as used by those authors, the accuracy of additional corrections (or, in some cases, lack thereof), the accuracy of determining the temperature increase of the calorimeter, the purity of the sample, etc., not to mention the initial calibration of the heat capacity of the calorimeter, which was performed by burning benzoic acid: the authors of Ref. 28 used a value of 6324 cal15/g (weighed in air) at 19.5 °C, implying 6331cal/g, to be compared with the best currently available value of 6319 cal/g (in vacuo) from ATcT.

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Afeefy, H. Y.; Liebman, J. F.; Stein, S. E. Neutral Thermochemical Data. In: NIST Chemistry WebBook. National Institute of Standards and Technology: Gaithersburg, MD 2018.

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Rossini, F. D.; Wagman, D. D.; Evans, W. H.; Levin, S.; Jaffe, I. Selected Values of Chemical Thermodynamic Properties. National Bureau of Standards Circ. 500, U.S. Dept. Commerce: Washington, DC 1952.

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Bradley, R. S.; Cotson, S. The Vapour Pressure and Lattice Energy of Hydrogen-bonded Crystals. Part II. - and Anhydrous Oxalic Acid and Tetragonal Pentaerythritol. J. Chem. Soc. 1953, 1684-1688.

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de Wit, H. G.; Bouwstra, J. A.; Blok, J. G.; de Kruif, C. G. Vapor Pressures and Lattice Energies of Oxalic Acid, Mesotartaric Acid, Phloroglucinol, Myoinositol, and Their Hydrates. J. Chem. Phys. 1983, 78, 1470-1475.

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Bilde, M.; Barsanti, K.; Booth, M.; Cappa, C. D.; Donahue, N. M.; Emanuelsson, E. U.; McFiggans, G.; Krieger, U. K.; Marcolli, C.; Topping, D.; et al. Saturation Vapor Pressures and Transition Enthalpies of Low-Volatility Organic Molecules of Atmospheric Relevance: From Dicarboxylic Acids to Complex Mixtures. Chem. Rev. 2015, 115, 41154156.

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Noyes, W. A., Jr.; Wobbe, D. E. The Vapor Pressure of Anhydrous Oxalic Acid. J. Am. Chem. Soc. 1926, 48, 18821887.

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Booth, A. M.; Barley, M. H.; Topping, D. O.; McFiggans, G.; Garforth, A.; Percival, C. J. Solid State and Sub-Cooled Liquid Vapour Pressures of Substituted Dicarboxylic Acids using Knudsen Effusion Mass Spectrometry (KEMS) and Differential Scanning Calorimetry. Atmos. Chem. Phys. 2010, 10, 4879-4892.

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Frenkel, M.; Chirico, R. D.; Diky, V. V.; Yan, X.; Dong, Q.; Muzny, C. NIST Thermodata Engine - Pure Compounds. NIST Std. Ref. Database #103A, Thermodynamics Research Center, National Institute of Standards and Technology: Boulder, CO 2004.

134

Ref 133 provides an empirically estimated 2nd virial coefficients for oxalic acid of B(340 K) = -1.1x10-2 m3/mol, B(350 K) = -8.9x10-3 m3/mol, B(360 K) = -7.6 x10-3 m3/mol, B(370 K) = -6.5x10-3 m3/mol, etc., with an estimated uncertainty of approximately ± 10%. These values imply an approximate equilibrium constant of Keq(298 K) = pdimer/p2monomer ~ 0.056 bar-1, which in turn suggests that pdimer(298 K) ~ 0.7 pPa, roughly 10 orders of magnitude less than pmonomer(298 K) ~ 11-12 mPa.

135

Burcat, A.; Ruscic, B. Third Millennium Ideal Gas and Condensed Phase Thermochemical Database for Combustion with Updates from Active Thermochemical Tables. Joint Report: ANL-05/20, Argonne National Laboratory, Argonne, IL, USA, and TAE 960, Technion, Haifa, Israel 2005.

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42


Enthalpy of Formation of C2H2O4 (Oxalic Acid) from High-Level

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