Theoretical Study on the Structure and Stabilities of Molecular Clusters

Article pubs.acs.org/JPCA

Theoretical Study on the Structure and Stabilities of Molecular Clusters of Oxalic Acid with Water Kevin H. Weber, Francisco J. Morales, and Fu-Ming Tao* Department of Chemistry and Biochemistry, California State University, Fullerton, 800 N. State College Boulevard, Fullerton, California 92834, United States S Supporting Information *

ABSTRACT: The importance of aerosols to humankind is well-known, playing an integral role in determining Earth’s climate and influencing human health. Despite this fact, much remains unknown about the initial events of nucleation. In this work, the molecular properties of common organic atmospheric pollutant oxalic acid and its gas phase interactions with water have been thoroughly examined. Local minima single-point energies for the monomer conformations were calculated at the B3LYP and MP2 level of theory with both 6-311++G(d,p) and aug-cc-pVDZ basis sets and are compared with previous works. Optimized geometries, relative energies, and free energy changes for the stable clusters of oxalic acid conformers with up to six waters were then obtained from B3LYP calculations with 6-31+G(d) and 6-311++G(d,p) basis sets. Initially, cooperative binding is predicted to be the most important factor in nucleation, but as the clusters grow, dipole cancellations are found to play a pivotal role. The clusters of oxalic acid hydrated purely with water tend to produce extremely stable and neutral core systems. Free energies of formation and atmospheric implications are discussed.

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INTRODUCTION Aerosols have tremendous influence on weather and health on Earth and often contain a substantial fraction of organic matter.1 Nonetheless, the role of organic compounds in new nanometer-sized particle formation remains ill defined.2 Low molecular weight carboxylic acids are quite soluble in water and enhance the hygroscopic properties of atmospheric particles. In particular, dicarboxylic acids have frequently been found in appreciable concentrations being produced by agricultural and industrial activities as well as secondary oxidation of volatile organic compounds from both anthropogenic and biogenic sources. These acids have been implicated as important participants in such processes as ice nucleation,3−6 cloud condensation, 7 and the production of fine particulate matter.8−10 Oxalic acid (ethanedioic acid, H2C2O4) represents the most prevalent dicarboxylic acid11−14 in the atmosphere and is the major constituent of organic matter in urban, rural, and remote background air,15−17 being found in concentrations 100−1000 times higher than those of ammonia. This proves significant as it has recently been predicted that oxalic acid is effective at binding with sulfuric acid, the dominant nucleating species in the atmosphere.18 Additionally, density functional theory (DFT) calculations suggest that oxalic acid significantly enhances the stability of ionic clusters catalyzing prenucleation clusters with positive charge.19 Of course, ionic and/or neutral forms of oxalic acid may be found in an atmospheric aerosol. However, the ionized form and the relative stabilities of clusters with an ionic core as compared to those of neutral core clusters © 2012 American Chemical Society

is of particular interest as the resulting negative ion may direct nucleation more powerfully than the neutral species.20 It should be noted that the relative importance of ionic versus neutral species continues to be debated. For example, Manninen et al.21 concluded in a recent study that approximately 10% of new particle formations were through an ion mediated pathway. However, Yu and Turco22 in analysis of the same data employing a different analytical approach concluded that most of the neutral particles observed were initially formed on ionic cores that were subsequently neutralized before growing to a detectable size (∼2 nm). Knowing the significance of oxalic acid in the atmosphere, it is of interest to study the stability of clusters with water and the initial ionization events at the molecular level. Aerosols in themselves are particularly interesting, having an interface of unlike phases giving rise to very different chemistries than the homogeneous bulk phase. Unfortunately, that same quality and the small size of embryos in new particle formations render them difficult to model (e.g., classical nucleation theory) and to research experimentally. Here, quantum chemical techniques have proven especially helpful for providing insight into these nanometer scale events. In this article, we detail the affect of water molecules on the structure and energies of clusters in the hydration of oxalic acid H2C2O4−(H2O)n (n = 0−6) as predicted by DFT calculations. Received: August 27, 2012 Revised: October 14, 2012 Published: October 22, 2012 11601

dx.doi.org/10.1021/jp308499f | J. Phys. Chem. A 2012, 116, 11601−11617

The Journal of Physical Chemistry A

Article

Figure 1. Optimized B3LYP/6-311++G(d,p) structures of the stable conformers of the oxalic acid monomer. Reported OH···O interaction distances and angles are in units of angstroms and degrees, respectively, and dipoles are in units of Debye. Mulliken charges are denoted in brackets for hydrogen and ν(OH) frequencies scaled by a factor of 0.952 are in cm−1.

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THEORETICAL METHODS The equilibrium geometries of the monomers and clusters constructed in this work were optimized using the Becke threeparameter exchange DFT23−25 method with the nonlocal correlation provided by Lee, Yang, and Parr (B3LYP)26 and additionally the Møller−Plesset second-order perturbation approximation (MP2)27 method for oxalic acid monomers. The calculations were carried out with 6-31+G(d) and 6-311+ +G(d,p) basis sets and, for oxalic acid monomers, with the augcc-pVDZ basis set as well. CCSD(T)/aug-cc-pVTZ calculations on B3LYP/6-311++G(d,p) optimized monomer geometries are also reported. All calculations were implemented with the Gaussian03 program package.28 The molecular clusters H2C2O4−(H2O)n (n = 1−6) were assembled from the optimized monomers H2C2O4 and H2O. Configurations of competing stability are examined, especially for the larger clusters where the most stable configurations are not necessarily obvious and competing assemblages must be considered. Relative energies, ΔEr, are defined as the electronic energy of the cluster in a given conformation with respect to that of the cluster in the most stable conformation. Different cluster configurations with the same number of water molecules (n) are arranged in order of increasing relative energy as determined by B3LYP/6-31+G(d) calculations, while relative energies from B3LYP/6-311++G(d,p) calculations are also reported. Intermolecular energies without and with zero-point energy correction, ΔE and ΔE0, respectively, are defined as the energy difference between the cluster and the corresponding infinitely separated monomers (H2C2O4 and H2O) in the cluster. Intermolecular enthalpies and Gibbs free energies, ΔH and ΔG, are defined in an analogous manner and calculated for a temperature of 298.15 K.

are presented in Figure 1. The nomenclature is the system introduced by Nieminen et al.29 where the lower case letters refer to cis (c) or trans (t) dihedral angles of H−O−C−C where cis gets priority (is denoted first) over the trans configuration. The uppercase letter refers to cis (C) or trans (T) dihedral for OC−CO. Absolute single-point energies of the most stable conformation, cTc, along with the relative energies of the four remaining conformers identified as local minima in this work (cTt, tTt, tCt, and cCt) are presented in Table 1 along with values from previous calculations for comparison. Zero-point Table 1. Single-Point Electronic Energies, Zero-Point Corrected (ZPC) Energies, and Free Energy Changes [ΔG] of the Local Minima Are in kcal mol−1 Relative to the Lowest Energy Conformer; Absolute Energies Are Reported in Hartrees

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RESULTS AND DISCUSSION Oxalic Acid Monomer. Geometries and Energies. The geometries of the oxalic acid monomer along with select interatomic distances and bond angles, dipole moments, Mulliken charges on hydrogen, and ν(OH) stretching frequencies calculated with a B3LYP/6-311++G(d,p) method

method

cTc

cTt

tTt

tCt

cCt

B3LYP/6-31G (d,p)a /6-31+G(d)b /6-311++G(d,p) (ZPC E0K) [ΔG298K] /6-311+G(2d,p)c /aug-cc-pVDZ MP2/6-311++G (d,p)e [ΔG458K]e /6-31G(d,p)a /aug-cc-pVDZ QCISD/6-31G (d,p)c CCSD(T)/aug-ccpVTZ∥B3LYP/6311++G(d,p)

−378.33068

2.42

3.63

3.99

5.48

−378.33958 −378.44899 −378.40471 [0.00] −378.46489 −378.38627 −377.54675

2.11 2.14 1.99 [1.40] 2.68 2.65 1.89

3.03 3.18 2.87 [1.20] 4.28 4.15 2.34

3.35 3.56 3.06 [1.45] 4.28 n.a.d 2.60

5.00 4.94 4.42 [3.56] 6.07 6.00 4.47

[0.22] −377.34796 −377.48124 −378.46489

[1.00] 2.03 2.62 1.82

[0.00] 2.51 3.89 2.29

n.a.f 2.96 4.43 2.75

[2.68] 5.00 5.69 4.78

−377.84672

2.75

4.31

4.99

6.02

a

Reference 30 includes ZPC. bReference 47 includes BSSE. cReference 31. dFound to lie on a repulsive surface. eReference 32. fThis conformer was considered an excited state of tTt at this temperature (for details, see ref 32). 11602



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the covalent O−H bond is shorter at 0.969 Å, indicating a stronger O−H bond. The cTt conformer has Cs symmetry, and the disruption of the higher order symmetry results in calculated dipoles of 3.14 and 3.31 D as predicted by B3LYP calculations with aug-cc-pVDZ and 6-311++G(d,p) basis sets, respectively. These values are comparable to the experimentally determined dipole value of 3.073 D.32 The tTt conformer is also predicted to be planar from B3LYP results and belong to the higher symmetry C2h point group. However, the MP2 level of theory with a 6-311++(d,p) basis set predicts a local minima with disruption of planarity (OC−CO dihedrals of 154° and 206°) resulting in C2 symmetry and a calculated dipole of 0.6 D.32 There are two intrafunctional group interactions with O−H···O angles of 74° and interatomic H···O distances of 2.324 Å, nearly identical to the parameters found for the intrafunctional interaction in cTt. Here, the charges on hydrogen are 0.292, the ν(OH) stretching frequencies are 3761 cm−1, and the O−H covalent bonds are 0.970 Å. The cis conformer tCt has two intrafunctional interactions present with H···O interaction distances of 2.334 Å and O− H···O angle of 73°. B3LYP/6-311++G(d,p) calculations predict a deviation from planarity of the OC−CO dihedral (C2 symmetry) with local minima predicted at 40° and 320°. The intrafunctional group interactions appear to be little affected by the rotation from tTt to tCt even though the cis conformation of the dicarbonyl backbone leads to a nonzero dipole of 2.77 D. The charges on hydrogen (0.294) and the O−H covalent bond distance (0.971 Å) are nearly identical to the tTt conformer, while the ν(OH) frequencies are slightly red-shifted compared to tTt at 3740 cm−1. Despite this redshift, the intramolecular interactions are predicted to be slightly attenuated from those in tTt due to steric repulsions of the adjacent carbonyls. The tTt and tCt isomers of oxalic acid have competitive stabilities with differences in energy in this work ranging from 0.38−0.68 kcal mol−1 depending on the computational method, with tCt being higher in energy. The highest energy conformer considered in this work is cCt (Cs symmetry). In this conformation, the intrafunctional interaction has a H···O distance of 2.357 Å, an O−H···O angle of 72°, and ν(OH) of 3753 cm−1, all consistent with the parameters observed in the other intrafunctional interactions. The charge on the free hydrogen of cCt, however, is significantly larger than in the other conformations at 0.323. In this conformation, the interfunctional group interaction is formed with the sp3 oxygen of an OH group instead of the carbonyl oxygen. The interfunctional H···O distance in cCt is 2.034 Å, the shortest interfunctional bond distance of any conformer, while the O−H···O bond angle is 116° as in the cTc conformation and the charge on hydrogen is similar at 0.259. Even though these intramolecular parameters remain little changed from the interfunctional hydrogen bonds in cTc and cTt, the stability of cCt is significantly diminished. Upon further inspection, the ν(OH) frequency of the interfunctional group interaction is 3759 cm−1, while the O−H covalent bond distance is 0.969 Å, both indicating that this interaction is not bound in the same manner as in the interfunctional hydrogen bonds found in cTc and cTt. In the analysis of Chen et al.,31 the interfunctional interaction in cCt is considered to be significantly attenuated compared to the hydrogen bond in cTc and cTt, by over 50%, to 4.1 kcal mol−1! A significantly stronger molecular dipole of 4.97 D results for this conformation.

corrected (ZPC) energies at 0 K and Gibbs free energy changes for these conformers at 298 K from B3LYP/6-311++G(d,p) calculations are also reported. The higher energy cGc isomer (∼14−16 kcal mol−1 less stable compared to cTc) identified theoretically by others29−31 was found to lie on a repulsive potential energy surface when diffuse functions were included as was observed by Chen et al.31 with B3LYP/6-311+(2d,p) calculations and Godfrey et al.32 with MP2/6-311++G(d,p). Results from these higher-level calculations (as opposed to previous HF calculations33) consistently predict the energetic stabilities at 0 K, from most stable to least, as cTc > cTt > tTt > tCt > cCt. From experiment, Nieminen et al.29 estimate that the cTt conformation is ∼7 kJ mol−1 higher in energy than cTc, while Maçôas et al.34 estimate the difference to be ∼10 kJ mol−1 and tTt to be ∼16 kJ mol−1 higher in energy than cTc. Because of the variation in the experimentally derived energy values, it is difficult to say which theoretical approach is most accurate. The relative stabilities can be understood, for the most part, by considering the intramolecular interactions present in these conformations. The most stable configuration, cTc, has a trans configuration with respect to the two dicarbonyl backbone and forms two interfunctional group strained hydrogen bonds35 with O−H···O angles of 116° and H···O hydrogen bond distances of 2.137 Å. Chen et al.31 estimate these hydrogen bonds in cTc (and the one in cTt) to have a strength on the order of 8.4 kcal mol−1 from B3LYP/6-31G(d,p) ZPC calculations. As expected for this conjugated system, the cTc conformer is planar. Because of the C2h symmetry, the local bond dipoles completely negate each other resulting in further stabilization. The unscaled ν(OH) stretching frequencies (see frequency analysis below for details) are red-shifted in comparison to non-hydrogen bound frequencies being 3661 cm−1 (symmetric) and 3664 cm−1 (antisymmetric). The charge on hydrogen is 0.261, significantly reduced from the charge in conformations where the hydrogen is not involved in hydrogen bonding and the O−H covalent bond distance is slightly elongated at a distance of 0.975 Å. In the cTt conformer, one hydroxyl group participates in an interfunctional hydrogen bond with approximately the same O−H···O angle of 117° as in cTc, while the H···O hydrogen bond distance slightly constricts to a value of 2.088 Å. The charge on that hydrogen is attenuated to 0.251 from the value of 0.261 in cTc indicating further electronic shielding for that atom, yet the ν(OH) frequency is slightly higher in energy at 3685 cm−1, and the covalent O−H bond distance is slightly shorter at 0.973 Å, all pointing to an overall dampening of the interfunctional interaction, as found in many two donor−one acceptor three center hydrogen bonding systems. The other OH group does not interact with the vicinal carbonyl group and instead is aligned with the dipole of geminal carbonyl. The H···O distance of 2.336 Å for this intrafunctional interaction in cTt is considerably larger than the interfunctional distances of ∼2.1 Å. The O−H···O angle is 73° and probably cannot be called a hydrogen bond according to the most recent guidelines of the International Union of Pure and Applied Chemistry.35 Nevertheless, this dipole−dipole interaction is quite energetically favorable being estimated to have an interaction strength of 5.7 kcal mol−1.31 A MP2/6-311++G(d,p) calculation reported by Godfrey et al.32 predicts the same type of interaction in formic acid (H2CO2) to be worth 4.8 kcal mol−1. The charge on this hydrogen is much larger (0.295) than found for those interfunctional hydrogen bonds, the ν(OH) stretching frequency is much larger (3750 cm−1), and 11603



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Nieminen et al. in the IR spectroscopy of matrix isolated oxalic acid.29 Similar results from spectroscopic studies of structurally related pyruvic acid38 and glyoxalic acid39,40 had previously been reported. By normalizing the absorption bands with IRinduced changes, the energy difference between the predicted two lowest energy conformers was estimated to be ∼7 kJ mol−1, correlating well with their calculated energies of cTc and cTt. More recent work32 not only confirmed the presence of a cTt conformer for these experimental conditions but additionally performed calculations of Gibbs free energies at elevated temperatures. The authors asserted that, although undetectable in their microwave spectroscopy experiment, the tTt conformer should also be present in considerable quantity under the conditions necessary to vaporize oxalic acid for matrix isolation and electron diffraction experiments (i.e., elevated temperatures ≥ ∼90 °C). The calculated free energies corrected to 458 K from Godfrey et al.32 are reported in Table 1 and, assuming a Boltzmann distribution, lead to mole fractions for conformers tTt, cTc, cTt, and cCt (the authors determine tCt to be an excited state of tTt at this temperature) of 0.46/0.36/0.15/0.02 in the heated nozzle before jet expansion cooling. Godfrey et al.32 (and references therein) have documented the significant cooling that occurs during the jet expansion inducing conformational relaxation when the barrier to the more stable state is less than ∼3 kcal mol−1. For this reason and from reports that population ratios, as dictated by an established thermal equilibrium, can be effectively trapped by the matrix isolation deposition (see, for example, Felder et al.41 and Räsänen et al.42), they conclude the mole fractions present after cooling to be 0.46/0.36/0.18 for conformers tTt, cTc, and cTt. Shortly after the Godfrey study, Maçôas et al. reported the IR induced conformational isomerization of oxalic acid isolated in an argon matrix and characterized by IR spectroscopy,34 which experimentally detected the tTt isomer in addition to the cTc and cTt isomers for the first time. In this work, when corrections for free energy are included for a temperature of 298.15 K, the tTt isomer is predicted to be more stable (ΔG = 1.20 kcal mol−1) than the cTt conformation (ΔG = 1.40 kcal mol−1) relative to the cTc zero reference of free energy, which leads to relative populations of 0.76/0.07/ 0.10/0.07 for the cTc, cTt, tTt, and tCt conformers. This would seem to indicate that, in the troposphere, these isomers may be present in significantly appreciable quantities. However, the work of Nieminen et al.29 report that the observed differences in frequencies and those predicted by theory clearly exclude the tTt and tCt conformers in the matrix isolated IR measurements. The evidence from this work and the spectra from Maçôas et al.34 are consistent with that assertion. At lower temperatures (i.e., matrix isolation and jet-cooled samples) cTc and possibly cTt conformations (depending on the experimental conditions) are present, and at higher temperatures or upon irradiation, the higher energy cTt and tTt (and possibly tCt) conformations should also be found. Clusters with H2O. Monohydrates. The optimized geometries of the monohydrated clusters are presented in Figure 2. Intramolecular and intermolecular interaction distances, O−H covalent bond distances of oxalic acid (in angstrom, Å), and the hydrogen bond angles (in degrees, °) denoted in the figure are from B3LYP/6-311++G(d,p) calculations as are the relative energies, ΔEr, intermolecular interaction energies, ΔE (italics), and free energy changes, ΔG (red). Relative energies, ΔEr, electronic and zero-point corrected intermolecular energies at 0 K, ΔE and ΔE0, and intermolecular enthalpy and free energy

The complete set of covalent bond distances and bond angles for the cTc, cTt, and tTt monomers from B3LYP and MP2 calculations with the aug-cc-pVDZ basis set, along with experimentally derived values and previous calculation for comparison are reported in Table S1 (Supporting Information). With the exception of the O−H bond distance and, to a lesser extent, the COH bond angle, the agreement with electron diffraction experiment for all three isomers is quite good. Significant intramolecular hydrogen bonding takes place in the cTc and cTt monomers but the elongated length of the OH bond and the slightly acute COH angle observed in experiment could imply that theoretical methods underestimate these interactions29 or that another type of interaction is occurring. It has been noted in experimental IR studies36 that removing all water from the commercial dihydrate is difficult; however, our investigation of hydrogen bonding with water (see below) and formation of oxalic acid dimers (not reported) are not consistent with experimentally observed values. Frequencies. Frequencies of the cTc, cTt, and tTt monomers were obtained at both the B3LYP and MP2 levels of theory with the 6-311++G(d,p) and aug-cc-pVDZ basis sets. The results for the cTc are provided in Table S2 (Supporting Information). The cTt modes are provided in Table S3 and the tTt modes in Table S4 (Supporting Information). MP2/6-311+ +G(d,p) results are omitted from Table S4 due to the breakage of symmetry (and poor performance). MP2/4-31G(d) calculations from Nieminen et al. 29 are included for comparison. The larger basis employed in this study provide vibrational frequencies in good agreement with experiment without scaling except for the ν(OH) stretching mode where anharmonicity becomes significant. The scaled values produced from MP2/4-31G(d) calculations are overall in reasonable agreement with experiment being corrected by a factor of 0.97. To obtain agreement with experiment for the OH stretching vibrations in this work, the following scaling factors were applied to the uncorrected values. For B3LYP and MP2 calculations with the aug-cc-pVDZ basis, the scaling values are 0.957 and 0.956, respectively, and the B3LYP and MP2 calculations with the 6-311++G(d,p) basis scaling factors are 0.952 and 0.936. Vibrational frequency analysis has also been reported by Higgens et al.30 at the B3LYP/6-31G(d,p) level of computation scaled by a factor of 0.9614. Their results are consistent with ours here in that the cTc, cTt, and tTt conformers should be clearly discernible due to distinct OH and CO stretching vibrations. Interpretation of Monomer Results. The work of Nahlovska, Nahlovsky, and Strand37 examined gas phase oxalic acid by electron diffraction and infrared (IR) spectroscopy at elevated temperature (ca. 150−160 °C). From the diffraction study, they concluded that oxalic acid possessed C2h symmetry corresponding to a trans dicarbonyl backbone with the cTc and tTt conformations fitting the data equally well. The IR spectroscopy of vapor phase oxalic acid proved experimentally challenging; however, it did produce a clear absorption band at 3475 cm−1, indicative of an intramolecular hydrogen bond, and oxalic acid was therefore considered to be in a cTc conformation. Redington and Redington36 observed both hydrogen bound and unbound OH stretching frequencies, separated by ∼100 cm−1, in the IR spectra of vapor phase and matrix isolated oxalic acid and ascribed them to the cTc and tTt conformations. More recently, the two conformers predicted to be the lowest in electronic energy by higher level theoretical calculations, cTc and cTt, were identified as present by 11604



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Table 2. B3LYP/6-311++G(d,p) Analysis for C2O4H2−H2O Clusters with Unscaled Frequencies in cm−1 and IR intensities in km mol−1 cluster

ν(O−H)

intensity

OHredshift

ΔIR

cTc-I

3200 3594 3290 3671 3337 3758 3262 3741 3346 3750

1302 174 1012 133 888 87 1072 199 1058 94

461 70 460 14 424 3 491 18 339 0

1302 −93 885 35 888 −102 916 258 931 35

cTt-Ia tTt-I cCt-I cTt-Ib

of water. One is an interior region, across which the interfunctional hydrogen bond can form. The other region type is exterior being created by the trans C−C−O−H dihedral (or alternatively, the cis (Z) carboxylic acid group). The cluster with the greatest overall electronic stability is cTc-I with an intermolecular interaction energy of ΔE = −10.61 kcal mol−1 and a free energy change ΔG = 0.00 kcal mol−1. The water molecule is bound to cTc by insertion into an interior binding region of the cTc monomer thus forming two new hydrogen bonds between water and oxalic acid. The COH angle of the acid group bound to water increases from 106° to 115° in accommodation of the intermolecular interactions present in the hydrogen bonded system allowing the O−H···O hydrogen bond angles to increase linearity. The strongest interaction formed in cTc-I is the hydrogen bond created between an acidic OH group of cTc and oxygen of water as evidenced by a short H···O bonding distance of 1.669 Å and very nearly linear O−H···O bond angle of 176°. Concomitantly, the water molecule forms a strained hydrogen bond with the adjacent carbonyl group with a H···O bond distance of 2.038 Å and O− H···O bond angle of 128°. The intramolecular hydrogen bond of cTc that remains appears to be slightly strengthened with a decrease in H···O distance from 2.139 Å to 1.988 Å and increase in O−H···O angle from 116° to 119°. The cTt-Ia cluster is 1.57 kcal mol−1 higher in relative energy, ΔEr, compared to the cTc-I complex. The total intermolecular interaction energy in the binding of water to the non-hydrogen bound exterior carboxylic group is slightly (0.57 kcal mol−1) stronger than found with the cTc-I cluster resulting in a ΔE = −11.18 kcal mol−1 and the ΔG = −0.69 kcal mol−1. Here, the oxygen of the water interacts with the acidic proton of the unbound external acid functional with a H···O hydrogen bond distance of 1.718 Å and O−H···O bond angle of 160°. From the increased hydrogen bond distances to water and smaller hydrogen bonding angles observed, it appears that the single water molecule does not fit as well into this exterior binding region as in the interior region of cTc-I. At the same time, one proton from water binds to the carbonyl group of that same carboxylic group with a 2.254 Å H···O bonding distance and O−H···O strained hydrogen bond angle of 120°. The intramolecular hydrogen bond present in the cTt monomer remains with little alteration of geometric parameters as evidenced by the H···O bond distance of 2.049 Å (very similar to the 2.088 Å distance in the cTt monomer) and the O−H···O intramolecular hydrogen bond angle of 118° (essentially identical to the 117° value for the monomer).

Figure 2. Optimized B3LYP/6-311++G(d,p) structures of the stable rotamers of the oxalic acid monohydrate. Relative energies, interaction energies (italics), and changes in Gibbs free energy at 298 K (red) are reported in kcal mol−1 from B3LYP/6-311++G(d,p) calculations.

changes at 298 K, ΔH and ΔG, are provided in Table S5 (Supporting Information) from B3LYP calculations with both the 6-31+G(d) and 6-311++G(d,p) basis sets. All the hydrated clusters are arranged in order of increasing relative single-point energy as determined from B3LYP/6-31+G(d) calculations. DFT methods have been a popular approach to calculating energies and are still believed to provide very good geometries; however, it has more recently been found to be lacking in the calculation of noncovalent interactions due to a lack of long range electron correlation.43 Wave function ab initio methods, configuration interaction methods, and couple cluster theory, although more computationally expensive, are widely believed to be capable of providing more accurate energies. To examine the reliability of the DFT results, MP2 and QCISD single-point energies were calculated for the B3LYP/6-311++G(d,p) optimized structures and are presented in Table S6 (Supporting Information) for comparison. In a recent study of density functional theory methods, it was found that the B3LYP method underestimated the intermolecular hydrogen bond strengths but was a best performer in calculating relative energies.44 In contrast, in this study, the B3LYP interaction energy values are in excellent agreement with the higher level calculated energies, and relative energies are only reasonably so, similar to the results obtained for the monomers in this study (see Table 1) and previous work.31 The work of Xie et al.45 concluded in the dissociation of the weak acid HF in water that B3LYP/6-311+G(d,p) was the most economical approach to reliable results for energetics, and the diffuse functions on hydrogen were found to be not important. Here, we focus on B3LYP/6-31+G(d) results that are economical as well as those from B3LYP/6-311++G(d,p) calculations, which are known to produce reliable frequencies (with some scaling) and find that the basis set can have profound effects on the equilibrium structure of the acid. One commonly employed method for monitoring hydrogen bonding strength is the magnitude of the redshift of the OH stretching vibration, and the enhancement of IR intensity and those parameters from B3LYP/6-311+ +G(d,p) calculation are reported in Table 2. Five stable monohydrate clusters of oxalic acid are identified in this work from B3LYP/6-311++G(d,p) calculations. In these clusters, there are two region types considered for the binding 11605



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The tTt-I conformer is 2.98 kcal mol−1 higher in relative energy, ΔEr. The intermolecular energy for this cluster, ΔE = −10.80 kcal mol−1, is 0.38 kcal mol−1 weaker than cTt-Ia that still possesses the intramolecular hydrogen bond and 0.19 kcal mol−1 stronger than cTc-I. The change in free energy is ΔG = −0.17 kcal mol−1. The general features of binding in the exterior region are as found in cluster cTt-Ia. However, in hydrogen bonding to water, the H···O bond distance is extended by 0.031 Å to 1.749 Å compared to that distance in cTt-Ia, while the O−H···O bond angle is nearly the same at 159°. Although the hydrogen bond between the acid and water appears to be diminished in strength, the water binding to the carbonyl group of the carboxylic acid has a H···O bonding distance of 2.150 Å, being constricted from the 2.254 Å distance in cTt-Ia, and the O−H···O bond angle of 126° as compared to 120° in cTt-Ia both indicate a strengthening of that interaction. The tCt-I cluster was found to lie on a repulsive surface with B3LYP/6-311++G(d,p) calculations and would optimize to the tTt-I configuration. When the 631+G(d) basis set is utilized, the results for the relative energy and interaction energy are very similar to the same parameters for the tTt-I cluster (for details see Table S5, Supporting Information). The monohydrate cluster cCt-I is 3.56 kcal mol−1 higher in relative energy, ΔEr, and has the largest calculated interaction energy of the monohydrates with an intermolecular energy ΔE = −11.99 kcal mol−1. The change in free energy iss ΔG = −1.23 kcal mol−1. This cluster has two adjacent cyclic hydrogen bonded networks as is found in the cTt-Ia cluster. The intramolecular hydrogen bond between the acidic proton of one carboxylic group and the OH group of the adjacent carboxylic acid group appears to be little affected as evidenced by a slightly shortened hydrogen bonding distance H···O of 2.018 Å, compared to 2.032 Å in the monomer and essentially the same O−H···O angle of 117°. The hydroxyl group that is intramolecularly hydrogen bound through its sp3 oxygen atom is also hydrogen bound to water through its acidic proton. The hydrogen bonding distance to water, H···O, is 1.707 Å, somewhat shorter than the distance found in the tTt-I complex (∼1.75 Å) but similar to the hydrogen bond distance found in cTt-Ia of 1.718 Å that has a similar bonding pattern. The hydrogen bonding angle of 159° is also essentially the same as found in the cTt-Ia complex. The acid bound water forms a strained hydrogen bound with the geminal carbonyl with a H···O distance of 2.212 Å and O−H···O angle of 122°. The least stable monohydrate cluster cTt-Ib has a ΔEr = 4.23 kcal mol−1, an intermolecular energy of ΔE = −8.52 kcal mol−1, and free energy change of ΔG = 1.42 kcal mol−1. In cluster cTtIb, the water binds to the interior region of cTt. Here, the COH bond angle expands from 106° in the monomer to 116°, similar to the change observed for the cTc-I cluster in which water is also bound to the interior region. The strongest hydrogen bond is formed by the acidic proton of the carboxylic group and the oxygen of water. The interatomic H···O distance of this hydrogen bond is 1.730 Å, and the O−H···O angle is nearly linear at 172°. The water is also bound to the adjacent carbonyl with a H···O bond distance of 2.122 Å and O−H···O angle of 123°. Table 2 reports the frequency analysis for the monohydrate clusters of oxalic acid. The OH stretching frequencies that participate in hydrogen bonding are all, in general, significantly red-shifted, on the order of ∼350−500 cm−1, with concomitant enhancement of the IR intensities, both characteristic of strong

hydrogen bonding. Additionally, the acidic OH bonds that are hydrogen bound to water in these clusters are found to have their bond distance extended to nearly 1.0 Å from the monomer values of 0.969−0.975 Å. In general, the elongation of the OH covalent bond distances are inversely related to the H···O interaction distance indicating that, as the hydrogen bond strength increases, the covalent OH bond is accordingly weakened. Correlation of the parameters in Table 2 to the binding energies becomes difficult due to the various intramolecular interactions. For example, although the largest redshift of OH stretching frequency observed in the cCt-I cluster has the strongest interaction energy (491 cm−1 and −11.99 kcal mol−1) and the smallest redshift of OH stretching frequency, cTt-Ib has the weakest interaction energy (339 cm−1 and −8.52 kcal mol−1), and the remaining three monohydrate clusters are only very loosely correlated. The two clusters with the weakest interaction energies and non-negative free energy changes are the clusters where water is bound to the interior binding region, cTc-I and cTt-Ib. The clusters with the strongest interaction with water are the cTt-Ia and cCt-I clusters. Both these clusters bind water to the exterior region and additionally have an intramolecular interaction participating in the hydrogen bonding network. The monohydrate tTt-I also binds water to the exterior region, however, the intramolecular dipole interaction is remote from the intermolecular binding region, and the result is the third weakest interaction energy of the monohydrates and the smallest negative free energy change. Note that the least stable conformation, cCt, has the largest interaction energy and most negative free energy change upon binding a single water molecule. Dihydrates. Ten stable dihydrates of oxalic acid were identified in this study. The optimized geometries of the monohydrated clusters are presented in Figure 3 with relative energies, ΔEr, interaction energies, ΔE (italics), and free energy changes at 298.15 K, ΔG (red), from B3LYP/6-311++G(d,p) calculations. Inter- and intramolecular hydrogen bond distances are denoted in the figure. Hydrogen bonding angles are all found to be relatively linear, with exception to the intramolecular bonding discussed previously. Relative single-point energies, ΔEr, intermolecular energies, ΔE and ΔE0, and intermolecular enthalpy and free energy changes, ΔH and ΔG, from B3LYP calculations with 6-31+G(d) and 6-311++G(d,p) basis sets are provided in Table S7 (Supporting Information). MP2 and QCISD single-point energies were calculated for the B3LYP/6-311++G(d,p) optimized structures and are presented in Table S8 (Supporting Information). Table 3 presents frequency analysis from B3LYP/6-311++G(d,p) calculations for O−H stretching modes (with associated redshifts from monomeric oxalic acid), IR intensities, and changes in intensity (km mol−1), for the dihydrated oxalic clusters, and a cooperativity ratio. This cooperativity factor46 is defined as the ratio of the νOH stretching mode redshift for the H2C2O4− (H2O)n−1 hydrate to the redshift for the H2C2O4−(H2O)n hydrate and is an index for the nonadditive nature of the binding observed. as such a value close to unity would be indicative of an interaction that is relatively unaffected by the binding of an additional water molecule, a value above one signifies a negative effect, while a value below one indicates an enhancement of binding in that cluster. As found for the monohydrate clusters, the OH stretching frequencies are typically found to be red-shifted, while the same enhancement 11606



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of IR intensity is observed for the strong intermolecular hydrogen bonding. Two clusters were found to be the most stable: cTc-IIa and cTt-IIa. In general, the dihydrate clusters with both the waters bound in a single ring are found to be more stable than those with each water in separate cycles. For comparison, the ZPC B3LYP/6-31+G(d) calculations47 have cTt-IIa as the most stable cluster and cTc-IIa as 0.56 kcal mol−1 higher in energy. Cluster cTc-IIa has an interaction energy of ΔE = −20.98 kcal mol−1 and the free energy change of ΔG = 0.06 kcal mol−1. In this cluster, two water molecules bind to one interior pocket of the cTc conformer. In the ring system created by the two waters, the bonding of the carboxylic acid to water appears the strongest, having the shortest bond distance of 1.624 Å. The acid bound water then binds to the second water of the cluster with a H···O distance of 1.772 Å. The second bound water binds to the carbonyl group of cTc through the adjacent carbonyl with a distance of 1.955 Å. The intramolecular hydrogen bond on the opposite side of the cTc monomer remains; however, the bond distance is shorter, from 2.109 Å in the monomer to 1.980 Å. The cooperativity factor for the carboxylic acid bound to water is 0.74 indicating a mild positive cooperation in the binding of these waters for this cluster configuration. Dihdrate cTt-IIa has an intermolecular interaction energy of ΔE = −23.11 kcal mol−1 and has two molecules of water both bound to the external carboxylic acid. For this cluster, ΔG = −1.87 kcal mol−1. The exterior carboxylic acid bonds strongly with water as evidenced by a short bond distance of 1.594 Å. The acid bound water then binds to the second water with a H···O hydrogen bond distance of 1.767 Å. The second water then completes a cyclic network binding to the carbonyl of cTt with a H···O bond distance of 1.894 Å. That same carbonyl also participates in the intramolecular hydrogen bond that has a 2.036 Å interaction distance, shortened from 2.088 Å in the monomer. The cTt monomer is approximately 2 kcal mol−1 higher in energy than cTc, while the intermolecular energies are the same, indicating that the two water interaction with the external acid is a couple of kcal mol−1 stronger than the interior binding region. For this cluster, the intermolecular hydrogen bonding produces a Δν/Δν′ value of 0.57, signifying strong positive cooperative binding. Cluster tTt-IIa has two waters binding the external carboxylic acid group as in cTt-IIa with an intermolecular energy ΔE = −23.45 kcal mol−1 and a resulting ΔEr = 0.70 kcal mol−1. The free energy change is ΔG = −1.38 kcal mol−1. The hydrogen bond between the acidic proton and oxygen of water in tTt-IIa appears strong with an interaction distance of 1.618 Å; however, the interaction distance is slightly elongated as compared to that in cTt-IIa that has an interfunctional hydrogen bond coupling to the acid bound water ring. The second water is bound to the first acid bound water with a H···O distance of 1.749 Å, and the cyclic hydrogen bonded circuit is closed by the binding of the second water to the carbonyl of the acid with a H···O distance of 1.862 Å. The tTtIIa cluster also engages in strong positive cooperative binding as evidenced by the 0.55 cooperativity factor. Cluster tCt-IIa is slightly skewed from true cis with respect to the carbonyl groups and has ΔEr = 0.80 kcal mol−1, intermolecular energy change of ΔE = −23.7 kcal mol−1, and free energy change of ΔG = −3.63 kcal mol−1, the largest negative change in free energy observed for the dihydrates. Both of the waters are bound to an external carboxylic acid

Figure 3. Optimized B3LYP/6-311++G(d,p) structures of oxalic acid dihydrate with select bond distances in Å. Relative energies, interaction energies (italics), and changes in Gibbs free energy at 298 K (red) are reported in kcal mol−1 from B3LYP/6-311++G(d,p) calculations.

Table 3. B3LYP/6-311++G(d,p) Analysis for C2O4H2− (H2O)2 Clusters with Frequencies in cm−1 and IR Intensities in km mol−1; Values in Parentheses Are from Intramolecular Interactions cluster

ν(O−H)

int.

OHred

ΔIR

Δν/Δν′

cTc-IIa

3037 3580 2945 3675 2990 3749 2971 3737 3251 3256 2899 3732 3351 3361 3341 3353 3320 3386 3178 3736

1676 198 1819 100 1702 81 1736 74 90 2390 1940 134 1707 1 1701 0 926 1002 1414 81

624 84 805 10 771 12 777 11 410 408 854 27 410 388 407 396 430 299 507 14

1676 −69 1692 2 1702 −107 1724 −72 90 2123 1784 75 1707 −187 1689 −146 799 904 1287 −17

0.74 (0.88) 0.57 (1.40) 0.55 (0.25) n.a.a n.a.a 1.12 1.13 0.57 (0.67) 1.03 1.09 n.a.a n.a.a 0.93 0.93 0.67 n.a.b

cTt-IIa tTt-IIa tCt-IIa cTc-IIb cCt-II tTt-IIb tCt-IIb cTt-IIb cTt-IIc a

tCt-I lies on a repulsive surface. bUndefined.

11607



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from the interior water has a shorter binding distance of 2.051 Å. Here, similar to the other IIb clusters, the cooperativity factor is close to unity, indicating nearly additive binding; however, in contrast to the cTc-IIb cluster, which has a mild negative binding effect, the values are slightly less than one for cTt-IIb (0.93), indicating a slight positive binding effect. The least stable cluster cTt-IIc has relative energy of ΔEr = 5.08 kcal mol−1, interaction energy of ΔE = −18.04 kcal mol−1, and free energy change of ΔG = 2.60 kcal mol−1. Both waters are bound to the interior binding region. The hydrogen bond between the carboxylic acid proton and the first water is 1.651 Å. That water hydrogen bonds to the second with a distance of 1.749 Å. The second water hydrogen bonds to the adjacent carbonyl of the acid completing the circuit with a distance of 1.945 Å. The cooperativity factor for cTt-IIc is 0.67, indicating strong positive cooperation in binding water in a ring; however, note that the cooperation is not as strong as in the cTt-IIa cluster in which binding is to the exterior acid group as opposed to the interior. Additionally, the clusters that have water bound to an interior region produce positive free energy changes. From the cooperativity ratios calculated, it is clear that the clusters in which two waters are bound into one ring, the most stable clusters with the exception of cTt-IIc, are highly cooperative as evidenced in the cTc-IIa, cTt-IIa, tTt-IIa, and tCt-IIa clusters. Interestingly, the tTt-IIa and tCt-IIa clusters break from planarity to minimize repulsions from the newly formed ring system in these clusters. Conversely, when one water each is bound to two opposed regions of tTt or tCt, the binding is basically additive. The two least stable configurations are cTt-IIb and cTt-IIc. In the cTt-IIb cluster, the waters are each bound to different regions, and even though they are adjacent to each other (both waters are bound to the same carbonyl group), the binding is not cooperative and thus does not compensate for the lack of dipole cancellation observed in the other clusters with individually bound water molecules. The least stable cluster cTt-IIc is found to have highly cooperative binding; however, because of geometric constraints, the bonds are found to be contorted as evidenced by the approximate 55° torsion from the trans configuration. As a side note, the interaction energies for the dihydrates track linearly with the redshifts (half of the interaction energy assigned to each redshift for the IIb clusters that have two separately bound waters) with an R2 value of 0.97. Trihydrates. Ten stable clusters of oxalic acid with three waters were identified in this work. The geometries and select parameters are presented in Figure 4 with relative energies, ΔEr, interaction energies, ΔE (italics), and free energy changes at 298.15 K, ΔG (red), from B3LYP-6-311++G(d,p) calculations. Relative single-point energies, ΔEr, intermolecular energies, ΔE and ΔE0, and intermolecular enthalpy and free energy changes, ΔH and ΔG, are provided in Table S9 (Supporting Information). Table 4 presents the frequency analysis for the trihydrate clusters where the cooperativity factor is the ratio of redshift in trihydrate compared to the corresponding dihydrate. The most stable cluster, cTt-IIIa, was found to have an intermolecular energy of ΔE = −36.44 kcal mol−1 and ΔG = −1.86 kcal mol−1. In this configuration, the three water molecules are arranged (approximately linearly) in a chain connecting the external acid OH group around to the carbonyl forming a cyclic ring structure. As found in the dihydrate, when all the waters are bound in a ring, the cooperativity factor is less than unity. Here, the value is 0.91 as compared to 0.57 in the dihydrate, evidence of significantly

group with the acid to water oxygen distance of 1.615 Å. The hydrogen bond between the two waters has a distance of 1.747 Å, and the back-bond distance to the carboxylic carbonyl group by the second water is 1.874 Å. The relative energies of the tCt and tTt monomers are very similar, as are the binding energies of their dihydrates. Although a tCt-I cluster is not available for calculation of the cooperativity factor, it is expected that the binding behavior of tCt be very similar to that of tTt. The cTc-IIb cluster has one water binding in each of the interior binding regions of cTc. The ΔEr of this cluster is 1.19 kcal mol−1, and the intermolecular energy is ΔE = −19.78 kcal mol−1, while the Gibbs free energy change is ΔG = 0.93 kcal mol−1. The binding here is approximately symmetric with an acidic proton bonding to the oxygen of water with a bonding distance of 1.698 and 1.699 Å. The water also binds to the carbonyl as in the cTc-I cluster, and the hydrogen bond distances are 2.029 and 2.033 Å. In the cTc-IIb cluster, the cooperative factor for the acid groups are 1.12 and 1.13, which are indicating that both groups are experiencing a mild negative cooperative binding effect. A relative energy of ΔE r = 1.37 kcal mol −1 and intermolecular interaction energy ΔE = −24.54 kcal mol−1 is observed for the cCt-II dihydrate cluster. The free energy change is found to be quite large with ΔG = −2.96 kcal mol−1. Here, two waters bind the open carboxylic acid group, while one intramolecular interaction is also present. That H···O interfunctional distance for cCt in this cluster is particularly short at 2.007 Å. The OH group of the external carboxylic acid group binds to water with an H···O distance of 1.585 Å. The hydrogen bonding to the second water has an H···O distance of 1.750 Å, while the second water hydrogen bonds to the carboxylic acid carbonyl with an H···O bond distance of 1.893 Å. As observed for the monohydrated clusters, the cCt conformation produces the largest interaction energy, and the cooperativity factor is 0.57 as cCt-II experiences a strong positive binding effect, just as in the other clusters in which the two waters are bound in a ring to an exterior acid group. The tTt-IIb and tCt-IIb clusters are very similar with the exception of the dicarbonyl dihedral. In these clusters, one water molecule is bound to each of the exterior binding sites. The hydrogen bond distance between the acidic proton and water oxygen are 1.761 Å for tTt-IIb and 1.760 Å for tCt-IIb. The relative energies for tTt-IIb and tCt-IIb are 2.65 and 2.86 kcal mol−1, respectively, intermolecular energies are ΔE = −21.50 kcal mol−1 and −21.67 kcal mol−1, and the changes in free energy are ΔG = 0.31 kcal mol−1 and ΔG = −1.32 kcal mol−1, respectively. The hydrogen bond between the water and the carbonyl were both found to be 2.109 Å for tTt-IIb, and for tCt-IIb, the distances were 2.109 and 2.114 Å. For the tTt-IIb cluster, the cooperative factors are 1.03 and 1.09 as the binding is essentially additive. The dihydrate cluster cTt-IIb has ΔEr = 4.44 kcal mol−1, ΔE = −18.68 kcal mol−1, and ΔG = 1.71 kcal mol−1. Here, one water is bound to an exterior carboxylic acid group, and additionally, one water binds to an interior binding site. The two hydrogen bonds involving the acidic proton of acid to water seem to both be of similar strength, evidenced by the 1.729 Å hydrogen bond of water to the exterior carboxylic acid and a 1.749 Å hydrogen bond to water for the interior region. The remaining two hydrogen bonds in this system are not similar with the water in the exterior carboxylic binding site creating a 2.265 Å hydrogen bond distance, while, in the interior hydrogen bonded system, the distance to the carbonyl 11608



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Table 4. B3LYP/6-311++G(d,p) Frequency Analysis for C2O4H2−(H2O)3 Complexes with Frequencies in cm−1; Values in Parentheses Are from Intramolecular Interactions complex

ν(OH)

ν(OH)red

·2H2Ored

Δν/Δν′

cTt-IIIa

2863 3675 3031 3343 2949 3570 2996 3328 2782 3730 3107 3228 2953 3755 2917 3740 2989 3426 3082 3728

887 10 730 418 712 94 752 420 977 23 554 436 808 6 931 8 770 327 603 22

805 10 710 12 624 84 777 11 954 27 624 84 771 12 777 12 854 27 507 14

0.91 (1.04) 0.97 n.a.a 0.88 (0.89) 1.03 n.a.a 0.98 (1.16) 1.13 n.a.a 0.95 (2.00) 0.94 (1.50) 1.11 n.a.a 0.84 (0.64)

tTt-IIIa cTc-IIIa tCt-IIIa cCt-IIIa cTc-IIIb tTt-IIIb tCt-IIIb cCt-IIIb cTt-IIIb a

Previously not intermolecularly bound.

to the external acid group has already been shown to be particularly stable due to positive cooperation, and the interatomic distances for the two water ring in the tTt-IIIa cluster are very similar to that of tTt-IIa. The bonding observed with one water on the other acid group is nearly indistinguishable from the bonding in tTt-I. The cooperativity ratio for tTt-IIIa is nearly unity, 0.97, indicating the binding in this cluster to be additive as therefore the additional stability is due to the partial cancellation of local dipoles. The cTc-IIIa cluster is found to have ΔEr = 0.26 kcal mol−1, intermolecular energy of ΔE = −31.25 kcal mol−1, and ΔG = −0.32 kcal mol−1. The three waters are all bound on one interior region of the internal acid configuration with the other acid group being intramolecularly hydrogen bound. Here, the intramolecular bond distance is slightly shorter, 1.979 Å, and the cooperativity factor calculated for the intramolecular interaction indicates a slightly positive bonding effect. The hydrogen bond from the acidic proton to the first water has a short bonding distance of 1.592 Å. The connection to the next water in the ring is 1.734 Å, very similar to the distance connecting the second water to the third of 1.773 Å. The final hydrogen bond connecting the third water to the carbonyl of the adjacent acid group is 1.877 Å. The binding of all three waters in a ring to cTc has a positive cooperative effect as evidenced by the Δν/Δν′ value of 0.88. The tCt-IIIa cluster has bonding parameters that are essentially indistinguishable from the tTt-III cluster with the exception of the OC−CO dihedral angle. The relative energy is ΔEr = 0.63 kcal mol−1, intermolecular energy change is ΔE = −34.43 kcal mol−1, and ΔG = −3.72 kcal mol−1. As found for the tTt conformer, the binding of two water molecules to one exterior acid group and one to the other results in a stronger interaction energy than that for the binding of all three waters in a ring. The bonding in the two water ring appears to be little affected by the binding of the opposing water molecule (Δν/Δν′ = 1.03) similar to tTt-IIIa.

Figure 4. Optimized B3LYP/6-311++G(d,p) structures of oxalic acid trihydrate with select bond distances in Å. Relative energies, interaction energies (italics), and changes in Gibbs free energy at 298 K (red) are reported in kcal mol−1 from B3LYP/6-311++G(d,p) calculations.

diminished positive cooperation in binding. The hydrogen bond distance to the first water is 1.565 Å, the second 1.691 Å, to the third 1.747 Å, and the final hydrogen bond connecting the third water to the carbonyl of the carboxylic acid moiety is 1.855 Å. In general, the strength of hydrogen bonding seems to decrease as we follow the circuit around the ring if one correlates the hydrogen bond distance with interaction energy. The second most stable cluster with 6-31+G(d) methods was found to be tTt-IIIa with a ΔEr = 0.57 kcal mol−1 and intermolecular energy ΔE = −34.11 kcal mol−1 according to B3LYP/6-311++G(d,p). The change in free energy for this cluster was found to be ΔG = −1.30 kcal mol−1. The water binding in this system comprises two waters bound to one external acid group and one water binding to the opposite external acid. Although the cooperativity ratio is greater than one for this cluster, indicating a small negative effect for this binding motif, based on the overall interaction energies, the cancellation of local dipoles appear to impart significant additional stability to the cluster. The binding of two waters 11609



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Figure 5. Optimized B3LYP/6-311++G(d,p) structures of the stable oxalic acid tetrahydrate complexes. Relative energies, interaction energies (italics), and changes in Gibbs free energy at 298 K (red) are reported in kcal mol−1 from B3LYP/6-311++G(d,p) calculations.

1.703 Å and then 1.744 Å. The final bond completing the cyclic hydrogen bond network is 1.836 Å. Here, ΔEr = 1.12 kcal mol−1 with a calculated ΔE = −33.56 kcal mol−1. The same three water ring binding to an external carboxylic acid group is exhibited in tCt-IIIb. The dihedral angle has little effect on the binding geometries and distances when compared to cluster tTt-IIIa. The resulting ΔEr and ΔE values are 1.46 and −33.61 kcal mol−1, respectively. The cooperativity in binding of the three waters in one ring for the tTt and tCt conformers is, perhaps, slightly positive with Δν/Δν′ values of 0.95 and 0.94, respectively, albeit not as positive as exihibited by the cTt-IIIa and cTc-IIIa clusters. The remaining two trihydrate clusters are much higher in relative energy compared to the preceding clusters. The cCtIIIb and cTt-IIIb clusters have ΔEr values of 4.78 and 5.03 kcal mol−1, respectively. The intermolecular energies, however, are found to be rather competitive ΔE = −31.66 and −28.61 kcal mol−1. In the cCt-IIIb cluster, two waters are bound to an external carboxylic acid group, while one water binds across the adjacent interior region. The interatomic bonding distances are increasing, 1.621, 1.735, and 1.873 Å as one follows the acidic proton through the two water network to the geminal carbonyl. The interior bound water has moderate hydrogen bond distances of 1.789 Å to the acid and 1.621 Å to the adjacent hydroxyl group of the exterior acid group. The cooperativity of binding in this cluster is slightly negative with a Δν/Δν′ of 1.11. The cTt-IIIb cluster has all three waters bound to the inner binding pocket with hydrogen bond distances of 1.626, 1.709, 1.757, and 1.881 Å as one follows the acid proton through the water molecules to the vicinal carbonyl. As found in all the trihydrate clusters with the waters bound in a ring, the

Trihydrate cCt-IIIa has similar binding as the most stable cTt-IIIa cluster with the exception of the cis OC−CO dihedral, causing the intramolecular hydrogen bond to be with the adjacent acid hydroxyl group instead of the carbonyl. The hydrogen bonding distance is similar, 2.004 Å, slightly tighter than 2.033 Å in cTt-IIIa. The relative energy of this cluster is ΔEr = 0.87 kcal mol−1 with an intermolecular energy of ΔE = −35.58 kcal mol−1. As in the cTt-IIIa and cTc-IIIa clusters (most stable), the binding of the waters are all in one ring; however, the cooperativity factor is essentially unity (Δν/Δν′ = 0.98), while producing the second largest interaction energy (within one kcal mol−1 of the cTt-IIIa interaction energy), whereas the cCt-II cluster produced the largest magnitude of interaction energy for the dihydrated clusters. Cluster cTc-IIIb has one water bound to the interior binding pocket with acidic proton to water oxygen distance of 1.690 Å and the water to carbonyl distance of 1.989 Å, similar to the cTc-Ia monohydrate. Two waters are bound to the other interior binding pocket with a short 1.660 Å hydrogen bond between the acidic proton and the first water. The water to water hydrogen bond distance is 1.786 Å, and the bond of the second water to the neighboring carbonyl is 1.927 Å. The resulting ΔEr = 1.16 kcal mol−1, and the intermolecular energy is ΔE = −30.34 kcal mol−1. The cooperativity factor for cTcIIIb is 1.13, which is denotive of negative cooperation for the two water bound ring upon binding of one more water to the opposing interior region of cTc. All three waters are bound to an external carboxylic acid group forming a cyclic ring in the tTt-IIIb trihydrate. The hydrogen bond between the first water and the acid is strong as evidenced by a 1.595 Å. The bonds to the adjacent waters are 11610



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interaction with water as evidenced by the ν(OH) redshift of 1834 cm−1, which produces the most positive cooperative binding (Δν/Δν′ = 0.48) exhibited for water bound ν(OH). The O···H hydrogen bond distance is a staggering 1.37 Å, while the covalent O−H bond is extended to 1.09 Å. The overall interaction energy for the cTt-IVa cluster is −44.03 kcal mol−1. Clusters of oxalic acid with 2 + 2 water clusters, cTc-IVa, cTtIVb, and tTt-IVc, are also found to be particularly stable. Note that, for cTc-IVa, the cooperativity factor indicates a strong negative effect (Δν/Δν′ = 1.47) on the acid binding; hence, the resulting stability of the cluster, ΔE = −40.88 kcal mol−1, would presumably be attributed to the cooperative stability of the 2 + 2 water clustering. Similarly, the cTt-IVb cluster has Δν/Δν′ of 1.31 with a ΔE = −42.82 kcal mol−1. Conversely, the tTt-IVc cluster (ΔE = −43.56 kcal mol−1) has a positive cooperative factor of 0.91. The clusters cCt-IV, tCt-IVb, and tTt-IVe have the four waters bound in a linear chain forming a large hydrogen bonded ring system on an exterior carboxylic acid group. For the cCt-IV cluster, the binding is cooperative with a Δν/Δν′ of 0.85. The cTc-IVd cluster was found to be least stable, and although it has four waters in a linear chain, they stretch from the interior binding region to the backside of the opposing carboxylic OH group through the sp3 oxygen. Pentahydrates. Optimized geometries of the 12 stable clusters of oxalic acid with five waters are presented in Figure 6 with relative energies, ΔEr, interaction energies, ΔE (italics), and free energy changes at 298.15 K, ΔG (red), from B3LYP-6311++G(d,p) calculations. Relative energies, ΔEr, intermolecular energies, ΔE and ΔE0, and intermolecular enthalpy and free energy changes, ΔH and ΔG, from B3LYP calculations with both 6-31+G(d) and 6-311++G(d,p) basis sets are provided in Table S11 (Supporting Information). The cluster cTt-Va, is found to spontaneously ionize into a solvated hydrogen oxalate−hydronium contact ion pair, as reported by Ramon et al.47 In this cluster, the deprotonated oxygen is separated from the proton of nascent hydronium ion by a distance of 1.41 Å, while the new O−H bond of hydronium experiences somewhat of a back interaction evidenced by a O− H covalent bond length of 1.06 Å. Although ionization typically results in a tremendous stabilization when compared to neutral clusters, here that is not the case. The neutral clusters tTt-Vb, cTt-Vb, and tTt-Va are all found within ∼0.5 kcal mol−1 of each other, and the ionic cluster cTt-Va is not necessarily the most stable due to the basis set dependence of the energy. The remaining clusters are all found to be neutral. As found in the tetrahydrate clusters, dipole cancellation leads to particular stability in these clusters. Additionally, the multiring hydrogen bonded networks, cTt-Va, cTt-Vb, tTt-Vb, tCt-Vb, tCt-Vc, and tCt-Vd, provide exceptional stability. Note from Table 6 that, with the exceptions cTt-Va that ionizes and cTt-Vb with additive binding, these clusters have strong positive cooperation identified by the low Δν/Δν′ values. For instance, the tTt-Vb has a Δν/Δν′ value of 0.50. The clusters cTt-Vc and tTt-Vc have oxalic acid binding to a cyclic five water ring system with cooperativity ratios of 0.51 and 0.72, respectively. However, the cCt-V cluster has the five waters bound to the internal acid bridging to the adjacent carbonyl in a chain in a manner that is decisively additive. Hexahydrates. A total of 24 clusters of oxalic acid with six waters were characterized. The geometries are presented in Figure 7 along with relative energies ΔEr, intermolecular energies, ΔE (italics), and free energy changes, ΔG (red), from B3LYP/6-311++G(d,p) calculations. Relative energies, ΔEr,

cooperativity is positive, and for cTt-IIIb, it is moderate at Δν/ Δν′ of 0.84. Tetrahydrates. The optimized geometries of the 16 stable clusters of oxalic acid with four waters are presented in Figure 5 with relative energies, ΔEr, interaction energies, ΔE, and free energy changes at 298.15 K, ΔG, from B3LYP-6-311++G(d,p) calculations (Table 5). Relative energies, ΔEr, intermolecular Table 5. B3LYP/6-311++G(d,p) Frequency Analysis for Select C2O4H2−(H2O)4 Complexes with Frequencies in cm−1; Values in Parentheses Are from Intramolecular Interactions complex

ν(OH)

ν(OH)red

·3H2Ored

Δν/Δν′

tTt-IVa

3029 3005 2995 3021 1916 3659 3176 3605 2968 3349 3072 3349 2983 3353 3017 3205 2604 3717 3088 3092 2875 3759 2767 3739

732 756 753 727 1834 26 485 60 793 412 678 412 765 395 644 459 1155 36 573 572 886 2 981 9

730 418 752 420 887 10 712 94 808 6 887 6 831 8 712 94 977 23 554 436 808 6 831 8

1.00 0.55 1.00 0.58 0.48 (0.38) 1.47 (1.57) 1.02 n.a.a 1.31 n.a.a 1.09 n.a.a 1.11 n.a.a 0.85 (0.64) 0.97 0.76 0.91 (3.00) 0.85 (0.89)

tCt-IVa cTt-IVa cTc-IVa tTt-IVb cTt-IVb tCt-IVb cTc-IVb cCt-IV cTc-IVc tTt-IVc tCt-IVc a

Previously not intermolecularly bound.

energies, ΔE and ΔE0, and intermolecular enthalpy and free energy changes, ΔH and ΔG, from B3LYP calculations with 631+G(d) and 6-311++G(d,p) basis sets are provided in Table S10 (Supporting Information). All of these clusters are found to be neutral, i.e., oxalic acid does not dissociate. Note that the relative energies are not as close to each other in magnitude as found for the smaller clusters due to the interaction energies becoming increasingly divergent. In these clusters the importance of dipole cancellations becomes more apparent when waters bind on opposing sides of the acid. For example, the two most stable tetrahydrate clusters are tTt-IVa and tCtIVa in which two waters bind each (opposing) exterior acid group. The cooperativity of the (particularly stable) two water bound ring is not affected by the incorporation of the second water to the opposing site for these conformers (Δν/Δν′ = 1.00 for both). However, the binding of the second water to the exterior acid has strong positive cooperation with Δν/Δν′ values of 0.55 and 0.58, respectively. Also, the cancellation of dipoles in these clusters introduces further stabilization. The overall interaction energies for tTt and tCt IVa clusters are −46.78 and −47.10 kcal mol−1. Other stable clusters display a bicyclic hydrogen bonded ring system such as cTt-IVa, tTt-IVe, and tTt-IVg. The cTt-IVa cluster has a particularly strong 11611



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Figure 6. Optimized B3LYP/6-311++G(d,p) structures of the stable oxalic acid pentahydrate complexes. Relative energies, interaction energies (italics), and changes in Gibbs free energy at 298 K (red) are reported in kcal mol−1 from B3LYP/6-311++G(d,p) calculations.

intermolecular energies, ΔE and ΔE0, and intermolecular enthalpy and free energy changes, ΔH and ΔG, from B3LYP calculations with 6-31+G(d) and 6-311++G(d,p) basis sets are provided in Table S12 (Supporting Information). Many configurations for the binding of these waters to oxalic acid were explored and are too numerous and varied to mention. As found in the pentahydrate clusters, the majority of the clusters are competitively stable and neutral. Two clusters were found to spontaneously ionize: cTt-VIb and cTc-VIa. From B3LYP/631+G(d) calculations, these clusters are 1.66 and 5.05 kcal mol−1 less stable than the neutral cluster cTt-VIa. Interestingly, from B3LYP/6-311++G(d,p) calculations, cTt-VIb is the most stable cluster; however, the neutral cluster tTt-VIa is only 0.03 kcal mol−1 less stable, essentially iso-energetic. The 6-311+ +G(d,p) basis set predicts for the cTt-VIa cluster to be less stable by 3.55 kcal mol−1. Interpretation of Hydration Results. The incremental binding energies from B3LYP/6-31+G(d) calculations for the cTc, cTt, tTt, tCt, and cCt conformers with their most stable configurations with up to six waters is presented in Table 7. Although cTc is the most stable conformer of oxalic acid, the clusters in which binding of water is to the exterior acid as opposed to an interior region experience somewhat stronger interactions with water. As a result, when the number of water molecules are increased, the clusters with cTc at their core become increasingly less stable compared to the other clusters. The clusters of cTc with up to three waters that are most stable

Table 6. B3LYP/6-311++G(d,p) Frequency Analysis for C2O4H2−(H2O)5 Complexes with Frequencies in cm−1; Values in Parentheses Are from Intramolecular Interactions complex

ν(OH)

ν(OH)red

·4H2Ored

Δν/Δν′

cTt-Va

n.a.a 3619 2999 3063 1846 3632 2962 3028 2429 3669 2995 3071 1997 3753 2529 3740 2604 3717 2616 3745

n.a.a 66 749 685 1904 53 799 733 1321 16 666 593 1764 8 1232 21 1155 36 1132 3

887 10 753 727 1834 26 732 756 678 13 554 436 886 2 886 2 1155 36 831 8

n.a.a (0.15) 1.01 1.06 0.96 (0.49) 0.92 1.03 0.51 (0.81) 0.83 0.74 0.50 (0.25) 0.72 (0.10) 1.00 (1.00) 0.73 (2.67)

tCt-Va cTt-Vb tTt-Va cTt-Vc cTc-V tTt-Vb tTt-Vc cCt-V tCt-Vb a

Complex cTt-Va is dissociated into H3O+−HC2O4−. 11612



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Figure 7. Optimized B3LYP/6-311++G(d,p) structures of the stable oxalic acid hexahydrate complexes. Relative energies, interaction energies (italics), and changes in Gibbs free energy at 298 K (red) are reported in kcal mol−1 from B3LYP/6-311++G(d,p) calculations

Table 7. Incremental Binding Energies in kcal mol−1 for the Most Stable Clusters of C2O4H2−(H2O)n from B3LYP/631+G(d) Calculations n

cTc

cTt

tTt

tCt

cCt

1 2 3 4 5 6

11.6 11.0 11.1 11.3 10.7 11.8

11.9 12.7 14.2 8.8 15.4 9.2

11.7 13.6 11.6 13.7 10.1 11.3

11.8 13.7 11.5 13.8 10.6 10.7

12.9 13.3 11.8 11.4 10.7 10.3

increase in binding energy, i.e., that six water dissociated structures are not more stable than the neutral ones. Although the nature of an acid is to simply protonate water, it can be complex in mechanism and dependent on the solvation of the hydrated protons.48 An example of this can be found in the theoretical work on the hydration of sulfuric acid49 where initially the poorly solvated ionized clusters are of competitive stability with hydrogen bound ones and become more stable as they are properly supported by a hydrogen bonding water network. The initial binding of one water to the exterior region of cTt in cluster cTt-Ia is found to be significantly stronger than for the interior region as in cTt-Ib. The binding of the second water is found to have an increase in binding energy large enough to make the cTt-IIa cluster as stable as the cTc-IIa cluster. In the cTt-IIa complex, two waters are bound in a single ring to the exterior region. This same binding pattern is also observed in cTt-IIIa, the most stable trihydrate cluster, where again the waters are bound in a single ring. This cluster has an especially strong incremental energy increase as reported in Table 7. In the most stable tetrahydrate cluster with a cTt core, cTt-IVa, two waters are bound to an exterior acid group and the remaining two waters from a ring with the two water plus acid cycle and the oxygen of the OH acid bond of that same exterior acid. The cTt-Va cluster has the largest incremental increase in binding energy of all the clusters examined here in this work. In this cluster, a contact ion pair spontaneously forms enhancing

all have a single ring of waters bound across an interior region of the acid being a highly cooperative configuration. In the most stable four water cluster, a 2 + 2 water tetramer is bound to an interior region, while the remaining acid functional intramolecularly hydrogen bonds the adjacent carbonyl. There is a slight drop in binding energy when five waters are bound in cTc-V, as that cluster accommodates three waters to one interior region and the two on the other side. However, the experienced destabilization is recovered in the binding of the sixth water where three waters are bound to each interior region of cTc in an approximately symmetric manner producing the greatest incremental interaction energy change observed for the cTc hydrate clusters. In the six water cluster cTc-VIa, oxalic acid spontaneously dissociates forming an ionic cluster. The nascent ionic cluster only produces a typical 11613



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Figure 8. Gibbs free energy changes in kcal mol−1 at 298.15 K for the hydrated clusters of the cTc, cTt, tTt, and cCt conformations of oxalic acid from B3LYP/6-311++G(d,p) frequency calculations.

binding energy. In the majority of clusters with tTt and tCt cores, the binding initiates various degrees of twisting from planarity to minimize electrostatic repulsions, the barrier to rotation of the C−C σ bond being less than 1 kcal mol−1. Disruption of planarity was observed in the optimization of geometry for the tCt monomer, which resulted in a OC− CO dihedral angle of 40° and C2 symmetry by B3LYP/6311++G(d,p) calculations, and presumably is the reason for the repulsive surface in the binding of tCt to one water. Note that the binding of one water to planar tTt monomer resulted in an OC−CO dihedral of 157°. The most stable dihydrates of these two conformers, tTt-IIa and tCt-IIa, have two waters bound cooperatively in a ring to an exterior acid group, while the other acid group undergoes a torsion of the C−C bond, at the same time maintaining the strong dipole−dipole intrafunctional group interaction. In the symmetric clusters tTt-IIb and tCt-IIb, one water binds on each end of the conformer, and the dihedral angle of the dicarbonyl is planar. The most stable three water clusters for tTt and tCt have two waters bound in a ring with one carboxylic acid group and one water bound to the other acid group. The degree of C−C σ bond torsion observed in these clusters is pronounced in response to the larger binding system. In the tTt-IIIa cluster, the dihedral of tTt dicarbonyl shifts to a value of 124°, and in tCt-IIIa, the OC− CO dihedral is nearly perpendicular at 83°! In the competively stable IIIb clusters (approximately one kcal mol−1 less stable), tTt and tCt bind all three waters in one ring. Here, we see dipole cancellation trump the cooperativity of linear ring binding with respect to overall stability introduced to the cluster. The planarity is again disrupted but not to the extent in the more stable IIIa clusters, τ = 148° for tTt-IIIb and τ = 50° for tCt-IIIb. In the binding of four waters to oxalic acid, the tTt and tCt conformers produce the most stable clusters as

the stability of the cluster. The geometry of this cluster has water bound to an exterior acid group with two waters on each side of that water creating a two ring cycle across the ionized carboxylate group. The remaining interfunctional hydrogen bond also coordinates to an oxygen of that nascent carboxylate group. There are a total of three stable pentahydrated clusters of cTt reported here, all within 1.5 kcal mol−1 of each other in relative energy. In cTt-Vb, there are two waters bound in a cycle to an exterior acid group that is being coordinated by the interfunctional hydrogen bond present. The remaining three waters bridge in a ring to the carbonyl group furthest away; the carboxylic group intramolecularly bound to the exterior acid group. The remaining cTt-Vc structure has a pentameric water ring bound to the exterior acid group of cTt. The cTt-VIa and cTt-VIb clusters are similar in stability, the answer to which one is more stable depends on the basis set employed, even though in cTt-VIa, the acid does not dissociate to form an ionic core and in cTt-VIb, it does ionize. This is a similar result as found with the cTc-VIa cluster that is also predicted to ionize and yet produce no additional stabilization compared with the neutral clusters. Three additional clusters are found to have similar stabilities for the hexahydrate cTt clusters. The tTt and tCt conformers are found to bind water quite similarly. These clusters bind one water to their exterior acid group with approximately the same strength as the cTt conformer. The binding of these conformers with two waters to their exterior acid group is highly cooperative and particularly strong as evidenced by the low cooperativity ratios discussed above and the large increases in incremental binding energy for n = 2 and 4 (Table 7) in which two waters are bound to each acid group. Note that the incremental binding energy for the most stable six water clusters, in which three waters are bound in a ring to each acid group, have only typical increase in 11614



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considered without diffusion and polarization function on hydrogen, nearly all (>93%) have a calculated negative free energy value at 298 K. The exceptions are the cTc-IVd, cTt-IIc, and cTt-IIIb clusters, which are found to have ΔG values >1 kcal mol−1 with the 6-31+G(d) basis. However, with the 6311++G(d,p) basis, nearly half (∼45%) of the clusters are predicted to have positive free energies (including the 0.00 free energy of the cTc-Ia and cCt-IIIb clusters). Interestingly, PCM calculations50 predict the relative populations in aqueous solution for cTc, cTt, tTt, tCt, and cCt conformers to be 0.597, 0.237, 0.080, 0.069, and 0.001, respectively. Compared to gas phase B3LYP/6-311++G(d,p) free energies for a temperature of 298 K with relative populations of 0.760, 0.071, 0.100, 0.066, and 0.002, a significant stabilization of the cTt conformer in bulk water is predicted. It is interesting to find the actual concentrations of the various hydrated clusters of oxalic acid under a given realistic atmospheric condition. The standard free energies ΔG give the equilibrium constants Kn (at 298.15 K) for the formation of the clusters from the respective monomers

a direct result of the large magnitude interaction energies. In tTt-IVa and tCt-IVa clusters, two waters bind in a ring to each carboxylic acid group, which allows for maximization of cooperative effects and dipole stabilization. In these clusters, the dicarbonyl dihedral experiences a torsion to τ = 114° for tTt and τ = 69° for tCt. Clusters tTt-IVb and tCt-IVb bind three waters to one carboxylic group and one to the other with a resultant energy that is ∼2.5−3 kcal mol−1 higher in energy. Clusters with all four waters bound in one ring, tTt-IVc and tCt-IVc, were also identified but some 3 kcal mol−1 less stable than the IVa clusters. The most stable clusters for tTt and tCt with five waters have three waters bound in a ring to one exterior acid group and two in a ring to the other acid group. In the six water clusters of tTt and tCt, the most stable have three on each side; however, the clusters with four waters on one side and two on the other are very competitive in stability. As these hydrated clusters of oxalic acid grow, the majority of stable clusters are produced with the more symmetric conformations tTt and tCt for which dipole cancellations can increase stability. Binding of one water molecule is most powerful with the cCt monomer due to the increased deshielding of the exterior acidic proton. Binding of the second water, with both waters in one ring, is even slightly stronger due to cooperativity, yet not as strong as in the two water ring systems of tTt and tCt. The stable clusters of cCt with three through six waters are all found to have the waters arranged in one ring to the exterior acid group. As can be seen from Table 7, the incremental binding to these waters becomes progressively weaker. Figure 8 plots the calculated free energy changes for the hydrated clusters of the conformers cTc, cTt, tTt, and cCt as determined by B3LYP/6-31+G(d) and B3LYP/6-311++G(d,p) methods. Although omitted, the tCt results are very similar to the tTt results. A striking feature of these calculations is the discrepancy predicted from the 6-31+G(d) versus 6-311+ +G(d,p) basis sets. For the cTc cluster, the difference in value for the free energy change for the two basis sets are dramatic. With the 6-31+G(d) basis, ΔG values are found to be negative, while for the 6-311++G(d,p) basis, the resulting values are positive! For cTc, the most stable conformation, although different growth pathways are possible, appears that the formation of the tetrahydrate corresponds to the most entropically prohibitive step in formation of higher order water clusters. For the cTt conformation, the initial waters are binding in a less favorable fashion than the larger clusters. Note that a greater number of stable clusters were identified with this conformer than for cTc, and additionally, the free energy values are more favorable. The clusters of water with the tTt conformation produce a multitude of stable configurations, and for the majority of clusters, both basis sets predict large negative free energy changes. Although not presented in the figure, tCt produces similar behavior to the tTt cluster producing many spontaneously forming clusters. The cCt conformation also appears to readily bind with water by both methods, although, in comparison to tTt and cTt, the number of stable clusters is fewer. Additionally, note that the larger clusters are less favored than the initial clusters. Overall, the conformations of oxalic acid that have an exterior carboxylic acid group produce more favored clusters with water. Considering the relative populations from previous discussion, it appears that the cTc, cTt, tTt, and tCt conformers should all be important in the initial hydration steps of oxalic acid. The majority of all clusters considered here in this work have calculated negative ΔG value at 298.15 K. For the clusters

ΔG = −RT ln(K n)

(1)

For reaction 2, the equilibrium constant Kn is defined by eq 3 C2O4 H 2 + nH 2O ↔ C2O4 H 2 ·nH 2O Kn =

[C2O4 H 2 ·nH 2O] [H 2O]n [C2O4 H 2]

(2)

(3)

where [C2O4H2], [C2O4H2·nH2O], and [H2O] are vapor pressures of oxalic acid, hydrated cluster, and water vapor, respectively. The saturated vapor pressure of water at 25 °C is 3.17 kPa (i.e., 100% relative humidity). With the vapor pressure of water and the equilibrium constant, it is allowed to calculate the relative population fraction (RPF), which is defined as RPF ≡

[C2O4 H 2 ·nH 2O] = K n[H 2O]n [C2O4 H 2]

(4)

These calculations were performed for all clusters. Only the mono- and dihydrated cluster fractions reported as the larger clusters with more waters are essentially zero. Observed concentrations of oxalic acid typically range from 0.2−1 ppb (rural) or 1−5 ppb (urban) and as large as 15 ppb during episodes. To access actual concentrations of hydrated clusters, we assumed a 5 ppb of oxalic acid total. From the relative populations of the monomers (see above) calculated using the Boltzmann distribution and the equilibrium constants Kn using ΔG values at 298.15 K, the RPF values and the concentrations of the hydrated clusters in the atmosphere are estimated in Table 8. Of course, the situation in the atmosphere is much more complicated, and this is a limited and simplistic approximation. However, we feel it is useful to get a feel for how important the various clusters are in their formations. From the table, we can see that, although the higher energy forms of oxalic acid are often more effective binders of water, their low populations of monomer conformation make them a small to negligible contribution to the atmospheric population, while the more stable configurations do lead to clusters. It must be considered that the thermal equilibrium concentrations may have no correlation to actual populations due to the formation mechanism(s). From these results, it seems clear that, considering the relative humidities encountered in the 11615



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Table 8. Gibbs Free Energya Values in kcal mol−1, Relative Bound Percentsb (RPF × 100%), Adjusted Bound Percents,c and Estimated Concentrations (Assuming a Total Oxalic Acid Concentration of 5 ppb) for the Most Stable Clusters of C2O4H2−(H2O) and C2O4H2−(H2O)2 (For Details, See Text) cluster

ΔG

rel. %

adj. %

ppt

cTc-I cTt-Ia tTt-I cCt-I cTt-Ib cTc-IIa cTt-IIa tTt-IIa tCt-IIa cTc-IIb cCt-II tTt-IIb tCt-IIb cTt-IIb cTt-IIc

0.00 −0.69 −0.17 −1.23 1.42 0.06 −1.87 −1.38 −3.63 0.93 −2.96 0.31 −1.32 1.71 2.60

3.17 10.17 4.23 25.32 0.29 0.09 2.37 1.04 46.25 0.02 14.92 0.06 0.94 0.01 0.00

2.41 0.72 0.42 0.05 0.02 0.07 0.17 0.10 3.05 0.02 0.03 0.01 0.06 0.00 0.00

120.6 36.1 21.1 2.5 1.0 3.5 8.4 5.2 152.6 0.8 1.5 0.3 3.1 0.0 0.0

acid, the clusters with the water in cyclic rings are found to be the most stable due to cooperative effects as found in the hydration of formic acid.51 As the clusters grow larger in size, the tTt and tCt conformers become increasingly important as they are able to maximize both cooperative effects and additionally stabilize by dipole cancellations leading to the most stable clusters. Of the numerous hydrated clusters with up to six waters examined in this work, only three were found to ionize with many of the neutral clusters being found to have comparable or greater stability indicating the propensity of oxalic to form highly stable neutral clusters with water. This work indicates that the formation of neutral cores are most important in the initial cluster formations with oxalic acid and water.

■

ASSOCIATED CONTENT

S Supporting Information *

Comparison of experimentally determined parameters and those for isomers cTc, cTt, and tTt from B3LYP and MP2 calculations with the aug-cc-pVDZ basis set. Vibrational mode frequencies for the cTc, cTt, and tTt conformers of oxalic acid from B3LYP and MP2 calculations with 6-311++G(d,p) and aug-cc-pVDZ basis sets. Relative energies (ΔEr), intermolecular energies (ΔE, ΔE0), intermolecular enthalpies (ΔH), and Gibbs free energies (ΔG) of the H2C2O4−(H2O)n, n = 1−6, complexes at various conformations from B3LYP/6-31+G(d) and B3LYP/6-311++G(d,p) calculations. Relative energies and intermolecular energies from MP2 and QCISD single-point calculations on B3LYP/6-311++G(d,p) optimized structures. This material is available free of charge via the Internet at http://pubs.acs.org.

a

B3LYP/6-311++G(d,p) results. bWith respect to the corresponding monomer. cCombined with the Boltzmann distribution factor for monomer.

atmosphere, the formation of new particles does not occur by an oxalic acid monomer and water addition alone. The preponderance of stable clusters identified in this study is found to be neutral. In fact, only one pentahydrate cluster cTc-Va ionizes to form an oxalic acid−hydronium ion pair hydrated on two opposing sides by water dimers reported by Ramon et al.,47 and of the 24 stable hexahydrates investigated, only in two clusters was ionization found to occur: cTt-VIb and cTc-VIa. Many of the neutral clusters are of competitive relative energy for the ionic pentahydrate cluster, and as for the hexahydrates, a good number of the neutral clusters are more stable than those that are ionized.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ ■

■

ACKNOWLEDGMENTS This work was supported by National Science Foundation (NSF Grant No. 1012994).

CONCLUSIONS Calculations of monomeric oxalic acid conformers were performed with DFT and ab initio methods with reasonably large basis sets. In general, the observed energies were in reasonable agreement with previous calculations, and the agreement with experimental geometric parameters was excellent. The majority of predicted frequencies for cTc and cTt conformers reported here are in agreement with matrix isolation experiments without scaling; however, the predicted OH stretching frequencies are in large discrepancy with experimental observed values without a corrective scaling factor. Considering the optimized geometries, calculated frequencies, and free energies at higher temperatures relevant to experiment, it seems likely that, in addition to the cTc and cTt conformers observed in matrix isolation studies, tTt (and possibly tCt) should also be present at elevated temperatures (vapor phase IR and electron diffraction studies). Oxalic acid adsorbs water powerfully. Oxalic acid dihydrate is a particularly stable form of oxalic acid, and in this work, dihydrate clusters with two waters bound in a cyclic ring produce much larger redshifts than those with the waters at separate binding sites, indicating more favorable binding. Generally, in the binding of the initial several waters to oxalic

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Theoretical Study on the Structure and Stabilities of Molecular Clusters

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