Accurate Thermochemistry of Hydrocarbon Radicals via an Extended

Article pubs.acs.org/JPCA

Accurate Thermochemistry of Hydrocarbon Radicals via an Extended Generalized Bond Separation Reaction Scheme Matthew D. Wodrich,*,† Clémence Corminboeuf,*,† and Steven E. Wheeler*,‡ †

Institut des Sciences et Ingénierie Chimiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland Department of Chemistry, Texas A&M University, College Station, Texas 77842, United States

‡

S Supporting Information *

ABSTRACT: Detailed knowledge of hydrocarbon radical thermochemistry is critical for understanding diverse chemical phenomena, ranging from combustion processes to organic reaction mechanisms. Unfortunately, experimental thermochemical data for many radical species tend to have large errors or are lacking entirely. Here we develop procedures for deriving high-quality thermochemical data for hydrocarbon radicals by extending Wheeler et al.’s “generalized bond separation reaction” (GBSR) scheme (J. Am. Chem. Soc., 2009, 131, 2547). Moreover, we show that the existing definition of hyperhomodesmotic reactions is flawed. This is because transformation reactions, in which one molecule each from the predefined sets of products and reactants can be converted to a different product and reactant molecule, are currently allowed. This problem is corrected via a refined definition of hyperhomodesmotic reactions in which there are equal numbers of carbon−carbon bond types inclusive of carbon hybridization and number of hydrogens attached. Ab initio and density functional theory (DFT) computations using the expanded GBSRs are applied to a newly derived test set of 27 hydrocarbon radicals (HCR27). Greatly reduced errors in computed reaction enthalpies are seen for hyperhomodesmotic and other highly balanced reactions classes, which benefit from increased matching of hybridization and bonding requirements. The best performing DFT methods for hyperhomodesmotic reactions, M06-2X and B97dDsC, give average deviations from benchmark computations of only 0.31 and 0.44 (±0.90 and ±1.56 at the 95% confidence level) kcal/mol, respectively, over the test set. By exploiting the high degree of error cancellation provided by hyperhomodesmotic reactions, accurate thermochemical data for hydrocarbon radicals (e.g., enthalpies of formation) can be computed using relatively inexpensive computational methods.

I. INTRODUCTION Chemists frequently rely upon chemical equations to interpret and quantify virtual chemical quantities, as well as to determine intrinsic molecular properties such as enthalpies of formation and heat capacities. Choosing appropriate chemical equations to obtain these quantities, however, is often a nontrivial task: numerous approaches matching various hybridization and bonding elements exist. Notably, these include isogyric,1 isodesmic,2−4 (hypo)homodesmotic,5−7 and hyperhomodesmotic,8,9 as well as other proposed reaction classes.10−18 The current state-of-the-art procedure for obtaining highly accurate enthalpies of formation is the application of basis set extrapolation techniques and coupled-cluster methodologies to atomization reactions. While these robust approaches yield data in excellent agreement with experiment,19−26 their immense computational expense renders them inapplicable to the vast majority of © 2012 American Chemical Society

molecular systems. For molecular systems that exceed the size for which these approaches are feasible, error-balanced chemical equations can provide a route to reliable thermochemical data without recourse to expensive computational methods.27,28 This follows the idea first advocated by Pople in the development of the isodesmic bond separation reaction (BSR), which provided improved thermochemical data using relatively inaccurate computational approaches.2 As the present paper examines the role of error balanced reactions in computational thermochemistry, a brief summary of various key reaction types follows. Isogyric reactions are transformations in which the number of electron pairs is conserved. 1,4 For closed shell hydrocarbons, Received: December 19, 2011 Revised: March 2, 2012 Published: March 2, 2012 3436

dx.doi.org/10.1021/jp212209q | J. Phys. Chem. A 2012, 116, 3436−3447

The Journal of Physical Chemistry A

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concept of isodesmic BSRs, defining “generalized bond separation reactions” (GBSRs) to accompany the hypohomodesmotic, homodesmotic, and hyperhomodesmotic reaction classes (see Charts 1−3). This GBSR scheme was designed to provide a unique chemical reaction for virtually any closed shell hydrocarbon that matches the various criteria of the corresponding hypohomodesmotic, homodesmotic, and hyperhomodesmotic reaction classes (RC3−5). Equations are constructed by choosing appropriate product and reactant compounds from predetermined sets of reference molecules (i.e., elementary reactants and products - compounds that cannot be broken down further while maintaining the constraints of the given reaction type). For example, for any closed shell hydrocarbon, the RC5 BSR can be constructed using the 27 closed shell product compounds given in Chart 2 and seven closed shell reactant compounds in Chart 3. In the course of unrelated work,37 we found that for some molecules, two RC5 reactions could be written based on the elementary reactants and products. For example, two RC5 reactions for cyclohexene are given by eqs 4 and 5. Subtracting eq 5 from eq 4 yields eq 6, which corresponds to the transformation of one elementary product (1-butene) to another (2-methylbutane) while still satisf ying the original hyperhomodesmotic criteria. A systematic test of all possible transformation reactions of this type (Reactant + Product → Reactant′ + Product′) revealed 12 instances where one elemental product could be transformed into another via reaction with an elemental reactant without violation of the rules defining hyperhomodesmotic equations (see Supporting Information (SI) for details). Thus, a deficiency exists in the current hyperhomodesmotic definition with respect to defining a unique GBSR scheme. The purpose of this paper is twofold: (i) identify the shortcomings of the current hyperhomodesmotic definition and provide a more rigorous set of requirements that eliminate possible transformation reactions and hence provide unique GBSRs and (ii) extend the GBSR concept to include open-shell hydrocarbon radicals and test the extent to which these balanced reactions provide routes to accurate thermochemical data.

isogyric reactions can be written in which the products are methane molecules, with molecular hydrogen used to balance the reactant side of the equation. The isodesmic BSR2−4 is defined as a reaction in which all bonds between heavy (nonhydrogen) atoms are separated into the simplest (or parent) molecule with this same type of bond. Reactions are then balanced by addition of the appropriate number of parent molecules, which for hydrocarbons includes methane, ethane, ethylene, acetylene, as well as the methyl, ethyl, vinyl, and ethynyl radicals. Equation 1 is the isodesmic BSR for benzene; the corresponding bond separation energy (BSE) represents the total molecular stability and has previously been used to assess the resonance energy of benzene.29−31

George, Trachtman, Bock, and Brett5 introduced homodesmotic reactions to better balance the bonding environments compared to isodesmic equations. However, two distinct definitions of these reactions appear in the literature.27 Homodesmotic reactions were originally defined5 as those having (a) equal numbers of carbon atoms in their various states of hybridization in reactants and products, and (b) a matching of carbon−hydrogen bonds in terms of the number of hydrogen atoms joined to individual carbon atoms in reactants and products. The same authors7 later defined homodesmotic reactions as those having (a) equal numbers of each type of carbon−carbon bond [C(sp3)−C(sp3), C(sp2)−C(sp2), C(sp2)C(sp2), and C(sp2)−C(sp3)] in reactants and products, and (b) equal numbers of each type of carbon atom (sp3, sp2, sp) with zero, one, two, and three hydrogens attached in reactants and products. In 2009, Wheeler, Houk, Schleyer, and Allen (WHSA)27 introduced the term “hypohomodesmotic” to describe reactions meeting the first, but not the second set of the criteria, and reserved the original “homodesmotic” term for reactions that meet the second set (which will necessarily also satisfy the hypohomodesmotic criteria). We employ these same definitions here. Equation 2 is a homodesmotic reaction for assessing the resonance energy of benzene.5,9,30−36 A subset of homodesmotic reactions can be classified as hyperhomodesmotic,9 designed by Hess and Schaad to provide an ameliorated matching of the bond environment where each carbon−carbon bond type (H3C−CH2, H3C−CH, H3C−C, H2C−CH2, H2C−CH, H2C−C, HC−CH, HC−C, C−C, H2CCH, HCCH, H2CC, HCC, CC, HCC, and CC) is conserved. Equation 3 represents a hyperhomodesmotic treatment. After clarifying the existing definitions of isogyric, isodesmic, (hypo)homodesmotic, and hyperhomodesmotic reactions, WHSA27 proposed a new hierarchy of reactions, subdivided into five reaction classes (RCs), where isogyric (RC1) ⊇ isodesmic (RC2) ⊇ hypohomodesmotic (RC3) ⊇ homodesmotic (RC4) ⊇ hyperhomodesmotic (RC5). Furthermore, WHSA extended the

II. A REFINED DEFINITION OF HYPERHOMODESMOTIC REACTIONS According to the original definition of Hess and Schaad,9 hyperhomodesmotic (RC5) equations have (a) equal numbers of carbon−carbon bond types and (b) equal numbers of each type of carbon atom (sp3, sp2, sp) with zero, one, two, and three hydrogen atoms attached. There are 16 types of carbon−carbon bonds (vide supra) possible in closed-shell hydrocarbons. The 27 closed-shell product compounds given in Chart 1 provide at least one compound with each of these bond types. Nonetheless, transformation reactions, such as eq 6, are still possible, with the breakdown of the uniqueness of the hyperhomodesmotic BSR scheme of WHSA27 arising from the definition of the carbon−carbon bond types (see Scheme 1). In particular, 3437



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Chart 1. Elemental Products for the GBSRs (Isogyric, Isodesmic, Hypohomodesmotic, Homodesmotic) of Closed Shell (Black) and Open Shell (Blue) Hydrocarbons

under the original hyperhomodesmostic definition, the CH3− CH bond in isobutane (indicated in red) is equivalent to the CH3−CH bond in propene, as are the CH2−CH bond in 1butene and the CH2−CH bond in 2-methylbutane (indicated in blue). The key difference in these bond types is the carbon atom hybridizations: the CH3−CH bond in isobutane is a C−C bond between two sp3 hybridized carbon atoms, while the CH3−CH bond in propene is a C−C bond between one sp3 and one sp2 carbon. This suggests a simple solution for eliminating this type of transformation reaction, and therefore defining truly unique GBSRs: include the hybridization of each carbon atom together with the type of carbon−carbon bonds. Thus, we propose that the definition of a hyperhomodesmotic reaction (RC5) be amended to an equation in which there are equal numbers of carbon−carbon bond types inclusive of carbon hybridization and number of hydrogens attached [H3C(sp3)−CH2(sp3), HC(sp2)C(sp2), etc.] in reactants and products. Note that this revised definition is similar in spirit to the recently proposed “generalized connectivity-based hierarchy” of Raghavachari and co-workers, which increasingly balances interactions based on the structure

and connectivity of a compound of interest38 and also to various group addivity methods, as demonstrated by Fishtik.39 In Scheme 2, the C−C bond types (including hybridization and number of attached hydrogens) are listed for eq 6. In contrast to the bonding and hybridization elements in Scheme 1, the balance required for our revised hyperhomodesmotic definition is not met, and this reaction no longer meets the criteria to be called hyperhomodesmotic. An examination of the 11 other transformation reactions identified in the SI shows that they are also forbidden under this refined hyperhomodesmotic definition. Similarly, eq 4 no longer fulfills the requirement of an RC5 reaction, and eq 5 is the unique hyperhomodesmotic BSR for cyclohexene.

III. EXTENSION OF THE GBSR SCHEME TO OPEN-SHELL HYDROCARBONS Hydrocarbon radicals are ubiquitous in the many reactions involved in combustion processes, and play key roles in soot formation pathways.40−76 As accurate enthalpies of formation for these species are crucial in combustion modeling, studies of soot formation pathways have turned increasingly to computed thermochemical data.77−81 On the other hand, accurate 3438



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Chart 2. Elemental Products for the GBSRs (Hyperhomodesmotic) of Closed Shell (Black) and Open Shell (Blue) Hydrocarbons

interactions, including hyperconjugation and branching.82−87 In light of studies showing that accurate thermochemical parameters

thermochemistry is also essential for studying fundamental aspects of radicals, such as the subtle roles played by stereoelectronic 3439



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Chart 3. Elemental Reactants for the GBSRs of Closed Shell (Black) and Open Shell (Blue) Hydrocarbons

Scheme 1. Carbon−Carbon Bond Types and Carbon Atom Hybridizations with Number of Attached Hydrogen Atoms for Eq 6a

Lewis structure form. While this assumption may appear unsettling upon preliminary examination, it should be noted that the Lewis representation of radicals simply serves as a book-keeping tool to facilitate the choice of reference compounds. In fact, for the higher RCs in particular, the reference compounds will contain delocalization patterns similar to those present in the target molcule.

a

The red and blue bonds indicate the shortcomings of the current hyperhomodesmotic definition.

can be obtained using relatively cheap levels of theory for closedshell hydrocarbons using WHSA’s GBSR scheme,27,28 we choose to extend this scheme to include doublet hydrocarbon radicals. In the construction of a GBSR scheme for hydrocarbon radicals, the definitions of specific reaction classes discussed above are retained. However, it is necessary to derive a new set of elemental reactants and products that meet the new bonding and hybridization criteria introduced by the inclusion of radicals. Note that the hybridization and bonding in radicals does not follow idealized “Lewis structures” to the same extent as closed shell hydrocarbons. As such, the product and reactant sets have been derived with the idea that there is no delocalization (i.e., the allyl radical is represented as H2C•−CHCH2), and that all radicals exist in their pure

For isogyric reactions, products include methane and the methyl radical with the reactions being balanced by addition of the necessary number of hydrogen molecules. The use of both methane and the methyl radical allows for conservation of the number of unpaired electrons in reactants and products. Isodesmic products consist of all sets of two carbon containing hydrocarbons, including their radical counterparts (Chart 1, blue compounds) and are balanced with a combination of methyl radicals and methane molecules. The more stringent requirements of hypohomodesmotic, homodesmotic, and hyperhomodesmotic reactions require product sets of 17, 40, and 96 compounds

Scheme 2. Refined Hyperhomodesmotic Definition Applied to eq 6

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Chart 4. The HCR27 Test Set of 27 Doublet Hydrocarbon Radicals

majority of large hydrocarbon radicals, computations have been benchmarked against estimated complete basis set (CBS) CCSD(T) electronic energies. These reference values were derived by separate extrapolations of Hartree−Fock (HF) and MP2 correlation energies appended to CCSD(T)/cc-pVTZ energies, all evaluated at B3LYP/6-311+G(d,p) optimized geometries. HF energies were extrapolated using an exponential form108 based on energies computed with the cc-pVXZ basis sets (X = T, Q, 5). The MP2 correlation energies were extrapolated using the functional form of Helgaker et al.109 using cc-pVTZ and cc-pVQZ energies. The following computational methods were tested to determine the extent of error cancellation provided by this extended GBSR scheme: the LDA SVWN5,110,111 the hybrid GGA B3LYP,112,113 the hybrid meta-GGAs M05-2X114 and M06-2X,115,116 the pure GGA PBE,117 the long-range corrected pure GGA LC-PBE,117,118 Grimme’s dispersion-corrected GGA B97-D,119,120 and Becke’s B97 functional119 augmented with a density dependent dispersion correction (B97-dDsC121−124), HF, MP2, and CCSD(T). The density functional theory (DFT) methods were paired with the 6-311+G(d,p) basis set, while ccpVTZ was used with the ab initio methods. The motivation for testing both long-range and dispersion corrected functionals arises from their recently demonstrated accuracy for the BSR energies of alkanes and other hydrocarbon compounds.101,122,125,126 The M05-2X and M06-2X computations made use of an “extrafine” integration grid,127 while all other functionals utilized the default grid in Gaussian09.128 HF, MP2, and CCSD(T) computations were executed using Molpro,129 while the B97-dDsC computations were carried out using an in-house development version of Q-Chem.130 HF, B97-dDsC, MP2, and CCSD(T) energies were evaluated at B3LYP/ 6-311+G(d,p) optimized geometries, while for the other DFT

(Charts 1 and 2, both black and blue compounds), respectively. These reactions are balanced with smaller sets of 6, 9, and 10 compounds (Chart 3, both black and blue compounds). Unique GBSRs for any doublet hydrocarbon radical can be constructed using only the sets of reactants and products in Charts 1−3. For example, eqs 7−11 are the unique GBSRs for 25TS (see Chart 4) hydrocarbon radical.

IV. ERRORS IN COMPUTATIONALLY DETERMINED THERMOCHEMISTRY As in the original work of WHSA,27 one of the primary motivations for extending the GBSR scheme to hydrocarbon radicals is for use in obtaining accurate thermochemical data.27 This is partially motivated by the troubling performance of many density functionals, including the widely employed B3LYP, to accurately describe reaction energies and enthalpies of formation of both closed- and open-shell organic species.88−106 Below, the focus is primarily on errors in computed electronic energies. This is motivated by two concerns: (1) errors in electronic energies are typically the largest source of errors in predicted reaction enthalpies27,107 and (2) WHSA previously showed27 that for closed-shell hydrocarbons, the homodesmotic hierarchy results in excellent cancellation of errors in computed ZPVEs and corecorrelation effects, as well as other effects typically included to derive highly accurate reaction enthalpies of formation. Cancellation of these errors for hydrocarbon radicals is expected to be similar to that observed for their closed-shell counterparts, so they are not considered here. Chart 4 provides a set of 27 doublet hydrocarbon radicals, which compose a newly derived test set, HCR27 (HydroCarbonRadicals-27), and for which the error-canceling abilities of the five reaction classes (RC1−5, see SI for the corresponding reactions) were tested. As experimental data is lacking for a 3441



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Figure 1. Errors of various methods versus estimated CCSD(T)/CBS values for isogyric (RC1), isodesmic (RC2), hypohomodesmotic (RC3), homodesmotic (RC4), and hyperhomodesmotic (RC5) reactions for the HCR27 test set.

MAD, even for isodesmic reactions, is less than 1 kcal/mol, while that of hyperhomodesmotic reactions approaches the kJ/mol range. The 95% confidence interval, which represents a more accurate picture of computational performance that is comparable to the error bars given by experiment, still has impressive accuracies of ±1.54 and ±0.90 for isodesmic and hyperhomodesmotic reactions when using the M06-2X functional. As would be expected, the CCSD(T) method has the smallest deviation from benchmark values, but its excellent performance comes at great computational expense, which make it inapplicable to all but the smallest systems. Mean signed deviations (MSDs) are provided in Figure 2 to highlight systematic errors exhibited by the various computational methods. For instance, the MSDs for the HF/cc-pVTZ computations each have a large negative value for RC1−RC3, indicating systematic underestimation of these reaction energies. In contrast, MP2/cc-pVTZ computations overestimate the same reaction energies (as indicated by positive MSD values). Density functional performance indicates some systematic deficiencies in the description of certain reaction classes. For example, B97-D appears particularly prone to overestimation, with positive MSD values for each reaction class. Other systematic errors are seen, but these are largely limited to RC1 and RC2. In general, if a sufficiently high reaction class is chosen, most computational methods appear free of significant systematic errors. Also of interest is the performance of these computational methods for various types of radicals, e.g., saturated alkyl and those possessing (hyper)conjugation between double/triple bonds or double/triple bonds and radicals. For alkyl radicals, no difference exists between RC3 (hypohomodesmotic) and RC4 (homodesmotic) reactions because all carbon atoms (aside from the radical center) have the same sp3 hybridization. For most (hyper)conjugated systems, each of the five reaction classes will be different due to the mix of carbon atom hybridizations. Figures 3 and 4 provide MADs from benchmark energies for both alkyl radicals and radicals containing (hyper)conjugation interactions, respectively. The following trends regarding various DFT functionals are apparent: (1) B3LYP fails to accurately

functionals geometries were optimized at their respective level of theory and confirmed to be energy minima by examination of harmonic vibrational frequencies. The error bars presented throughout the paper respresent a confidence level of 95%. Figure 1 shows the mean absolute deviation (MAD) of reaction energies from the benchmark values of various computational methods over the five reaction classes, providing a powerful demonstration of the error cancellation provided by the homodesmotic hierarchy when applied to hydrocarbon radicals. A clear trend of error reduction is observed as one ascends the hierarchy (RC1→ RC5). For example, for isogyric (RC1) reactions, HF leads to an average error of 30.4 (±60.9 at the 95% confidence interval) kcal/mol. However, when applied to hyperhomodesmotic transformations, the same method yields a drastically reduced MAD of only 1.5 (±5.6 at the 95% confidence interval) kcal/mol. Density functionals describe the energetics of the radical species more accurately than HF, but still benefit immensely from the error cancellation provided by the more balanced reaction classes. The dispersion-corrected functional B97-dDsC performs very well for isogyric reactions, with a MAD of 1.3 (±3.0 at the 95% confidence interval) kcal/mol, but performs less well for isodesmic reactions with a MAD of 3.1 (±8.5 at the 95% confidence interval) kcal/mol. Interestingly, another dispersion corrected functional, B97-D, as well as the long-range corrected LC-PBE, are among the worst performers from isogyric (RC1) reactions, but improve dramatically for isodesmic reactions (RC2) and higher rungs on the homodesmotic hierarchy. This may arise from the small molecular size of reference compounds used in RC1, which do not fully benefit from the improved long-range description, relative to the larger reference molecules used in RC2 and higher reaction classes. Indeed, for hyperhomodesmotic reactions, LC-PBE is outperformed only by M06-2X and CCSD(T). B97-D and B97dDsC also perform particularly well for this reaction class. Myriad studies have shown the excellent performance of the Minnesota family of density functionals (including M06-2X) for energetics.131−134 Similar results are revealed here, where M06-2X is among the best performing functionals tested. Remarkably, the 3442



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Figure 2. (A) MSDs for isogyric (RC1), isodesmic (RC2), hypohomodesmotic (RC3), homodesmotic (RC4), and hyperhomodesmotic (RC5) reactions at various computational levels. (B) Detailed view of mean signed deviations for RC4 and RC5.

Figure 3. Mean absolute deviations for isogyric (RC1), isodesmic (RC2), homodesmotic (RC3/RC4), and hyperhomodesmotic (RC5) reactions of alkyl radicals (1TS−5TS) versus estimated CCSD(T)/CBS values using various computational methods.

reproduce energies of systems containing branched alkanes while performing satisfactorily for conjugated systems; (2) LC-PBE, B97-D, and B97-dDsC show excellent performance for alkyl radicals and less satisfactory performance for (hyper)conjugated system (aside from B97-dDsC, this is particularly true for isogyric (RC1) reactions); and (3) the Minnesota density functionals show a high level of robustness, performing well for both types of systems tested, particularly so for M06-2X.

to determine enthalpies of formation for radicals for which experimental data is lacking or dubious. For instance, to compute ΔHf for the 2-hexyl radical, GBSR reactions 12−15 can be written, representing RC1−5. Similarly, application of the extended GBSR scheme to the 2-methylhex-2-yl radical would employ eqs 16−19. Computing ΔHf for a target compound requires the enthalpy of the reaction corresponding to the chosen RC as well as experimental (or previously determined computational) enthalpies of formation of the reference compounds. Here, experimental ΔHf298K values for the reference compounds were taken from the NIST Neutral Thermochemical Database.135 While the use of RC1 and, to a lesser extent, RC2 leads to a wide variation in the determined enthalpies of formation among different computational methods,

V. SAMPLE APPLICATIONS Enthalpies of Formation of Hydrocarbon Radicals. Of key interest for computational thermochemistry is the determination of intrinsic molecular properties, most notably enthalpies of formation. The radical GBSR scheme can be used 3443



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Figure 4. Mean absolute deviations for isogyric (RC1), isodesmic (RC2), hypohomodesmotic (RC3), homodesmotic (RC4), and hyperhomodesmotic (RC5) reactions of radicals containing (hyper)conjugation interactions (see SI for specific compounds) versus estimated CCSD(T)/CBS values using various computational methods.

presumably though accurate computational predictions. These simple examples demonstrate the utility of the homodesmotic hierarchy in the determination of accurate enthalpies of formation for medium and even large hydrocarbon radicals.

homodesmotic (RC4) and hyperhomodesmotic (RC5) treatments lead to enthalpies of formation that vary little with the choice of method (Figure 5). CCSD(T)/cc-pVTZ computations

Figure 5. Mean deviations, MADs, and range of deviation from CCSD(T)/cc-pVTZ in the computationally determined ΔHf298K values for all tested methods (see computational details and SI) based on RC1−5 equations for the 2-hexyl and 2-methylhex-2-yl radicals.

give ΔHf298K = 5.91 ± 1.06 and −3.54 ± 1.34 kcal/mol for 2hexyl radical and the 2-methylhex-2-yl radical, respectively. For even the poorest performing methods, the deviation from CCSD(T)/cc-pVTZ is only 0.21 kcal/mol for the 2-hexyl radical (M05-2X) and 0.38 kcal/mol for the 2-methylhex-2-yl radical (PBE) when hyperhomodesmotic (RC5) equations are used. To provide a more comprehensive picture of the accuracies associated with the computed heats of formation, the error bars of both the computational and experimental data must be included. Table 1 shows these errors, indicating that, despite the small deviations of the DFT reaction energies from CCSD(T)/ cc-pVTZ values, errors bars on the predicted enthalpies of formation are still in the range of ±1−2 kcal/mol. However, the largest component of these errors arises from experimental uncertainties, and not from the computed reaction energies. In other words, much smaller error bars would result if ΔHf298K values for the reference species were known to greater accuracy,

BSR Energies using Hyperhomodesmotic Equations. Isodesmic BSEs can be used to assess the total amount of (de)stabilization within a molecule of interest in order to, for instance, quantify various stereoelectronic interactions. However, many DFT methods perform poorly when applied to isodesmic BSRs.95,100,104 In part, this is due to the inability of HF and many density functionals, to accurately reproduce energies associated with the alkyl group branching interactions that are lost during the bond separation procedure. For example, the BSEs for the two alkyl radicals (2-hexyl, eq 13, and 2-methylhex-2-yl, eq 17) discussed in the previous section are given in Table 2 (Direct BSE) along with their deviations from CCSD(T)/cc-pVTZ values. Shortcomings of HF and several density functionals in this context are apparent, as indicated by the large deviations in Table 2 (over 4 kcal/mol for 2-hexyl and over 6 kcal/mol for the 2-methylhex-2-yl radical). However, 3444



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(Table 2). This approach has been shown previously to work well for large systems.136

Table 1. 95% Confidence Level Errors for Computed Heats of Formation of the 2-Hexyl and 2-Methylhex-2-yl Radicals Using Hyperhomodesmotic Reactions (Eqs 15 and 19)a CCSD(T)/cc-pVTZ

M06-2X

B97-dDsC

±0.08 2(±0.12) 2(±0.16) 1(±0.5) ±1.06

±0.54 2(±0.12) 2(±0.16) 1(±0.5) ±1.60

±0.35 2(±0.12) 2(±0.16) 1(±0.5) ±1.41

±0.08 2(±0.12) 2(±0.16) 1(±0.7) ±1.34

±0.54 2(±0.12) 2(±0.16) 1(±0.7) ±1.80

±0.35 2(±0.12) 2(±0.16) 1(±0.7) ±1.61

2-Hexyl Radical computational Error 2 propane 2 n-butane 1 2-butyl radical total 2-Methylhex-2-yl Radical computational error 2 propane 2 n-butane 2-methylbut-2-yl radical total

VI. CONCLUSION In this work, we extended the generalized bond separation scheme27 for the purpose of determining accurate thermochemical data for hydrocarbon radicals. This extended GBSR scheme employs a revised definition of hyperhomodesmotic reactions that includes matching of hybridization and bonding elements together in order to provide a unique BSR for each reaction class (RC1−5) based on a set of elemental reactants and products. As a result of these stricter requirements, this refined definition eliminates the possibility of writing multiple hyperhomodesmotic BSRs for a given hydrocarbon, and is recommended for applications to both closed-shell hydrocarbons and hydrocarbon radicals The new hydrocarbon radical GBSR scheme was shown to provide accurate thermochemical predictions even with HF and DFT methods. The high accuracy of data obtained using inexpensive computational methods relies upon the enhanced error cancellation provided by the upper rungs of the homodesmotic hierarchy, which, by construction, more closely match the hybridization and bonding elements present in a molecule being examined. Consequently, accurate thermochemical parameters can be obtained for large, previously inaccessible hydrocarbon radicals by combining the error canceling ability of hyperhomodesmotic equations with standard DFT methods. Such data should be tremendously beneficial in fields where accurate radical thermochemistry is needed, ranging from classical physical organic chemistry to combustion modeling.

The “computational error” values are taken as 2 times the root mean square deviation of RC5 reactions over the five alkyl radicals in the HCR27 test set (see SI Table 17), while errors from reference compounds are taken from the NIST database (see SI Table 20). All quantities are in kcal/mol.

a

hyperhomodesmotic equations can be used to derive markedly improved BSE values via an “indirect” procedure. Such indirect BSE values are obtained by first computing the enthalpy of formation of a molecule of interest via hyperhomodesmotic GBSR (see previous section), followed by using experimental heats of formation of BSE products and reactants along with the computed heat of formation of the compound of interest to assess the isodesmic equation (Figure 6). These “indirect” BSE

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ASSOCIATED CONTENT

S Supporting Information *

Electronic energies for relevant compounds, reaction energies used to derive Figures 1−5, Cartesian coordinates of test set compounds, and full citations for refs 128−130 are provided. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 6. Flowchart for assessing BSEs via the “direct” and “indirect” procedures. The direct procedure directly uses DFT energies to calculate the BSE, while the indirect first computes the heat of formation via a hyperhomodesmotic equation and then uses experimental heats of formation to calculate the BSE.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M.D.W.); clemence. [email protected] (C.C.); [email protected] (S.E.W.).

assessments vary little based on the level of theory used, and reproduce the CCSD(T)/cc-pVTZ data to within 0.5 kcal/mol

Table 2. BSE and Deviations from CCSD(T)/cc-pVTZ Values (Deviation) for the 2-Hexyl and 2-Methylhex-2-yl Radicals at Various Computational Levels Using the Direct and Indirect Proceduresa 2-Hexyl Radical Direct B3LYP B97-D PBE LC-PBE SVWN5 M05-2X M06-2X B97-dDsC HF MP2 CCSD(T) a

2-Methylhex-2-yl Radical Indirect

Direct

Indirect

BSE

deviation

BSE

deviation

BSE

deviation

BSE

deviation

5.19 8.23 6.07 8.45 9.52 8.46 8.56 9.30 4.77 10.08 9.14

−3.94 −0.91 −3.07 −0.69 0.38 −0.68 −0.58 0.16 −4.37 0.94

9.24 9.34 9.20 9.18 9.18 9.13 9.18 9.49 9.26 9.31 9.34

−0.10 0.00 −0.14 −0.16 −0.16 −0.21 −0.16 0.15 −0.08 −0.03

4.42 9.35 5.70 9.48 10.53 9.48 10.08 10.90 4.56 12.44 10.99

−6.57 −1.64 −5.30 −1.51 −0.47 −1.51 −0.91 0.09 −6.43 1.44

11.76 11.99 11.71 11.73 11.77 11.91 11.76 12.12 11.88 12.08 12.09

−0.34 −0.10 −0.38 −0.36 −0.32 −0.18 −0.33 0.03 −0.21 −0.01

Values in kcal/mol. 3445



Article

Notes

(35) Suresh, C.; Koga, N. J. Org. Chem. 2002, 67, 1965. (36) Glukhovtsev, M. N.; Bach, R. D.; Laiter, S. J. Mol. Struct. THEOCHEM 1997, 417, 123. (37) Wodrich, M. D.; Corminboeuf, C. Abstracts of Papers (ORGN243), 237th ACS National Meeting, Salt Lake City, UT, March 22−26, 2009. (38) Ramabhadran, R. O.; Raghavachari, K. J. Chem. Theory Comput. 2011, 7, 2094. (39) Fishtik, I.; Datta, R. J. Phys. Chem. A 2003, 107, 6698. (40) Kinney, R. E.; Crowley, D. J. Ind. Eng. Chem. 1954, 46, 258. (41) Stehling, F. C.; Frazee, J. D.; Anderson, R. C. Proc. Combust. Inst. 1956, 6, 247. (42) Bittner, J. D.; Howard, J. B. Proc. Combust. Inst. 1981, 56, 1105. (43) Weissman, M.; Benson, S. W. Int. J. Chem. Kinet. 1984, 16, 307. (44) Frenklach, M.; Clary, D. W.; Gardiner, J., W. C.; Stein, S. E. Proc. Combust. Inst. 1984, 20, 887. (45) Frenklach, M.; Clary, D. W.; Yuan, T.; Gardiner, J., W. C.; Stein, S. E. Combust. Sci. Technol. 1986, 50, 79. (46) Frenklach, M.; Warnatz, J. Combust. Sci. Technol. 1987, 51, 265. (47) Colket, M. B. Proc. Combust. Inst. 1987, 21, 851. (48) Harris, S. J.; Weiner, A. M.; Blint, R. J. Combust. Flame 1988, 72, 91. (49) Glassman, I. Proc. Combust. Inst. 1989, 22, 295. (50) Westmoreland, P. R.; Dean, A. M.; Howard, J. B.; Longwell, J. P. J. Phys. Chem. 1989, 93, 8171. (51) Bastin, E.; Delfau, J.-L.; Reuillon, M.; Vovelle, C.; Warnatz, J. Proc. Combust. Inst. 1989, 22, 313. (52) Wang, H.; Frenklach, M. J. Phys. Chem. 1994, 98, 11465. (53) Wang, H.; Frenklach, M. Combust. Flame 1997, 110, 173. (54) Anderson, H.; McEnally, C. S.; Pfefferle, L. D. Proc. Combust. Inst. 2000, 28, 2577. (55) Miller, J. A.; Melius, C. F. Combust. Flame 1992, 91, 21. (56) Adamson, J. D.; Morter, C. L.; DeSain, J. D.; Glass, G. P.; Curl, R. F. J. Phys. Chem. 1996, 100, 2125. (57) Kern, R. D.; Singh, H. J.; Wu, C. H. Int. J. Chem. Kinet. 1988, 20, 731. (58) Wu, C. H.; Kern, R. D. J. Phys. Chem. 1987, 91, 6291. (59) Leung, K. M.; Lingstedt, R. P. Combust. Sci. Technol. 1995, 102, 129. (60) Marinov, N. M.; Pitz, W. J.; Westbrook, C. K.; Castaldi, M. J.; Senkan, S. M. Combust. Sci. Technol. 1996, 116−117, 211. (61) Miller, J. A. Proc. Combust. Inst. 1996, 26, 461. (62) Castaldi, M. J.; Marinov, N. M.; Melius, C. F.; Huang, J.; Senkan, S. M.; Pitz, W. J.; Westbrook, C. K. Proc. Combust. Inst. 1996, 26, 693. (63) Lindstedt, R. P.; Skevis, G. Combust. Sci. Technol. 1997, 125, 73. (64) Marinov, N. M.; Castaldi, M. J.; Melius, C. F.; Tsang, W. Combust. Sci. Technol. 1997, 128, 295. (65) D’Anna, A.; Violi, A. Proc. Combust. Inst. 1998, 27, 423. (66) Marinov, N. M.; Pitz, W. J.; Westbrook, C. K.; Lutz, A. E.; Vincitore, A. M.; Senkan, S. M. Proc. Combust. Inst. 1998, 27, 605. (67) Marinov, N. M.; Pitz, W. J.; Westbrook, C. K.; Vincitore, A. M.; Castaldi, M. J.; Senkan, S. M.; Melius, C. F. Combust. Flame 1998, 114, 192. (68) Violi, A.; D’Anna, A.; D’Alessio, A. Chem. Eng. Sci. 1999, 54, 3439. (69) Pope, C. J.; Miller, J. A. Proc. Combust. Inst. 2000, 28, 1519. (70) Miller, J. A. Faraday Discuss. 2001, 119, 461. (71) Miller, J. A.; Klippenstein, S. J. J. Phys. Chem. A 2001, 105, 7254. (72) D’Anna, A.; D’Alessio, A.; Kent, J. H. Combust. Flame 2001, 125, 1196. (73) D’Anna, A.; Kent, J. H. Combust. Flame 2003, 132, 715. (74) Miller, J. A.; Klippenstein, S. J. J. Phys. Chem. A 2003, 107, 2680. (75) McEnally, C. S.; Pfefferle, L. D.; Robinson, A. G.; Zwier, T. S. Combust. Flame 2000, 123, 344. (76) Olten, N.; Senkan, S. M. Combust. Flame 2001, 125, 1032. (77) Wheeler, S. E.; Allen, W. D.; Schaefer, H. F. J. Chem. Phys. 2004, 121, 8800.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS S.N. Steinmann, J.F. Gonthier, and D.F. Jana are acknowledged for insightful discussion and computational assistance. The Swiss NSF and EPFL are acknowledged for financial support (C.C.), as well as the ACS Petroleum Research Fund (ACS PRF 50645-DNI6, S.E.W.).

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REFERENCES

(1) Minkin, V. I. Pure Appl. Chem. 1999, 71, 1919. (2) Hehre, W. J.; Ditchfield, R.; Radom, L.; Pople, J. A. J. Am. Chem. Soc. 1970, 92, 4796. (3) Radom, L.; Hehre, W. J.; Pople, J. A. J. Am. Chem. Soc. 1971, 93, 289. (4) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; John Wiley & Sons: New York, 1986. (5) George, P.; Trachtman, M.; Bock, C. W.; Brett, A. M. Theor. Chim. Acta 1975, 38, 121. (6) George, P.; Trachtman, M.; Bock, C. W.; Brett, A. M. Tetrahedron 1976, 32, 317. (7) George, P.; Trachtman, M.; Bock, C. W.; Brett, A. M. J. Chem. Soc., Perkin Trans. 2 1976, 1222. (8) Hess, B. A. Jr.; Schaad, L. J. J. Am. Chem. Soc. 1971, 93, 305. (9) Hess, B. A. Jr.; Schaad, L. J. J. Am. Chem. Soc. 1983, 105, 7500. (10) George, P.; Bock, C. W.; Trachtman, M. Theor. Chim. Acta 1987, 71, 289. (11) Zhao, M.; Gimarc, B. M. J. Phys. Chem. 1993, 97, 4023. (12) Warren, D. S.; Gimarc, B. M. J. Phys. Chem. 1993, 97, 4031. (13) Gimarc, B. M.; Zhao, M. J. Phys. Chem. 1994, 98, 1596. (14) Nyulaszi, L.; Varnai, P.; Vespremi, T. J. Mol. Struct. THEOCHEM 1995, 358, 55. (15) Chesnut, D. B.; Davis, K. M. J. Comput. Chem. 1997, 18, 584. (16) Vianello, R.; Liebman, J. F.; Maksic, Z. B. Chem.Eur. J. 2004, 10, 5751. (17) Baric, D.; Kovacevic, B.; Maksic, Z. B.; Muller, T. J. Phys. Chem. A 2005, 109, 10594. (18) El-Nahas, A. M.; Bozzelli, J. W.; Simmie, J. M.; Navarro, M. V.; Black, G.; Curran, H. J. J. Phys. Chem. A 2006, 110, 13618. (19) Tajti, A.; Szalay, P. G.; Csázár, A. G.; Kállay, M.; Gauss, J.; Valeev, E. F.; Flowers, B. A.; Vazquez, J.; Stanton, J. F. J. Chem. Phys. 2004, 121, 11599. (20) Bomble, Y. J.; Vazquez, J.; Kallay, M.; Michauk, C.; Szalay, P. G.; Csaszar, A. G.; Gauss, J.; Stanton, J. F. J. Chem. Phys. 2006, 125, 064108. (21) Harding, M. E.; Vazquez, J.; Ruscic, B.; Wilson, A. K.; Gauss, J.; Stanton, J. F. J. Chem. Phys. 2008, 128, 114111. (22) Martin, J. M. L.; de Oliveira, G. J. Chem. Phys. 1999, 111, 1843. (23) Boese, A. D.; Oren, M.; Atasoylu, O.; Martin, J. M. L.; Kallay, M.; Gauss, J. J. Chem. Phys. 2004, 120, 4129. (24) Karton, A.; Rabinovich, E.; Martin, J. M. L.; Ruscic, B. J. Chem. Phys. 2006, 125, 144108. (25) East, A. L. L.; Allen, W. D. J. Chem. Phys. 1993, 99, 4638. (26) Csaszar, A. G.; Allen, W. D.; Schaefer, H. F. J. Chem. Phys. 1998, 108, 9751. (27) Wheeler, S. E.; Houk, K. N.; Schleyer, P. v. R.; Allen, W. D. J. Am. Chem. Soc. 2009, 131, 2547. (28) Wheeler, S. E. WIREs Comput. Mol. Sci. 2012, 2, 204. (29) Cyranski, M. K. Chem. Rev. 2005, 105, 3773. (30) Mo, Y.; Schleyer, P. v. R. Chem.Eur. J. 2006, 12, 2009. (31) Wodrich, M. D.; Wannere, C. S.; Mo, Y.; Jarowski, P. D.; Houk, K. N.; Schleyer, P. v. R. Chem.Eur. J. 2007, 13, 7731. (32) Chesnut, D. B.; Davis, K. M. J. Comput. Chem. 1996, 18, 584. (33) Kovacevic, B.; Baric, D.; Maksic, Z. B.; Mü l ler, T. ChemPhysChem 2004, 5, 1352. (34) Hohlneicher, G.; Packschies, L.; Weber, J. ChemPhysChem 2005, 6, 2541. 3446



Article

(118) IIkura, H.; Tsuneda, T.; Yanai, T.; Hirao, K. J. Chem. Phys. 2001, 115, 3540. (119) Becke, A. D. J. Chem. Phys. 1997, 107, 8554. (120) Grimme, S. J. Comput. Chem. 2006, 27, 1787. (121) Steinmann, S. N.; Corminboeuf, C. J. Chem. Phys. 2011, 134, 044117. (122) Steinmann, S. N.; Corminboeuf, C. J. Chem. Theory Comput. 2010, 6, 1990. (123) Steinmann, S. N.; Corminboeuf, C. Chimia 2011, 65, 240. (124) Steinmann, S. N.; Corminboeuf, C. J. Chem. Theory Comput. 2011, 3567. (125) Song, J.-W.; Tsuneda, T.; Sato, T.; Hirao, K. Org. Lett. 2010, 12, 1440. (126) Steinmann, S. N.; Csonka, G.; Corminboeuf, C. J. Chem. Theory Comput. 2009, 5, 2950. (127) Wheeler, S. E.; Houk, K. N. J. Chem. Theory Comput. 2010, 6, 395. (128) Frisch, M. J. et al. Gaussian 09, revision B.01; Gaussian, Inc.: Wallingford CT, 2009. See Supporting Information for full citation. (129) Werner, H.-J. et al. MOLPRO, A Package of Ab Initio Programs, version 2010.1; see http://www.molpro.net. See Supporting Information for full citation. (130) Shao, Y.; et al. Phys. Chem. Chem. Phys. 2006, 8, 3172 See Supporting Information for full citation. (131) Zhao, Y.; Truhlar, D. G. J. Phys. Chem. A 2008, 112, 1095. (132) Zhao, Y.; Truhlar, D. G. Chem. Phys. Lett. 2011, 502, 1. (133) Zhao, Y.; Truhlar, D. G. J. Chem. Theory Comput. 2011, 7, 669. (134) Also see references cited within refs 124−126. (135) Afeefy, H. Y.; Liebman, J. F.; Stein, S. E. In NIST Chemistry WebBook, NIST Standard Reference Database Number 69; Linstrom, P. J., Mallard, W. G., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, 2004; (http://webbook.nist.gov). (136) Gonthier, J. F.; Wodrich, M. D.; Steinmann, S. N.; Corminboeuf, C. Org. Lett. 2010, 12, 3070.

(78) Wheeler, S. E.; Robertson, K. A.; Allen, W. D.; Schaefer, H. F.; Bomble, Y. J.; Stanton, J. F. J. Phys. Chem. A 2007, 111, 3819. (79) Diau, E. W.; Lin, M. C.; Melius, C. F. J. Chem. Phys. 1994, 101, 3923. (80) Hemelsoet, K.; Moran, D.; van Speybroeck, V.; Waroquier, M.; Radom, L. J. Phys. Chem. A 2006, 110, 8942. (81) Tokmakov, I. V.; Lin, M. C. J. Am. Chem. Soc. 2003, 125, 11397. (82) Matsunaga, N.; Rogers, D. W.; Zavitsas, A. A. J. Org. Chem. 2003, 68, 3158. (83) Zavitsas, A. A. J. Org. Chem. 2008, 73, 9022. (84) Poutsma, M. L. J. Org. Chem. 2008, 73, 8921. (85) Zavitsas, A. A.; Rogers, D. W.; Matsunaga, N. J. Org. Chem. 2010, 75, 5697. (86) Poutsma, M. L. J. Org. Chem. 2011, 76, 270. (87) Wodrich, M. D.; McKee, W. C.; Schleyer, P. v. R. J. Org. Chem. 2011, 76, 2439. (88) Choi, C. H.; Kertesz, M.; Karpfen, A. Chem. Phys. Lett. 1997, 276, 266. (89) Curtiss, L. A.; Raghavachari, K.; Redfern, P. C.; Pople, J. A. J. Chem. Phys. 2000, 112, 7374. (90) Woodcock, H. L.; Schaefer, H. F.; Schreiner, P. R. J. Phys. Chem. A 2002, 106, 11923. (91) Check, C. E.; Gilbert, T. M. J. Org. Chem. 2005, 70, 9828. (92) Izgorodina, E. I.; Coote, M. L.; Radom, L. J. Phys. Chem. A 2005, 109, 7558. (93) Grimme, S. Angew. Chem., Int. Ed. 2006, 45, 4460. (94) Izgorodina, E. I.; Coote, M. L. J. Phys. Chem. A 2006, 110, 2486. (95) Wodrich, M. D.; Corminboeuf, C.; Schleyer, P. v. R. Org. Lett. 2006, 8, 3631. (96) Schreiner, P. R.; Fokin, A. A.; Pascal, R. A. Jr.; de Meijere, A. Org. Lett. 2006, 8, 3635. (97) Grimme, S.; Steinmetz, M.; Korth, M. J. Chem. Theory Comput. 2007, 3, 42. (98) Izgorodina, E. I.; Brittain, D. R. B.; Hodgson, J. L.; Krenske, E. H.; Lin, C. Y.; Namazian, M.; Coote, M. L. J. Phys. Chem. A 2007, 111, 10754. (99) Schreiner, P. R. Angew. Chem., Int. Ed. 2007, 46, 4217. (100) Wodrich, M. D.; Corminboeuf, C.; Schreiner, P. R.; Fokin, A. A.; Schleyer, P. v. R. Org. Lett. 2007, 9, 1851. (101) Wodrich, M. D.; Jana, D. F.; Schleyer, P. v. R.; Corminboeuf, C. J. Phys. Chem. A 2008, 112, 11495. (102) Brittain, D. R. B.; Lin, C. Y.; Gilbert, A. T. M.; Izgorodina, E. I.; Gill, P. M. W.; Coote, M. L. Phys. Chem. Chem. Phys. 2009, 11, 1138. (103) Wheeler, S. E.; Moran, A.; Pieniazek, S. N.; Houk, K. N. J. Phys. Chem. A 2009, 113, 10376. (104) Wodrich, M. D.; Corminboeuf, C. J. Phys. Chem. A 2009, 113, 3285. (105) Steinmann, S. N.; Wodrich, M. D.; Corminboeuf, C. Theor. Chem. Acc. 2010, 127, 429. (106) Wodrich, M. D.; Gonthier, J. F.; Steinmann, S. N.; Corminboeuf, C. J. Phys. Chem. A 2010, 114, 6705. (107) Feller, D.; Peterson, K. A.; Crawford, T. D. J. Chem. Phys. 2006, 124, 054107. (108) Feller, D. J. Chem. Phys. 1993, 98, 7059. (109) Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. J. Chem. Phys. 1997, 106, 9639. (110) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (111) Slater, J. C. Quantum Theory of Molecular Solids: The SelfConsistent Field for Molecules and Solids; McGraw-Hill: New York, 1974; Vol. 4. (112) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (113) Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1988, 37, 785. (114) Zhao, Y.; Schultz, N. E.; Truhlar, D. G. J. Chem. Theory Comput. 2006, 2, 364. (115) Zhao, Y.; Truhlar, D. G. Theor. Chem. Acc. 2008, 120, 215. (116) Zhao, Y.; Truhlar, D. G. Acc. Chem. Res. 2008, 41, 157. (117) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865. 3447


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