Benchmark thermochemistry of the C_nH_{2n+2} alkane isomers (n=2--8) and performance of DFT and composite ab initio methods for dispersion-driven isomeric equilibria
aa r X i v : . [ phy s i c s . c h e m - ph ] M a y Benchmark thermochemistry of the C n H n +2 alkane isomers(n=2–8) and performance of DFT and composite ab initiomethods for dispersion-driven isomeric equilibria ∗ Amir Karton, David Gruzman, and Jan M. L. Martin*
Department of Organic Chemistry, WeizmannInstitute of Science, IL-76100 Reh. ovot, Israel † (Dated: J. Phys. Chem. A MS jp-2009-04369h : Received May 11, 2009; Accepted May 15, 2009) ∗ Dedicated to Prof. Yitzh. ak Apeloig on the occasion of his 65th birthday bstract The thermochemistry of linear and branched alkanes with up to eight carbons has beenreexamined by means of W4, W3.2lite and W1h theories. ‘Quasi-W4’ atomization energies havebeen obtained via isodesmic and hypohomodesmotic reactions. Our best atomization energies at 0K (in kcal/mol) are: 1220.04 n -butane, 1497.01 n -pentane, 1774.15 n -hexane, 2051.17 n -heptane,2328.30 n -octane, 1221.73 isobutane, 1498.27 isopentane, 1501.01 neopentane, 1775.22 isohexane,1774.61 3-methylpentane, 1775.67 diisopropyl, 1777.27 neohexane, 2052.43 isoheptane, 2054.41neoheptane, 2330.67 isooctane, and 2330.81 hexamethylethane. Our best estimates for ∆ H ◦ f, K are: -30.00 n -butane, -34.84 n -pentane, -39.84 n -hexane, -44.74 n -heptane, -49.71 n -octane, -32.01isobutane, -36.49 isopentane, -39.69 neopentane, -41.42 isohexane, -40.72 3-methylpentane, -42.08diisopropyl, -43.77 neohexane, -46.43 isoheptane, -48.84 neoheptane, -53.29 isooctane, and -53.68hexamethylethane. These are in excellent agreement (typically better than 1 kJ/mol) with theexperimental heats of formation at 298 K obtained from the CCCBDB and/or NIST ChemistryWebBook databases. However, at 0 K a large discrepancy between theory and experiment (1.1kcal/mol) is observed for only neopentane. This deviation is mainly due to the erroneous heatcontent function for neopentane used in calculating the 0 K CCCBDB value. The thermochemistryof these systems, especially of the larger alkanes, is an extremely difficult test for density functionalmethods. A posteriori corrections for dispersion are essential. Particularly for the atomizationenergies, the B2GP-PLYP and B2K-PLYP double-hybrids, and the PW6B95 hybrid-meta GGAclearly outperform other DFT functionals. † Electronic address: [email protected] . INTRODUCTION The fundamental importance of alkanes as organic chemistry building blocks and inindustrial chemistry (particularly petrochemistry) is self-evident to any chemist.Linear and branched lower alkanes are the principal components of gaseous and liquidfossil fuels. Accurate knowledge of their thermodynamic properties is essential for reliablecomputational modeling of combustion processes. (We note that one of us, JMLM, is amember of a IUPAC task group working in this area.[1])Aside from their practical relevance, alkanes present some intriguing methodologicalissues. The importance of accurate zero-point vibrational energies and diagonal Born-Oppenheimer corrections has been discussed previously[2], and this applies to bothwavefunction ab initio and density functional methods. While post-CCSD(T) computationalthermochemistry methods like W4 theory[3, 4] or HEAT[5, 6, 7] have no trouble dealingwith systems that, from an electronic structure point of view, are much more taxing thanalkanes, their steep cost scaling makes application to higher alkanes (or higher hydrocarbonsin general) impractical at present.Density functional theory seems to be the obvious alternative. However, in recent years anumber of authors[8, 9, 10, 11, 12, 13, 14, 15] have pointed to a disturbing phenomenon[16]:the error in computed atomization energies of n -alkanes grows in direct proportion tothe chain length. In addition, these same authors found that popular DFT methodshave significant problems with hydrocarbon isomerization energies in general, and alkaneisomerization energies in particular. This latter problem appears to be related to the poordescription of dispersion by most DFT functionals and can be remedied to a large extent byempirical dispersion corrections[11].For molecules as chemically systematic as alkanes, a computationally more cost-effectiveapproach than brute-force atomization energy calculations is the use of bond separationreactions, such as isodesmic[17] and homodesmotic[18] reactions. Recently, Schleyer andcoworkers[19, 20] discussed the concepts of ‘protobranching’ and of ‘hypohomodesmoticreactions’, i.e., reactions which, in addition to being isodesmic (i.e., conserving numbersof each formal bond type), conserve the number of C atoms in each hybridization state andhapticity (primary, secondary, tertiary, quaternary). The latter is a refinement of the earlier‘homodesmotic reaction’ concept[18]. 3hey established a consistent hierarchy of hydrocarbon reaction types that successivelyconserve larger molecular fragments: atomization ⊇ isogyric ⊇ isodesmic ⊇ hypohomod-esmotic ⊇ homodesmotic ⊇ hyperhomodesmotic which provides a converging sequence inthe sense that the energetic components of the reaction cancels to a larger extent betweenreactants and products as the reaction hierarchy is traversed.In the present work we obtain ‘quasi-W4’ atomization energies for C –C alkanes throughthe use of isodesmic and hypohomodesmotic reaction cycles that involve only methane,ethane, and propane in addition to one larger alkane. The reaction energies are calculatedat the W3.2lite or W1h levels, while for methane, ethane, and propane W4 benchmarkvalues are used. We shall show that the reaction energies of hypohomodesmotic reactions andjudiciously selected isodesmic reactions are well converged even at the W1h level. The use ofhypohomodesmotic reactions leads to near-perfect cancellation of valence correlation effects,and the use of judiciously selected isodesmic reactions leads to near-perfect cancellation ofpost-CCSD(T) correlation effects.We will then proceed to evaluate a number of DFT functionals and composite ab initiothermochemistry methods against the reference values obtained, both for the atomizationand for the isomerization energies. II. COMPUTATIONAL METHODS
All calculations were carried out on the Linux cluster of the Martin group at Weizmann.DFT geometry optimizations were carried out using Gaussian 03 Revision E.01[21]. TheB3LYP[22, 23, 24] DFT hybrid exchange-correlation (XC) functional was used in conjunctionwith the pc-2[25] polarization consistent basis set of Jensen. All large-scale self-consistentfield (SCF), CCSD and CCSD(T) calculations[26, 27] were carried out with the correlationconsistent family of Dunning and coworkers[28, 29, 30, 31, 32] using version 2006.1 of theMOLPRO[33] program system. All single-point post-CCSD(T) calculations were carried outusing an OpenMP-parallel version of M. K´allay’s general coupled cluster code MRCC[34]interfaced to the Austin-Mainz-Budapest version of the ACES II program system[35]. Thediagonal Born-Oppenheimer correction (DBOC) calculations were carried out using theCFOUR program system[36].The computational protocols of W n h theories W1[37, 38], W3.2lite[39], and W4[3] used in4he present study have been specified and rationalized in great detail elsewhere[3, 37, 38, 39].(Throughout, W3.2lite refers to variant W3.2lite(c) as described in ref. [39].) The use of theW n h variants of the W n methods, in which the diffuse functions are omitted from carbon andless electronegative elements, is of no thermochemical consequence for neutral alkanes[38],but computer resource requirements are substantially reduced.For the sake of making the paper self-contained, we will briefly outline the various stepsin W3.2lite theory and in W4h theory for first-row elements: • reference geometry and ZPVE correction are obtained at the B3LYP/pc-2 level oftheory for W3.2lite, and at the CCSD(T)/cc-pVQZ level for W4h. • the ROHF-SCF contribution is extrapolated using the Karton-Martin modification[40]of Jensen’s extrapolation formula[41]: E HF ,L = E HF , ∞ + A ( L + 1) exp( − √ L ) (1)For W3.2lite and W4h, the extrapolations are done from the cc-pV { Q,5 } Z and cc-pV { } Z basis set pairs, respectively. • the RCCSD valence correlation energy is extrapolated from these same basis sets.Following the suggestion of Klopper[42], E corr , RCCSD is partitioned in singlet-coupledpair energies, triplet-coupled pair energies, and ˆ T terms. The ˆ T term (which exhibitsvery weak basis set dependence) is simply set equal to that in the largest basis set, whilethe singlet-coupled and triplet-coupled pair energies are extrapolated using A + B/L α with α S =3 and α T =5. • the (T) valence correlation energy is extrapolated from the cc-pV { T,Q } Z basis setpair for W3.2lite, and from cc-pV { Q,5 } Z for W4h. For open-shell systems, theWerner-Knowles-Hampel (a.k.a., MOLPRO) definition[43] of the restricted open-shellCCSD(T) energy is employed throughout, rather than the original Watts-Gauss-Bartlett[27] (a.k.a. ACES II) definition. • the CCSDT − CCSD(T) difference, ˆ T − (T), in W3.2lite is obtained from the empiricalexpression 2.6 × cc-pVTZ(no f d )(no p on H)–1.6 × cc-pVDZ(no p on H), where theCCSDT energy is calculated using ACES II. In W4h it is instead extrapolated using A + B/L from cc-pV { D,T } Z basis sets.5 the difference between ACES II and MOLPRO definitions of the valence RCCSD(T)definition is extrapolated from cc-pVDZ and cc-pVTZ basis sets. One-half of thiscontribution is added to the final result, as discussed in the appendix of ref. [3]. • post-CCSDT contributions in W3.2lite are estimated from UCCSDT(Q)/cc-pVDZ(no p on H)–UCCSDT/cc-pVDZ(no p on H) scaled by 1.1. In W4h, the connectedquadruples are obtained as 1.1 × [UCCSDT(Q)/cc-pVTZ − UCCSDT/cc-pVTZ +UCCSDTQ/cc-pVDZ − UCCSDT(Q)/cc-pVDZ], while the contribution of connectedquintuple excitations is evaluated at the CCSDTQ5/cc-pVDZ(no d ) level. • the inner-shell correlation contribution, in both cases, is extrapolated fromRCCSD(T)/cc-pwCVTZ and RCCSD(T)/cc-pwCVQZ calculations. • the scalar relativistic contribution, again in both cases, is obtained from thedifference between nonrelativistic RCCSD(T)/cc-pVQZ and second-order Douglas-Kroll RCCSD(T)/DK-cc-pVQZ calculations. • atomic spin-orbit coupling terms are taken from the experimental fine structure. • finally, a diagonal Born-Oppenheimer correction (DBOC) is obtained at the ROHF/cc-pVTZ level.The main changes in W1h relative to W3.2lite are: (a) the SCF component is extrapolatedfrom the cc-pV { T,Q } Z basis sets, using the formula A + B/L ; (b) The valence RCCSDcomponent is extrapolated from the same basis sets, using A + B/L . ; (c) the valenceparenthetical triples, (T), component is extrapolated from cc-pV { D,T } Z basis sets, using A + B/L . ; (d) inner-shell correlation contributions are evaluated at the CCSD(T)/MTsmalllevel; and (e) post-CCSD(T) correlation effects as well as the DBOC are completelyneglected.The CCSDTQ5/cc-pVDZ(no d ) calculation for propane proved to be too taxing even forour strongest machine (8-core, Intel Cloverton 2.66 GHz, with 32 GB of RAM). For thealkanes for which we do have this term, CH and C H , it is practically zero (0.00 and 0.01kcal/mol, respectively), therefore for propane it was safely neglected.The anharmonic zero-point vibrational energy (ZPVE) of propane, propene, propyne,and allene was calculated using the following equation[44],ZPVE = 12 X i ω i − X ijk φ iik φ kjj ω k − X ijk φ ijk ω i + ω j + ω k + 132 X ij φ iijj + Z kinetic , (2)6here the cubic, quartic, and kinetic energy terms were computed at the MP2/cc-pVTZ levelof theory, and the harmonic term was partitioned into valence and core-valence contributionswhich were calculated at the CCSD(T)/cc-pVQZ and CCSD(T)/MTsmall levels of theory,respectively. (For propane we resorted to a CCSD(T)/cc-pVTZ calculation since theCCSD(T)/cc-pVQZ proved too daunting; based on the results for the other systems thisis expected to have little effect, e.g., the differences between the harmonic ZPVE calculatedwith the two basis sets are: 0.02, 0.02, 0.02, and 0.01 kcal/mol for methane, ethane, propene,and allene, respectively.)Unless noted otherwise, experimental data for the heats of formation at 0 K weretaken from the NIST Computational Chemistry Comparison and Benchmark Database(CCCBDB)[45]. The atomization energies quoted in CCCBDB assume CODATA[46] valuesfor the atomic heats of formation at 0 K: however, particularly for carbon atom, theATcT value[47] (170.055 ± ± H ◦ f, K toatomization energy raises the atomization energy over the CCCBDB value by m × . m carbon atoms. Thus, throughout the paper, the experimentalTAE were obtained from the heats of formation at 0 K using ATcT atomic heats offormation at 0 K (C 170.055 ± ± n -heptane, n -octane, isoheptane, and isooctane), they were first converted to 0 K using H − H for H (g) 2.024 ± ± H ◦ f, K .Rather than mix our calculated atomization energies with the TRC enthalpy functions,we have calculated our own H − H for the alkanes. The translational, rotational, andvibrational contributions were obtained within the RRHO (rigid rotor-harmonic oscillator)approximation from the B3LYP/pc-2 calculated geometry and harmonic frequencies.Internal rotation corrections were obtained using the Ayala-Schlegel method[49], again on theB3LYP/pc-2 potential surface. This leaves us with the issue of correcting for the ensemble7f low-lying conformers of the alkanes: by way of illustration, n -butane through n -octanehave 2, 4, 12, 30, and 96 unique conformers, respectively. The relative energies of theseconformers (which are surprisingly sensitive to the level of theory as they are strongly drivenby dispersion) were the subject of a recent benchmark study by our group[50]. While largebasis set CCSD(T) calculations for all conformers of the heptanes and octanes proved toocostly (primarily for those without any symmetry), it was found in Ref.[50] that the B2K-PLYP-D double-hybrid functional[51] with an empirical dispersion correction, in conjunctionwith a sufficiently large basis set, tracks the CCSD(T) reference data[50] for n-butane, n-pentane, and n-hexane exceedingly closely, and this is the approach we have followed for allsystems with more than one conformer in this work.Dispersion corrections for the DFT energies (denoted by the suffix ”-D”) were appliedusing our implementation of Grimme’s expression[52, 53]. E disp = − s N at − X i =1 N at X j = i +1 C ij R ij f dmp ( R ij ) (3)where the damping function is taken as f dmp ( R ij ) = (cid:20) (cid:18) − α ( R ij s R R r − (cid:19)(cid:21) − (4)and C ij ≈ q C i C j , R r = R vdW ,i + R vdW ,j is the sum of the van der Waals radii of the twoatoms in question, and the specific numerical values for the atomic Lennard-Jones constants C i and the van der Waals radii have been taken from Ref.[52], whereas the length scaling s R =1.0 and hysteresis exponent α =20.0 as per Ref.[53].Eq.(3) has a single functional-dependent parameter, namely the prefactor s . This wastaken from Refs.[52, 53] for BLYP, B3LYP, and PBE, from Ref.[54] for the double hybrids,and optimized in the present work for the remaining functionals. These were, for the mostpart, optimized against the S22 benchmark set of weakly interacting systems[55]. III. RESULTS AND DISCUSSIONA. Overview of diagnostics for nondynamical correlation
The percentages of the nonrelativistic, clamped-nuclei W1h total atomization energyat the bottom of the well (TAE e ) accounted for by SCF, and (T) triples contributions8re reported in Table S1 of the Supporting Information, together with the coupled cluster T and D diagnostics[56, 57], and the largest T amplitudes. The percentage of the totalatomization energy accounted for by parenthetical connected triple excitations, %TAE e [(T)],has been shown to be a reliable energy-based diagnostic for the importance of nondynamicalcorrelation effects[3]. Ref.[3] gives useful criteria for assessing the extent of nondynamicalcorrelation effects, e.g., %TAE e [SCF] ≥
67% and/or %TAE e [(T)] <
2% indicate systems thatare dominated by dynamical correlation.As expected, all the alkanes considered in the present study exhibit very mildnondynamical correlation effects, and can be regarded as dominated by dynamicalcorrelation. 77–79% of the atomization energy is accounted for at the Hartree-Fock level,and 0.7–1.3% by the (T) triples. Table S1 (see Supporting Information) shows that the%TAE e [(T)] slightly increases with the degree of branching.In systems with very mild nondynamical correlation, CCSD(T) is generally very closeto the full CI limit, as the higher order triple excitations, ˆ T − (T), and the connectedquadruple excitations, ˆ T , tend to largely cancel one another (they are of similar orders ofmagnitude, but connected quadruples universally increase atomization energy while higher-order triples generally decrease it[3, 4]). For the smaller systems (of up to five carbons), forwhich we were able to explicitly calculate the ˆ T − (T) and (Q) contributions, the percentageof the atomization energy accounted for by post-CCSD(T) excitations, %TAE[post-(T)],varies between -0.01 and -0.02%. Small as these numbers may seem in relative terms,the atomization energy for an alkane with more than three carbons already exceeds 1,200kcal/mol, 0.02% of which amounts to 1 kJ/mol. B. Linear alkanes
The gas phase heat of formation (viz. the atomization energy) of an arbitrary linearalkane can be obtained from the following hypohomodesmotic reaction involving ethane andpropane. C m H m +2 + ( m − H → ( m − H (5)As the atomization energy of ethane is very well established both by W4 theory and byATcT, it is highly desirable to also have a W4 reference value for propane. A componentbreakdown of the W4 atomization energies for methane, ethane, and propane is given in9able I, while the final atomization energies at 0 K are compared with ATcT and CCCBDBexperimental values in Table I. There is good agreement between W4 and the availableATcT values (to within the sum of the uncertainties).W3.2lite component breakdowns from methane, ethane, propane, n -butane, and n -pentane (as well as for the relevant hypohomodesmotic reactions) are given in Table II.Let us briefly consider the raw components of the W3.2lite atomization energies. First,we note that there is a perfect linear relationship (R > T − (T)contribution reduces the atomization energies by amounts ranging form 0.12 kcal/mol inmethane to 1.07 kcal/mol in n -pentane, and the (Q) contribution, which increases theatomization energy, ranges from 0.09 kcal/mol in methane to 0.79 kcal/mol in n -pentane.The overall post-CCSD(T) contributions, which increase linearly with the size of the system,reduce the atomization energies by 0.03–0.3 kcal/mol. We finally note that, while the directlycomputed W3.2lite total atomization energies in the hypothetical motionless state (“at thebottom of the well”) agree well with the available W4 data (to within 0.2 kcal/mol or better),agreement at 0 K is rather less pleasing (differences of 0.3, 0.6, and 0.7 kcal/mol are seenfor methane, ethane, and propane, respectively). The chief part of these differences (0.2,0.4, and 0.5 kcal/mol, respectively) comes from the comparatively low-level approximationused for the zero-point vibrational energies: in molecules like n -pentane, ZPVE reachesthe 100 kcal/mol regime, and even a 1% error due to neglect of explicit anharmonicitywill translate into a 1 kcal/mol error in the final ZPVE. We made the point earlier[2]that, at least for species containing many hydrogens, the factor limiting accuracy of W n and similar thermochemical protocols will increasingly be the quality of the zero-pointvibrational energy. Nevertheless, for the ‘quasi-W4’ data obtained in the present paperfrom hypohomodesmotic or isodesmic cycles, the ZPVEs that enter the final results are thevery accurately known ZPVEs of methane, ethane, and propane on the one hand, and thesmall ZPVE component (nearly two orders of magnitude smaller than for the brute-forceTAE calculation) of the hypohomodesmotic or isodesmic reaction energy on the other hand.The relatively low-level approximation to the latter simply cannot wreak as much ”damage”in the latter case.Turning to the hypohomodesmotic reactions of n -butane and n -pentane, the moststriking feature of Table II is the near-perfect cancellation of all the valence correlation10ontributions between reactants and products. For n -butane, the CCSD, (T), T − (T), and(Q) contributions to the hypohomodesmotic reaction energy amount to merely, -0.03, -0.03, -0.01, and 0.00 kcal/mol, respectively. While for the hypohomodesmotic reactioninvolving n -pentane they amount to: +0.02, -0.02, -0.03, and +0.01 kcal/mol, respectively.The scalar relativistic and DBOC contributions are, likewise, basically null. The dominantcontributions to the reaction energies come from the SCF, inner-shell, and ZPVE components(specifically, +0.20, +0.07, and +0.13 kcal/mol, respectively, for the n -butane reaction, and+0.21, +0.11, and +0.22 kcal/mol, respectively, for the n -pentane reaction). Overall, thesehypohomodesmotic reactions are very slightly endothermic; the reaction energies at 0 K are0.32 and 0.52 kcal/mol for n -butane and n -pentane, respectively.Having established that valence post-CCSD(T) and DBOC contributions to thehypohomodesmotic reaction energies are thermochemically negligible, we proceed tocalculate the atomization energies of larger n -alkanes from W1h hypohomodesmotic reactionenergies. W1h component breakdowns from linear alkanes up to n -octane (as well as forthe relevant hypohomodesmotic reactions) are gathered in Table S2 of the SupportingInformation. First, let us consider the components of the atomization energies for thefive species for which a comparison with W3.2lite can be made. The SCF component,extrapolated from the cc-pV { T,Q } Z basis sets, is well converged. The valence CCSDcontribution, extrapolated from the same basis sets, systematically overestimates the cc-pV { Q,5 } Z results (by 0.4, 0.5, 0.7, 0.9, and 1.1 kcal/mol for methane, ethane, propane, n -butane, and n -pentane, respectively). These differences are partially compensatedby the fact that the W1h inner-shell contribution calculated with the MTsmall basisset systematically underestimates the cc-pwCV { T,Q } Z results (by ∼ ∼ { D,T } Z basis sets, is reasonably converged—thelargest deviation (of 0.1 kcal/mol) from the cc-pV { T,Q } Z results is seen for neopentane.Overall, the W1h atomization energies at 0 K overestimate the W3.2lite values by 0.2,0.3, 0.4, 0.6, and 0.7 kcal/mol for methane, ethane, propane, n -butane, and n -pentane,respectively. This demonstrates the limitations of directly computing TAEs with lower levelcompound thermochemistry methods such as W1h without the aid of hypohomodesmotic orisodesmic cycles. (We note that most of the difference comes from basis set incompletenessin the W1h valence CCSD contribution, and that its nefarious effects is actually mitigated11y error compensation with neglect of DBOC and post-CCSD(T) components. By wayof illustration, the W2.2h numbers for n-butane and n-pentane are 1220.71 and 1497.75kcal/mol, respectively, very close to W3.2lite.)Turning to the W1h components of the hypohomodesmotic reaction energies (Table S2,Supporting Information) the SCF, valence CCSD, valence (T), and inner-shell contributionsto the hypohomodesmotic reaction energies are very similar to their counterparts at theW3.2lite level. The overall reaction energies at 0 K differ by merely 0.02 and 0.03 kcal/mol forthe n -butane and n -pentane hypohomodesmotic reactions, indicating that the basis sets usedin W1h are sufficiently large to ensure adequate convergence of these hypohomodesmoticreaction energies. For the larger n -alkanes, again we see that the valence (T) contributionpractically cancels out between reactants and products. The valence CCSD contribution tothe reaction energy becomes thermochemically significant for the larger n -alkanes, reaching0.30 kcal/mol for the n -octane reaction. The overall reaction energies at 0 K range from0.34 to 1.50 kcal/mol for the n -butane and n -octane reactions, respectively.Using W4 atomization energies for ethane and propane and assuming that thehypohomodesmotic reaction energies stay constant between W1h and W4 theories, weobtain ‘quasi-W4’ atomization energies. Table III compares the ‘quasi-W4’ atomizationenergies obtained from W3.2lite and W1h reaction energies with experimental data takenfrom CCCBDB[45] (adjusted for the revised, ATcT, heat of formation of carbon atom).First of all, for the species where such a comparison can be made, the ‘quasi-W4’ valuesobtained from W3.2lite and W1h reaction energies are in very close agreement with eachother. The hypohomodesmotic ‘quasi-W4’ and adjusted CCCBDB values agree very well(0.06–0.32 kcal/mol of one another, where the largest deviations are seen for n -pentane).Finally, we note that the W1h TAE increasingly overestimate the ‘quasi-W4’ TAE (by 1.3–2.0 kcal/mol) on going from n -butane to n -octane, and even the W3.2liteTAE for n -butane and n -pentane overestimate the ‘quasi-W4’ TAE by 0.7 and 0.8kcal/mol, respectively. These large differences demonstrate the obvious advantage of usinghypohomosesmotic reactions rather than atomization reactions when one is limited tocomparatively low-level compound thermochemistry methods such as W1h. The differencebetween the W3.2lite and best values mostly derives from neglect of explicit anharmonicityin the ZPVE. (The use of scale factors, of course, to some degree accounts implicitly foranharmonicity.) 12 . Branched alkanes In the nature of things, for an arbitrary alkane, there are more isodesmic reactionsinvolving small hydrocarbon prototypes then there are hypohomodesmotic ones, renderingthe former more useful for thermochemical applications. If we consider only prototypes forwhich we have explicit W4 reference values (namely, methane, ethane, and propane) thenfor an arbitrary alkane two linearly-independent isodesmic reactions are: reaction 5 and thebond-separation reaction 6,C m H m +2 + ( m − → ( m − H (6)Table II gives W3.2lite component breakdowns from isobutane and neopentane (as wellas for the said isodesmic reaction energies). We note that for both isodesmic reactions thepost-CCSD(T) contributions are not only fairly small, but partly cancel out between higher-order triples and connected quadruples, something more pronounced for reaction 5 than forreaction 6.Table S2 of the Supporting Information gives W1h total atomization energies componentbreakdowns from branched alkanes of up to eight carbons and Table S3 of the SupportingInformation gives W1h component breakdowns from the said isodesmic reaction energies.As was the case for the hypohomodesmotic reactions, the W1h components of reaction5 are practically identical to their counterparts at the W3.2lite level, indicating that thebasis sets used in W1h theory are large enough to ensure convergence of the componentsof this isodemic reaction. The components of the bond-separation reaction 6 are not assimilar to their counterparts at the W3.2lite level—the largest deviations are seen for theCCSD component (namely, 0.14 and 0.20 kcal/mol for the reactions involving isobutane andneopentane). Again we find that the DBOC contributions, which are neglected in W1h, arefairly small, something more pronounced for reaction 5 than for reaction 6.Perusing the energy components of reactions 5 and 6 in Table S3 (see SupportingInformation), several systematic features emerge: (a) the valence CCSD and (T) componentsof reaction 5 are substantially lower than those of reaction 6; (b) inner-shell correlationeffects are somewhat more pronounced in reaction 5 than in reaction 6; (c) scalar relativisticand DBOC contributions are fairly small, but in contrast to reaction 5, in reaction 6 theysystematically increase with the size of the alkane; and (e) the ZPVE contribution to reaction5 is substantially lower than to reaction 6. 13n effect, the energy components (except for inner-shell) of reaction 5 cancel out to a largerextent between reactants and products than in the bond-separation reaction 6. Overall, thereaction energies at 0 K for reaction 5 are considerably lower (by 6–17 kcal/mol) than thoseof reaction 6. The superiority of reaction 5 is ascribed to the fact that the numbers of CH and CH groups are roughly equal on both sides of the reaction (with the notable exceptionof hexamethylethane, a.k.a. 2,2,3,3-tetramethylbutane), while in reaction 6 the differencein CH and CH groups between reactants and products increases systematically with thesize of the alkane. Furthermore, in reaction 5 the number of 1,3-interactions is roughlyequal on both sides, while in reaction 6 1,3-interactions occur only on the left-hand-side,and thus the imbalance increases systematically with the size of the alkane. The differencebetween the two reactions is further emphasized when considering the linear alkanes forwhich reaction 5 is hypohomodesmotic, and thus the numbers of CH groups, CH groups,and 1,3-interactions are perfectly balanced on both sides of the reaction. For example, for n -octane, the SCF, CCSD, and (T) components of reaction 6 are 4.79, 6.87, and 1.56 kcal/mol,while for reaction 5 they are merely 0.27, 0.30, and 0.03 kcal/mol, respectively (see TableS2 of the Supporting Information).From the linearly independent reactions 5 and 6 we can easily construct new isodesmicreactions—any linear combination of isodesmic reactions is also isodesmic—in which thenumber of 1,3-interactions, CH groups, or CH groups are perfectly balanced on bothsides of the reaction. In the remainder of the paper, these are referred to as (1,3), (CH ),and (CH ) reactions, respectively. The question that naturally arises is: which of thefive isodemic reactions should be used to derive the ‘quasi-W4’ atomization energy ofthe branched alkanes? As far as the electronic structure is concerned, we can divide thequestion into two parts: convergence of the n-particle space (”correlation treatment”) andone-particle space (”basis set”). (In the present study we are not analyzing errors arisingfrom anharmonic corrections to the ZPVE.)We have already seen for isobutane and neopentane (Table II) that convergence of then-particle space is faster in reaction 5 than in reaction 6. Table II also shows that theisodesmic reaction that balances the 1,3-interactions exhibits the fastest convergence, i.e.,the ˆ T − (T) and (Q) contributions are 0.00 and 0.00 kcal/mol for the isobutane reaction,and -0.01 and -0.02 kcal/mol for the neopentane reaction.Convergence of the one-particle space can be explored in a more systematic manner.14able S4 (given in the Supporting Information) shows the basis set convergence of theSCF, CCSD, and (T) components of the said isodesmic reactions. In general, the SCFcomponent converges quite rapidly with the basis set size, for all the isodesmic reactionsthe cc-pV { D,T } Z results are close to the basis set limit, something more pronounced forreactions 5 and (1,3) (we note that reaction 6 exhibits anomalous convergence for some ofthe systems). Convergence of the valence CCSD component becomes markedly slower. Formost of the systems the isodesmic reactions 5 and (1,3) the cc-pVTZ basis set yields resultsclose to the basis set limit; nevertheless, they exhibit anomalous, nonmonotonous, basis setconvergence, the cc-pV { D,T } Z results being further away from the basis set limit. As forthe valence (T) component, again, these two reactions converge more rapidly, i.e., for mostsystems convergence is obtained even with the cc-pVDZ basis set.Table S3 of the Supporting Information gives W1h component breakdowns from the fiveisodesmic reaction energies. Several systematic trends are observed: (a) for any branchedalkane the reaction energies both in the hypothetical motionless state (”at the bottom ofthe well”) and at 0 K increase in the following order: RE[(1,3)] < RE[5] < RE[(CH )] < RE[(CH )] < RE[6]; (b) the SCF, valence CCSD, valence (T), relativistic, and DBOCreaction components, generally, increase in the same order; (c) the inner-shell and ZPVEcorrections decrease in the same order.From the above discussion, it seems that reactions 5 and (1,3) offer the greatest similaritybetween the electronic structure of the reactants and products. The former has the additionaladvantage that it results in atomization energies with smaller uncertainties (Table S3,Supporting Information). Therefore, we select it as the ‘best’ reaction for deriving themost accurate ‘quasi-W4’ atomization energies.Table S3 (see Supporting Information) gives the TAE of the branched alkanes obtainedfrom the five said isodesmic reactions (by assuming that the isodesmic reaction energy stayconstant at the W1h and W4 levels, and using the W4 TAE for methane, ethane andpropane). In practice, the resulting TAE vary by 0.3–3.2 kcal/mol depending on whichreaction is used and the standard deviation varies between 0.2 and 1.1 kcal/mol. It isinteresting to note, however, that for any alkane the TAE are ordered in the same way asthe reaction energies, namely, TAE[6] < TAE[(CH )] < TAE[(CH )] < TAE[5] < TAE[(1,3)].Table III compares the final ‘quasi-W4’ atomization energies obtained from isodesmicreaction 5 with experimental data taken from the CCCBDB[45]: these were adjusted for the15evised (ATcT) heat of formation of carbon atom[3]. First, the ‘quasi-W4’ TAE obtainedfrom W3.2lite and W1h reaction energies are in close agreement with each other. Theisodesmic ‘quasi-W4’ and adjusted CCCBDB values agree very well (to within 0.1–0.3kcal/mol) for all the systems but neopentane. The discrepancy of 1.1 kcal/mol seen forneopentane is too large to be easily explainable in terms of issues with the calculations. Wenote, however, that a discrepancy of 0.7 kcal/mol exists between our best calculated H − H and the value that Scott[58] used in TRC and propagated into CCCBDB; moreover, thecomputed and observed ∆ H ◦ f, K are in much more plausible agreement. (We also notethat neopentane does not have conformers and therefore these cannot introduce uncertaintyin H − H . In addition, the Ayala-Schlegel[49] internal rotation correction is an order ofmagnitude smaller than the discrepancy.) We would argue that the experimental data forneopentane bear reexamination.Table III also compares the final ‘quasi-W4’ heats of formation at 298 K with theavailable experimental data. There is reasonable agreement between theory and experiment(generally, to within 0.0–0.4 kcal/mol). In general, the available TRC values are 0.1–0.3kcal/mol higher than the ‘quasi-W4’ values, and the Rossini values are 0.0–0.4 kcal/molhigher than the theoretical values.The equilibrium geometry of isooctane (a.k.a. 2,2,4-trimethylpentane, the “100%”fixpoint of the “octane rating” scale) has no symmetry: for this reason we were only ableto obtain a W1h value for the first-order saddle point (which has C s symmetry) possessingan imaginary frequency (37.5i cm − ) that corresponds to an internal rotation. The W1hTAE e for the idealized structure (Table S2, Supporting Information) is 2483.28 kcal/mol.For the deformation energy difference between C isooctane and the C s saddle point weobtain 0.25 kcal/mol at the CCSD(T) limit (0.135, 0.100, and 0.015 kcal/mol from the SCF,valence CCSD, and valence (T) components, respectively.) Assuming that the core-valenceand relativistic contributions to the deformation energy will be zero, we obtain an estimatedW1h TAE e of 2483.53 kcal/mol for the equilibrium structure of isooctane. Inclusion of theZPVE from a scaled B3LYP/pc2 harmonic calculation (150.69 kcal/mol) results in a W1hTAE of 2332.84 kcal/mol. Using the reaction energy of reaction 5 at the W1h level, andW4 atomization energies for ethane and propane, we obtain a ‘quasi-W4’ atomization energyfor isooctane of 2330.67 kcal/mol, in reasonable agreement with the NIST value of 2330.94kcal/mol (adjusted for the revised, ATcT, heat of formation of carbon atom).16 . Performance of compound thermochemistry methods for alkanes Table IV presents root mean square deviations (RMSD) for atomization energiesrelative to our best values for more approximate compound thermochemistry methodssuch as G2(MP2)[59], G2[60], G3[61], G3B3[62], G4[63], G4(MP2)[64], CBS-QB3[65], CBS-APNO[66], W1h[38], W2.2h, and W3.2lite. Application of the more expensive W2.2h andW3.2lite methods was possible only for a subset of small systems. Starting with the zero-point exclusive (‘bottom of the well’) data, the performance of the empirically corrected G n methods systematically improves as one proceeds along the series (the RMSDs are: G1 11.2,G2 4.3, G3 3.7, and G4 1.3 kcal/mol). G2(MP2) performs significantly worse (RMSD=5.9)than the standard G2 procedure; interestingly, standard G4 offers no improvement overG4(MP2). We note that, while G2(MP2) does not include post-MP2 correlation effects atall, the somewhat confusingly named G4(MP2) does include a small basis set CCSD(T)step: the main differences with full G4 theory are the absence (vs. presence) of an explicitinner-shell correlation step and of valence MP4 steps with some intermediate-sized basissets. As in the n-alkanes the inner-shell correlation term scales quite linearly with n , this isan optimal scenario for absorbing its effect into the empirical ‘high-level correction’, whilethe systems are also sufficiently dominated by dynamical correlation (as well as apolar) thatthe MP n series converges well and a single CCSD(T)/6-31G* step can adequately handlepost-MP2 correlation effects.W1h gives a RMSD of 0.9 kcal/mol for the whole set and 0.7 kcal/mol for the subset ofsmall systems. Using more elaborate basis sets for the extrapolations of the SCF, CCSD,and (T) contributions in W2.2h cuts the RMSD for the smaller subset to 0.3 kcal/mol.Including post-CCSD(T) correlation effects in W3.2lite further reduces the RMSD to 0.2kcal/mol. For the empirical methods the RMSD for the isomerization energies are lowerthan for the atomization energies: all the methods show similar performance with RMSD of0.7–1.2 kcal/mol. Interestingly, while G4 outperforms both CBS-QB3 and CBS-APNO forthe ‘bottom of the well’ atomization energies, CBS-QB3 and CBS-APNO both surpass G4’sperformance for the isomerization energies.We have already stressed that for molecules containing many hydrogens the principalfactor limiting the accuracy of W n methods lies in the quality of the zero-point vibrationalenergy. This is in accord with the deterioration in performance of the W n methods when17ero-point corrections are included, the RMSD increase by 0.6 kcal/mol for W1h, W2.2h,and W3.2lite compared to the zero-point exclusive RMSD. Despite that both G n and W n methods use ZPVEs at relatively low levels of theory (scaled HF/6-31G* and B3LYP/pc-2 harmonic frequencies, respectively) the performance of the G n methods significantlyimproves due to the fact that these methods were parametrized against experimentalatomization energies at 0 K. Thus, some correction for the zero-point energy is evidentlyabsorbed in the empirical corrections. What’s more, the systematic convergence of the G n methods seen for the zero-point exclusive results is not observed for the zero-point inclusiveresults: G3 shows the best performance with a RMSD of 0.5 kcal/mol, while G2 and G4give RMSDs of 0.8 and 0.7 kcal/mol, respectively. Again we see that G4 and G4(MP2) showsimilar performance and that G2(MP2) performs substantially worse than G2. E. Performance of density functional theory for alkanes
Recent studies[8, 9, 10, 11, 13, 15] have shown that DFT methods fail to adequatelypredict dissociation[8, 9, 10], isomerization[11], and isodesmic[13] reaction energies involvingalkanes. Grimme[11] showed that DFT methods generally predict the wrong sign for the n -octane → hexamethylethane isomerization, with errors ranging from 5.4 (B2-PLYP) to 11.8(BLYP). Schleyer and coworkers[13] investigated the performance of various DFT functionalsin reproducing the experimental reaction energy of the isodesmic reaction 6 for linear alkanesof up to n -decane. They showed that van der Waals corrected DFT functionals (suchas MPWB1K and MPW1B95) underestimate the protobranching stabilization energy asdefined by reaction 6 by ∼ ∼ s p d f ] quality but optimized for Hartree-Fock and DFT, whilefor MP2 and the double hybrids which exhibit slower basis set convergence we also use thepc-3 basis set.To ensure we are “comparing apples to apples”, so to speak, secondary effects that arenot explicitly included in the DFT calculations such as relativity, deviations from the Born-Oppenheimer approximation, and zero-point vibrational corrections are excluded from thereference values. For methane, ethane, and propane we use nonrelativistic, clamped-nuclei,zero-point exclusive TAEs from W4 theory, and for the remaining systems our best ‘quasi-W4’ values (given in the second or third column of Table III). The B3LYP/pc-2 referencegeometries, the said reference TAEs, and the individual errors of the various functionals canbe found in the Supporting Information. The empirical s scaling factors for each functionalwere taken from ref. [52, 53, 54] or otherwise for SVWN5, PBE, HCTH407, BLYP, TPSS,B97-1, B97-2, B97-3, TPSSh, TPSS1KCIS, PW6B95, and B1B95 optimized using the sameprocedure as detailed in Ref. [54]. The optimized s values, as well as error statistics over theS22 benchmark set of weakly interacting systems by Hobza and coworkers[55], can be foundin Table S5 of the Supporting Information. The RMSD, mean signed deviations (MSD),and mean average deviations (MAD) for the atomization reactions are gathered in Table V.Table VI lists the RMSD, MSD, and MAD for the n -alkane → branched-alkane isomerizationreactions with and without dispersion corrections. Finally, we consider the performanceof DFT for the isodesmic reaction 6. Table VII reports the RMSD for the n -, iso-, andneo-alkanes as well as for the entire set.Before proceeding to a detailed discussion of the specific performance of the various DFTfunctionals for our four evaluation sets, a few general remarks are in order: • The S22 benchmark set for weak interactions contains 22 model complexes involvingtypical noncovalent interactions, such as hydrogen bonding (e.g., H O and NH dimers), dipolar and multipolar dispersion interactions (e.g., pyrazine and benzenedimers). We have previously[54] made the point that small s values can be seen as anindication of a functional’s ability to cope with dispersion. This is demonstrated byan almost perfect linear correlation between the magnitude of the s values and the19ncorrected RMSD over the S22 set seen in Table S5 (see Supporting Information).Excluding the M06 family of functionals (M06, M06-L, and M06-2X) and the doublehybrids (B2K-PLYP, B2GP-PLYP, B2-PLYP, and mPW2-PLYP) all the functionalsperform rather poorly without the dispersion correction: RMSD vary between 3–6 kcal/mol. Correcting for dispersion dramatically reduces the RMSD to 0.3–1.0kcal/mol. For comparison, Hartree-Fock gives a RMSD of 0.8 kcal/mol after correctingfor dispersion. • Of our four evaluation sets, the atomization energies are clearly the toughest nut tocrack. In general, a posteriori correction for dispersion are essential for an accurateestimation of the atomization energies. As expected, the dispersion corrections increasewith the number of carbons in the alkane, and for isomers with the number of 1,3-interactions present. Nevertheless, the magnitude of the corrections are somewhatsurprising: for instance, with s =1.0, the dispersion energy correction ranges from 0.6(methane) to 27.0(!) kcal/mol (hexamethylethane). The latter amounts to 1% of thetotal atomization energy. • Without the dispersion correction the RMSD for the n -alkane → branched-alkaneisomerization reactions (Table VI) ranges from 0.2 kcal/mol (M06-2X) to 5.4 kcal/mol(HCTH407). Inspection of the individual errors reveals that most functionals predictthe wrong sign for the isomerization reactions of the larger alkanes: n -hexane, n -heptane, and n -octane (i.e., that the linear isomer is more stable than the branchedisomer). MP2, SVWN5 (!), and the M06 family of functionals (M06, M06-L, andM06-2X) are the only ones to obtain the right sign across the board. The correctedresults are much more encouraging: without exception, all the functionals predict thatthe branched isomers are more stable than their linear counterparts, and for most ofthe functionals the overall RMSD is below 0.5 kcal/mol, specifically X3LYP, B97-3,M06-2X, B2K-PLYP, and B2GP-PLYP with a RMSD below 1 kJ/mol. • As for the performance of DFT for the isodesmic reaction 6 (Table VII), we findthat all functionals other than LDA underestimate the reaction energy (whetherit involves linear or branched alkanes), confirming the findings of Schleyer andcoworkers[13] for linear alkanes. Also we find that without correction for dispersionthe RMSD increases with the number of 1,3-interactions present, i.e., in the order20 -alkanes < isoalkanes < neoalkanes. This trend is of course attributed to the increasein dispersion interactions with the degree of branching.We shall start with the nonempirical functionals SVWN5 LDA, PBE GGA, and TPSSmeta-GGA functionals and their corresponding hybrid functionals PBE0 and TPSSh. TheSVWN5 functional fails miserably for the alkane atomization energies, with a RMSD ofover 200 kcal/mol, and will be omitted from further discussion, other than to say that itperforms surprisingly well for the isomerization and isodesmic reactions. For the S22 setthe nonempirical functionals perform rather poorly with uncorrected RMSD between 4–5kcal/mol and corrected RMSD between 0.7–1.0 kcal/mol. For the atomization reactions(Table V) the nonempirical functionals TPSS, PBE, TPSSh, and to a lesser extent PBE0systematically overbind the alkanes (and in fact, are the only functionals other than LDAthat do so) as evident from MSD ≈ MAD. Obviously, there is no point in ‘correcting’ fordispersion then, as it can only increase the errors further. Indeed, if the empirical s scalingfactors are reoptimized by minimizing the RMSD for the atomization energies, then negative(anomalous) s values are obtained. The uncorrected RMSD for the atomization reactionsare 4.2, 13.7, 3.7, and 1.7 kcal/mol for TPSS, PBE, TPSSh, and PBE0, respectively (we notethat without the dispersion correction PBE0 exhibits the best performance of the functionalsconsidered). For the linear → branched isomerization reactions the uncorrected RMSD are onthe order of 3 kcal/mol and the corrected RMSD vary between 0.3–0.9 kcal/mol, where PBE0and PBE are the best performers. For the isodesmic reaction 6, the uncorrected RMSD varybetween 5–7 kcal/mol and the corrected RMSD between 0.4–1.3 kcal/mol, again with PBEand PBE0 as best performers.In the following discussion the empirical functionals are conveniently divided into threecategories: lightly, moderately, and heavily parameterized. The GGA BLYP, hybrid GGAB3PW91 and B3LYP, and hybrid meta GGA TPSS1KCIS and B1B95 functionals belong tothe first category. The uncorrected RMSD over our four validation sets (S22, atomization,isomerization, and isodesmic reactions) are unacceptably large ranging from 2–37 kcal/mol.After applying the dispersion correction, B3LYP emerges as the best performer with RMSDof 0.7, 1.9, 0.5, and 0.6 kcal/mol for the said four validations sets, respectively, followedby B1B95 with RMSD of 0.5, 2.5, 0.8, and 0.9 kcal/mol, respectively. We note thatBLYP, B3PW91 and TPSS1KCIS yield comparable RMSDs for the S22, isomerization, andisodesmic reactions, but the former grossly underestimates the atomization energies and the21atter two largely overestimate the atomization energies.The X3LYP and PW6B95 functionals may be regarded as belonging to the secondcategory. Again the RMSD before correcting for dispersion are unacceptably large (2–13 kcal/mol). After correcting for dispersion both functionals perform relatively well,particularly PW6B95 with RMSD of 0.5, 0.2, 0.3, and 1.6 kcal/mol for the S22, atomization,isomerization, and isodesmic reactions, respectively.The heavily parametrized functionals include the GGA HCTH407, meta GGA M06-L,hybrid GGAs B97-1, B97-2, B97-3, and hybrid meta GGAs BMK, M06, and M06-2X. TheM06 family of functionals (M06-L, M06, and M06-2X) have exceptionally low s values(0.20, 0.25, and 0.06, respectively). Without the dispersion correction M06-2X performsrelatively well for the S22 and isomerization reactions, but grossly underestimates theatomization energies. All the other functionals yield unacceptably large errors withoutthe dispersion correction. After correcting for dispersion, BMK and B97-2 seem to offer thebest performance with BMK RMSDs of { } kcal/mol and B97-2 RMSDs of { } kcal/mol for the S22, atomization, isomerization, and isodesmic datasets.Finally, we come to the double-hybrid class of functionals (which may be consideredlightly parameterized). It is perhaps not surprising that (for the BLYP-based double hybrids)the s values decrease with the percentage of MP2 correlation included. Furthermore,the performance of the double hybrides (over the four evaluation test sets) systematicallyimproves with the percentage of MP2 correlation included: most notably for the atomizationenergies RMSD of 12.5, 8.7, 8.0, 5.5, and 4.2 kcal/mol are obtained with the mPW2-PLYP(25% MP2), B2-PLYP (27% MP2), B2T-PLYP (31% MP2), B2GP-PLYP (36% MP2), andB2K-PLYP (42% MP2) functionals, respectively. (For this narrow application, the original‘thermochemistry and H-transfer barriers’ parametrized double hybrid B2K-PLYP[51] thusslightly outperforms the more ‘general-purpose’ parametrized B2GP-PLYP[54].) Similartrends are seen for the other three evaluation sets. After correction for dispersion, theperformance of the double hybrids is quite remarkable: for the four validation sets (S22,atomization, isomerization, and isodesmic reactions), B2GP-BLYP yields RMSD of 0.4, 0.2,0.2, and 0.5 kcal/mol, respectively, while B2K-PLYP slightly bests it for the isomerizationenergies and slightly underperforms it for the isodesmic reactions (0.1 kcal/mol). We notethat B2T-BLYP and B2-PLYP yield similar RMSD for the S22, isomerization, and isodesmicreactions, but an RMSD of ∼ s scaling factor to minimizethe RMSD for the atomization reactions (Table V). Obviously this approach also correctsfor deficiencies other than dispersion, such as basis set incompleteness and/or limitations ofthe XC functional. Nevertheless, for PW6B95, B2GP-PLYP, and B2K-PLYP there was nosignificant change in the s values or RMSD, indicating that (a) the original s values areoptimal; and (b) that these functionals do not exhibit any systematic errors. We note thatfor a few other functionals, namely X3LYP, B3LYP, BMK, B2T-PLYP, and B2-PLYP, thereoptimized s values are very similar to the original ones but the RMSD is improved by0.6–1.2 kcal/mol. F. Hydrocarbon prototype reactions
As discussed by Allen and coworkers[20], the heat of formation at 0 K (viz.the atomization energy) of an arbitrary acyclic hydrocarbon can be obtained fromhypohomodesmotic reactions involving a very limited number of branching prototypes:methane, ethane, ethene, acetylene, propane, propene, propyne, allene, isobutane,neopentane, and isobutene. As the atomization energies of the first four species are very wellestablished both by W4 theory and by ATcT, this leaves seven species for which it wouldbe highly desirable to have highly accurate reference values. W4 component breakdownsfrom methane, ethane, ethene, acetylene, propane, propene, propyne, and allene are givenin Table I. The final W4 atomization energies (Table I) are: 392.47, 666.18, 532.09, 388.70,942.95, 811.53, 670.45, and 669.37 kcal/mol, respectively. For isobutane, neopentane, andisobutene we obtain ‘quasi-W4’ TAE from isodesmic reaction energies at the W3.2levelby using reaction 5 for the former two and isobutene+CH → C H +C H for the latter(component breakdowns from the said reactions are given in Table II). Using W4 referencevalues for ethane, propane, and ethene we obtain atomization energies of 1221.73, 1501.01,and 1092.10 kcal/mol for isobutane, neopentane, and isobutene, respectively. IV. CONCLUSIONS
Benchmark post-CCSD(T) atomization energies and heats of formation for C n H n +2 (allisomers up to n =6 inclusive, plus selected isomers for n =7 and n =8) have been obtained.23xcellent agreement with the best available experimental data has been observed exceptthat an issue with the experimental enthalpy function of neopentane has been identified.The hypohomodesmotic reaction energy (eq. 5, in conjunction with linear alkanes)converges very rapidly with the level of theory. Valence post-CCSD correlation effectsare thermochemically negligible (at least for the small alkanes considered in the presentwork). Basis set requirements are likewise rather modest the SCF component convergeswith the cc-pV { D,T } Z basis sets and CCSD component with the cc-pV { T,Q } Z basis sets.Scalar relativistic and DBOC contributions are thermochemically insignificant. Inner-shellcorrelation accounts for 25–30% of the reaction energy at the ‘bottom of the well’.For isodesmic reactions convergence of the one- and n-particle spaces depends on thenature of the reaction: it is found that balancing the number of 1,3-ineractions betweenreactants and products significantly accelerates the convergence.We evaluated the performance of popular compound thermochemistry methods inreproducing the atomization energies. Post-CCSD(T) corrections are necessary for obtainingsub kJ/mol accuracy for the ‘bottom of the well’ atomization energies, as W3.2lite is the onlymethod that achieves this goal. W2.2h and W1h are capable of sub kcal/mol predictions;and G4, G4(MP2), and CBS-APNO afford ∼ n methods increase by 0.6 kcal/mol when zero-point corrections are included (specifically,the RMSDs are 1.7, 0.9, and 0.8 kcal/mol for W1h, W2.2h, and W3.2lite, respectively).The performance of the empirical G n methods (G2, G3B3, G3, G4(MP2), and G4), on theother hand, improves upon inclusion of ZPVE corrections (with RMSD ranging between0.4–0.6 kcal/mol), presumably due to the fact that their empirical correction terms werefitted against experimental atomization energies at 0 K.The performance of different DFT exchange-correlation functionals in predictingatomization, isomerization and isodesmic reaction energies was evaluated. The atomizationreactions are clearly the most daunting test; taking dispersion corrections into account threebest performers emerge: PW6B95, B2K-PLYP, and B2GP-PLYP, all of which attain anear-zero RMSD of 0.2 kcal/mol. For the isodesmic reactions the three said best performersattain RMSD of 1.7, 0.6, and 0.5 kcal/mol, respectively. And for the isomerization reactions24for which almost all the functionals perform exceptionally well) said best functionals attainRMSD of 0.3, 0.1, and 0.2 kcal/mol, respectively. Acknowledgments
Research at Weizmann was funded by the Israel Science Foundation (grant 709/05), theHelen and Martin Kimmel Center for Molecular Design, and the Weizmann AlternativeEnergy Research Initiative (AERI). JMLM is the incumbent of the Baroness ThatcherProfessorial Chair of Chemistry and a member ad personam of the Lise Meitner-MinervaCenter for Computational Quantum Chemistry.
Supporting Information
Diagnostics for the importance of nondynamical correlation (Table S1); componentbreakdown of W1h atomization energies and from hypohomodesmotic and isodesmicreactions (Tables S2 and S3); basis set convergence of hypohomodesmotic and isodesmicreaction energy components (Table S4); root mean square deviations for the S22 weakinteraction reference set for various DFT functionals (Table S5); component breakdownof computed enthalpy functions H − H (Table S6); errors of DFT methods for individualatomization energies, isomerization reactions, and isodesmic reactions (Tables S7 throughS9); B3LYP/pc-2 (and, where available, CCSD(T)/cc-pVQZ) optimized geometries inCartesian coordinates. 25 ABLE I: Component breakdown of the final W4 total atomization energies at the bottom ofthe well (TAE e ) and comparison between W4 total atomization energies at 0 K (TAE ), ActiveThermochemical Tables benchmarks, and CCCBDB reference data (in kcal/mol) a SCF CCSD (T) ˆ T − (T) ˆ T ˆ T inner relativ. spin DBOC ( a ) M-A b TAE e ZPVE c TAE d ATcT d,e
CCCBDB f CCCBDB g shell orbitmethane 331.55 84.72 2.89 -0.08 0.08 0.00 1.27 -0.19 -0.08 0.10 -0.05 0.02 420.21 27.74 392.47 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± h N/A 3.61 -0.58 -0.25 0.20 -0.11 0.06 1007.15 64.20 942.95 ± ± ± ± h i ± ± ± i ± ± ± i ± ± ± ( a )∆DBOC=DBOC[CCSD/cc-pVDZ] − DBOC[HF/cc-pVDZ] correction. a W4 values for methane, ethane, ethene, and acetylene are from ref. [2], and for propyne and allene from ref. [54] (except forthe ZPVE corrections which were calculated in the present work, see computational details section); W4h values for propaneand propene are from the present work. Note that the atomization energies include a post-SCF contributions to the diagonalBorn-Oppenheimer correction, cfr. footnote ( a ). b Difference between the ACES II and MOLPRO definitions of the valence ROCCSD(T), half of this contribution is added tothe final TAE as discussed in the appendix of ref. [3]. c Methane, ethane, ethene, and acetylene from ref [2]; propane, propene, propyne, and allene this work (see computationaldetails section). d The adjunct uncertainties correspond to 95% confidence intervals. e ATcT reference values for methane, ethane, ethene, acetylene, and propane are taken from refs. [88]. f Using ATcT atomic heats of formation at 0 K in converting the CCCBDB[45] molecular heat of formation at 0 K into anatomization energy (see computational details section). g Using CODATA atomic heats of formation at 0 K in converting the CCCBDB[45] molecular heat of formation at 0 K intoan atomization energy. h Here ˆ T ≈ . E [CCSDT(Q)/cc-pVTZ(no d on H)] − E [CCSDT/cc-pVTZ(no d on H)] + E [CCSDTQ/cc-pVDZ(no p on H)] − E [CCSDT(Q)/cc-pVDZ(no p on H)]. i The ˆ T contribution approximated as CCSDTQ(5) Λ /cc-pVDZ(no d ) − CCSDTQ/cc-pVDZ(no d ). ABLE II: Component breakdown of the final W3.2lite total atomization energies and fromhypohomodesmotic and isodesmic reactions (in kcal/mol).
SCF valence valence ˆ T − (T) (Q) inner relativ. spin DBOC M-A a TAE e ZPVE b TAE CCSD (T) shell orbitmethane 331.54 84.76 2.94 -0.12 0.09 1.28 -0.19 -0.08 0.10 0.02 420.32 27.57 392.75ethane 558.00 146.40 6.43 -0.33 0.25 2.45 -0.39 -0.17 0.14 0.04 712.81 45.98 666.83propane 785.21 209.08 10.21 -0.56 0.43 3.59 -0.58 -0.25 0.20 0.06 1007.35 63.69 943.67 n -butane 1012.62 271.72 13.96 -0.82 0.61 4.81 -0.77 -0.34 0.25 0.08 1302.08 81.26 1220.83 n -pentane 1239.85 334.45 17.75 -1.07 0.79 5.99 -0.96 -0.44 0.30 0.10 1596.73 98.88 1497.87isobutane 1012.93 272.63 14.17 -0.84 0.62 4.81 -0.76 -0.34 0.26 0.08 1303.51 81.01 1222.51neopentane 1240.03 337.34 18.41 -1.16 0.82 5.97 -0.95 -0.42 0.32 0.10 1600.41 98.56 1501.86isobutene 895.34 244.67 15.09 -0.89 0.74 4.92 -0.71 -0.34 0.24 0.08 1159.10 66.44 1092.67Hypohomodesmotic reactions, eq. 5 n -butane 0.20 -0.03 -0.03 -0.01 0.00 0.07 0.00 [0] 0.00 [0] 0.19 0.13 0.32 n -pentane 0.21 0.02 -0.02 -0.03 0.01 0.11 0.00 [0] -0.01 [0] 0.30 0.22 0.52Isodesmic reactions, eq. 5isobutane 0.51 0.88 0.18 -0.03 0.01 0.07 0.01 [0] 0.00 [0] 1.62 0.39 2.01neopentane 0.39 2.90 0.64 -0.11 0.04 0.09 0.02 [0] 0.02 [0] 3.99 0.53 4.52Isodesmic reactions, eq. 6isobutane 2.00 2.96 0.75 -0.09 0.04 0.03 0.02 [0] 0.03 [0] 5.73 1.80 7.53neopentane 2.63 6.03 1.49 -0.20 0.08 0.03 0.03 [0] 0.05 [0] 10.15 2.65 12.80Isodesmic reactions balance the number of 1,3-interactions on both sidesisobutane+3C H → CH +3C H -0.24 -0.17 -0.11 0.00 0.00 0.09 0.00 [0] -0.01 [0] -0.43 -0.32 -0.76neopentane+8C H → +6C H -1.85 -0.23 -0.22 -0.02 -0.01 0.15 0.00 [0] -0.02 [0] -2.18 -1.59 -3.77Isodesmic reactions balance the number of CH groups on both sides2isobutane+CH → H → H → C H +C H a The difference between the ACES II and MOLPRO definitions of the valence ROCCSD(T). b B3LYP/pc-2 harmonic frequencies scaled by 0.985 ABLE III: Total atomization energies at 0 K and heats of formation at 298 K (in kcal/mol).‘Quasi-W4’ atomization energies are estimated from the hypohomodesmotic reaction eq. 5 forlinear alkanes and from the isodesmic reaction eq. 5 for branched alkanes.
TAE ea TAE H − H ∆ H ◦ f, Quasi-W4 b Quasi-W4 c W1h W3.2lite Quasi-W4 b Quasi-W4 c Expt. d Theor. o TRC n ‘Quasi-W4’ CCCBDB WebbookTRC m Rossini e Other n -butane 1302.90 1302.89 1221.42 1220.83 1220.06 1220.04 1220.10 ± ± ± ± g n -pentane 1597.84 1597.81 1498.61 1497.87 1497.04 1497.01 1497.33 ± N/A 5.70 5.78 -34.84 -35.09 -35.00 ± ± h -35.08 ± i n -hexane 1892.79 1775.91 1774.15 1774.28 ± N/A 6.79 6.86 -39.84 -39.89 -39.96 ± j n -heptane 2187.76 2053.14 2051.17 2051.40 ± f ± k n -octane 2482.72 2330.46 2328.30 2328.45 ± N/A f ± l isobutane 1304.35 1304.31 1223.13 1222.51 1221.76 1221.73 1221.98 ± ± ± ± g isopentane 1598.74 1499.83 1498.27 1498.48 ± N/A 5.30 5.26 -36.49 -36.74 -36.92 ± ± h -36.73 ± i neopentane 1601.53 1601.46 1502.63 1501.86 1501.06 1501.01 1502.14 ± N/A 4.84 5.54 -39.69 -40.13 -39.67 ± ± h -40.14 ± i isohexane 1893.67 1776.99 1775.22 1775.53 ± N/A 6.28 6.29 -41.42 -41.73 -41.66 ± ± N/A 6.37 6.23 -40.72 -41.11 -41.02 ± ± N/A 6.07 5.85 -42.08 -42.04 -42.49 ± ± N/A 5.98 6.01 -43.77 -43.98 -44.35 ± ± N/A f ± k neoheptane 2190.19 2056.38 2054.41 7.01 6.98 -48.84 -49.29 ± k hexamethylethane 2484.09 2332.98 2330.81 7.52 7.53 -53.68 -53.99 ± l isooctane 2484.26 2332.84 2330.67 2330.86 ± f ± a W4 zero-point exclusive, nonrelativistic, clamped-nuclei atomization energies for testing/parametrization of DFTfunctionals. b Using W1h reaction energies and W4 values for ethane and propane. c Using W3.2lite reaction energies and W4values for ethane and propane. d n -octane and isooctane from ref. [89], all the rest from the experimental data section of ref.[45]; using ATcT atomic heats of formation at 0 K in converting the molecular heat of formation at 0 K into an atomizationenergy (see computational details section). e n -butane and isobutane data were instead taken from E. J. Prosen, F. W.Maron, and F. D. Rossini, J. Res. NBS , 106 (1951), higher alkanes from E. J. Prosen and F. D. Rossini, J. Res. NBS ,263 (1945). f Heats of formation at 298 K were converted to 0 K using the TRC enthalpy functions, H − H , for themolecules (see table) and the CODATA[46] enthalpy functions for the elemental reference states: H − H [H (g)]=2.024 ± H − H [C(cr,graphite)]=0.251 ± g D. A. Pittam and G. Pilcher J.Chem. Soc. Faraday Trans. 68, 2224-2229 (J. Chem. Soc. Faraday Trans. 1, 1972, 68, 2224-2229). h G. Pilcher and J. D. M.Chadwick, Trans. Faraday Soc. 63, 2357-2361 (1967). i W. D. Good, J. Chem. Thermodyn. 2, 237-244 (1970). j W. D. Goodand N. K. Smith J. Chem Eng. Data 14, 102-106 (1969); liquid combined with Rossini vaporization enthalphy (D. R.Burgess). k G. F. Davis and E. C. Gilbert J. Am. Chem. Soc. 63, 2730-2732 (1941); liquid combined with Rossinivaporization enthalphy (D. R. Burgess). l W. D. Good, J. Chem. Thermodyn. 4, 709-714 (1972); liquid combined withRossini vaporization enthalphy (D. R. Burgess). m TRC (Thermodynamic Research Center) database[48]. n Ref.[48] All valuesexcept for n -butane appear to have been taken from Ref.[58]. For n -butane, the latter source lists 4.71 kcal/mol. o Thetranslational, rotational, and vibrational contributions were obtained within the RRHO (rigid rotor-harmonic oscillator)approximation from the B3LYP/pc-2 calculated geometry and harmonic frequencies. Internal rotation corrections wereobtained using the Ayala-Schlegel method[49], again on the B3LYP/pc-2 potential surface. Conformer corrections were takenfrom a recent benchmark study[50]. The specific data given here were obtained at theB2K-PLYP-D/pc-2//PW6B95/6-311G** level. See Computational Methods section for further details. ABLE IV: Root mean square deviations (relative to our best values, in kcal/mol) of popularcompound thermochemistry methods for the atomization and isomerization energies of the alkanesconsidered in the present work. zero-point exclusive zero-point inclusiveLarge set a Small set b Isomerization c Large set a Small set b Isomerization c G1 11.21 7.61 1.19 7.56 4.98 0.97G2(MP2) 5.87 3.90 1.17 2.24 1.31 0.96G2 4.27 2.82 1.16 0.77 0.47 0.94G3B3 3.16 2.13 0.94 0.73 0.46 0.86G3 3.73 2.57 1.05 0.45 0.36 0.83G4(MP2) 1.31 1.03 1.00 0.77 0.61 1.01G4 1.29 0.95 0.93 0.74 0.53 0.95CBS-QB3 2.78 1.76 0.79 2.50 1.53 0.79CBS-APNO d a Includes all the C –C alkanes considered in the present work. b Includes all the C –C alkanes (w/o isopentane). c For the linear → branched isomerization reactions. d Not including isooctane, for which the calculation was deemed too demanding in computer time. ABLE V: Performance statistics (in kcal/mol) of various exchange-correlation functionals withand without the s correction for the atomization energies of the C –C alkanes considered in thepresent work. uncorrected correctedFunctional Basis set s RMSD MSD MAD RMSD MSD MAD s , opt RMSD opt
SVWN5 pc-2 -0.25 231 218 218 227 215 215 -16 44PBE pc-2 0.75 13.70 12.65 12.71 24.07 22.08 22.10 -0.93 3.75HCTH407 pc-2 1.10 23.95 -21.90 21.90 8.49 -8.07 8.07 1.69 1.50BLYP pc-2 1.20 36.70 -33.86 33.86 19.93 -18.77 18.77 2.58 4.03TPSS pc-2 1.00 4.22 3.56 3.91 16.81 16.13 16.13 -0.16 3.53M06-L pc-2 0.20 12.27 -11.58 11.58 9.50 -9.06 9.06 0.86 2.05PBE0 pc-2 0.70 1.70 0.55 1.29 10.57 9.35 9.57 -0.04 1.61B3PW91 pc-2 1.10 7.02 -6.00 6.00 8.63 7.83 7.85 0.49 0.90X3LYP pc-2 0.85 13.53 -11.91 11.95 1.57 -1.22 1.39 0.96 0.43B97-3 pc-2 0.90 15.63 -14.29 14.29 3.03 -2.97 2.97 1.10 0.91B3LYP pc-2 1.05 16.60 -14.69 14.74 1.86 -1.49 1.67 1.17 0.64B97-2 pc-2 1.05 17.03 -15.77 15.77 2.59 -2.57 2.57 1.20 1.48B97-1 pc-2 0.65 20.41 -14.29 14.29 11.38 -2.97 2.97 1.43 2.85TPSSh pc-2 0.90 3.69 2.82 3.38 14.66 14.14 14.14 -0.11 3.34TPSS1KCIS pc-2 0.90 6.33 -5.02 5.18 6.64 6.29 6.29 0.44 1.28PW6B95 pc-2 0.50 7.03 -6.27 6.27 0.21 0.02 0.15 0.50 0.21BMK pc-2 0.65 7.39 -6.82 6.82 1.94 1.36 1.54 0.52 0.65M06-2X pc-2 0.06 7.42 -7.13 7.13 6.59 -6.37 6.37 0.51 1.63M06 pc-2 0.25 7.56 -7.12 7.12 4.16 -3.97 3.97 0.52 1.51B1B95 pc-2 0.75 12.87 -11.89 11.89 2.52 -2.46 2.46 0.91 1.18B2K-PLYP pc-3 a a a a a b b b b b a (frozen core) pc-3 basis set combined with a CBS extrapolation where Nmin=15 as recommended in Ref. [90]. b (frozen core) pc-2 basis set combined with a CBS extrapolation where Nmin=10 as recommended in Ref. [90]. ABLE VI: Performance statistics (in kcal/mol) of various exchange-correlation functionals withand without the s correction for the reaction energies of the isomerization linear-alkane → branched-alkane for the C –C alkanes considered in the present work. uncorrected correctedFunctional Basis set s RMSD MSD MAD RMSD MSD MADSVWN5 pc-2 -0.25 0.67 -0.56 0.56 0.44 0.30 0.30PBE pc-2 0.75 2.90 2.34 2.34 0.36 -0.25 0.28HCTH407 pc-2 1.10 5.39 4.26 4.26 0.68 0.46 0.46BLYP pc-2 1.20 4.52 3.65 3.65 0.68 -0.50 0.51TPSS pc-2 1.00 3.50 2.90 2.90 0.87 -0.55 0.60M06-L pc-2 0.20 1.07 0.94 0.94 0.48 0.25 0.40PBE0 pc-2 0.70 2.79 2.26 2.26 0.26 -0.15 0.19B3PW91 pc-2 1.10 3.82 3.11 3.11 0.95 -0.68 0.69X3LYP pc-2 0.85 3.65 2.94 2.94 0.13 0.00 0.10B97-3 pc-2 0.90 3.89 3.15 3.15 0.17 0.04 0.12B3LYP pc-2 1.05 3.99 3.22 3.22 0.56 -0.41 0.42B97-2 pc-2 1.05 4.18 3.38 3.38 0.38 -0.24 0.29B97-1 pc-2 0.65 3.09 2.51 2.51 0.33 0.26 0.28TPSSh pc-2 0.90 3.42 2.83 2.83 0.55 -0.28 0.36TPSS1KCIS pc-2 0.90 3.74 3.06 3.06 0.27 -0.04 0.19PW6B95 pc-2 0.50 2.09 1.76 1.76 0.28 0.03 0.20BMK pc-2 0.65 1.52 1.27 1.27 1.31 -0.98 0.98M06-2X pc-2 0.06 0.21 0.19 0.19 0.15 -0.02 0.10M06 pc-2 0.25 0.26 -0.15 0.18 1.31 -1.01 1.01B1B95 pc-2 0.75 2.47 2.09 2.09 0.83 -0.50 0.54B2K-PLYP pc-3 a a a a a b b b b b a (frozen core) pc-3 basis set combined with a CBS extrapolation where Nmin=15 as recommended in Ref. [90]. b (frozen core) pc-2 basis set combined with a CBS extrapolation where Nmin=10 as recommended in Ref. [90]. ABLE VII: Performance statistics (RMSD, kcal/mol) of various exchange-correlation functionalswith and without the s correction for the isodesmic reaction 6 for the C –C alkanes consideredin the present work. uncorrected corrected n -alkane iso-alkane neo-alkane all n -alkane iso-alkane neo-alkane allFunctional Basis set s (6 species) (4 species) (3 species) (13 species) (6 species) (4 species) (3 species) (13 species)VWN5 pc-2 -0.25 0.38 0.80 0.90 0.68 0.61 0.98 1.03 0.86PBE pc-2 0.75 3.35 5.53 6.12 4.88 0.46 0.31 0.39 0.40HCTH407 pc-2 1.10 4.42 8.18 9.16 7.09 0.19 0.49 0.72 0.46BLYP pc-2 1.20 4.99 8.34 9.24 7.33 0.37 0.22 0.17 0.29TPSS pc-2 1.00 4.94 7.74 8.47 6.87 1.09 0.84 0.84 0.96M06-L pc-2 0.20 3.29 4.45 4.75 4.07 2.52 3.07 3.22 2.88PBE0 pc-2 0.70 3.39 5.55 6.03 4.87 0.69 0.67 0.67 0.68B3PW91 pc-2 1.10 4.53 7.48 8.16 6.56 0.29 0.32 0.33 0.31X3LYP pc-2 0.85 4.13 6.87 7.54 6.03 0.86 0.93 1.03 0.93B97-3 pc-2 0.90 4.65 7.60 8.32 6.69 1.19 1.32 1.43 1.29B3LYP pc-2 1.05 4.52 7.52 8.26 6.60 0.48 0.25 0.25 0.37B97-2 pc-2 1.05 4.65 7.81 8.55 6.83 0.61 0.50 0.52 0.55B97-1 pc-2 0.65 3.93 6.30 6.90 5.57 1.43 1.76 1.92 1.67TPSSh pc-2 0.90 4.79 7.54 8.21 6.68 1.33 1.29 1.33 1.31TPSS1KCIS pc-2 0.90 4.76 7.67 8.42 6.78 1.30 1.40 1.53 1.39PW6B95 pc-2 0.50 3.43 5.18 5.66 4.64 1.50 1.70 1.84 1.65BMK pc-2 0.65 3.34 4.69 4.92 4.22 0.84 0.44 0.29 0.63M06-2X pc-2 0.06 1.35 1.71 1.62 1.55 1.12 1.29 1.16 1.19M06 pc-2 0.25 1.74 1.88 1.52 1.75 0.78 0.41 0.44 0.60B1B95 pc-2 0.75 3.86 5.93 6.46 5.28 0.97 0.76 0.74 0.85B2K-PLYP pc-3 a a a a a b b b b b a (frozen core) pc-3 basis set combined with a CBS extrapolation where Nmin=15 as recommended in Ref. [90]. b (frozen core) pc-2 basis set combined with a CBS extrapolation where Nmin=10 as recommended in Ref. [90].
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