Ab Initio Coupled Cluster Determination of the Heats of Formation of C

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Ab Initio Coupled Cluster Determination of the Heats of Formation of C2H2F2, C2F2, and C2F4 David Feller,*,† Kirk A. Peterson,† and David A. Dixon‡ † ‡

Department of Chemistry, Washington State University, Pullman, Washington 99164-4630, United States Department of Chemistry, The University of Alabama, Shelby Hall, Box 870336, Tuscaloosa, Alabama 35487-0336, United States ABSTRACT: Heats of formation at 298.15 K were computed for cis- and trans-1,2-difluoroethylene, 1,1-difluoroethylene, difluoroethyne, and tetrafluoroethylene using conventional coupled cluster theory through perturbative triples and basis sets up through augmented 6ζ or 7ζ quality. As an independent check, calculations were also performed with the explicitly correlated F12b method and basis sets up through 4ζ quality. F12b calculations converge to the basis set limit much more rapidly than the conventional method. Both approaches were subsequently extrapolated to the complete basis set limit using a variety of simple formulas. After the 1-particle basis set limits were established, corrections were applied for core/valence, scalar relativistic, non-Born-Oppenheimer, and higher order correlation effects through explicit quadruple excitations. A final correction for the remaining error relative to full configuration interaction theory, the exact result for a given basis set, was included. Although this general approach produced excellent agreement with recently reported semiexperimental structures for the difluoroethylene compounds, the level of agreement with the available tabulated experimental heats of formation for 1,1-difluoroethylene is considerably poorer. The best theoretical ΔHf0(298K) estimates (with experimental values from tabulations given in parentheses) are 1,1-difluoroethylene = -84.1 ( 0.6 kcal/mol (-80.5 ( 0.2); cis-1,2-difluoroethylene = -74.4 ( 0.6 kcal/mol (-73.3 ( 1.2); trans-1,2-difluoroethylene = -73.5 ( 0.6 kcal/mol (-72.6 ( 1.2). In the other two cases, where Active Thermochemical Table values were available, the agreement was much better: C2F2 = 1.0 ( 0.3 (0.7 ( 0.4) and C2F4 = -161.3 ( 0.8 kcal/mol (-161.8 ( 0.2). Considering the demonstrated accuracy of the approach in previous studies for the three difluoroethylene equilibrium structures and its performance in predicting the enthalpies of formation of over 100 small-to-medium size chemical systems, we believe the magnitude of disagreement between theory and experiment for 1,1-difluoroethylene merits a re-examination of the experimental value.

’ INTRODUCTION New gas phase infrared (IR) spectra of CF2dCHD, CF2dCD2 and five isotopic species have recently been published, together with a semiexperimental equilibrium structure for 1,1-difluoroethylene, CF2dCH2.1 The latter offers a more direct comparison with theoretical “bottom of the well” re structures than typical (r0, rg, rR, rz, etc.) structures obtained from spectroscopic or diffraction-based experimental measurements.2-5 Semiexperimental structures are derived from a combination of vibration-rotation interaction constants (spectroscopic Re’s) generated by low-level perturbation theory or density functional theory and experimental data.6-10 Similar structures have also been reported for the cis and trans forms of 1,2-difluoroethylene. High level coupled cluster structures, which are available for all three compounds, are in excellent agreement with experiment with the observed deviations for bond lengths less than 0.001 Å and for bond angles less than 0.05°.11 Several of these molecules have been the subject of a large number of previous studies. A comprehensive review of the literature on them is beyond the scope of this investigation. We will briefly mention only a few relevant findings. In a 1986 study based on configuration interaction wave functions at the singles r 2011 American Chemical Society

and doubles level (CISD) with a double-ζ basis set augmented by d functions on the carbons, Dixon et al. reported geometries and heats of formation from isodesmic reactions at 298 K (cisC2H2F2 = -71.0 kcal/mol, trans-C2H2F2 = -70.0 kcal/mol).12 Additional calculations showed that the effects of electron correlation on the cis/trans energy difference was very small.13 In 2007 Hrenar et al. reported anharmonic vibrational frequencies for trans-1,2-difluoroethylene using local coupled cluster methods.14 Harding et al.15 predicted the 19F nuclear magnetic shielding constants of the C2H2F2 isomers using coupled cluster theory through perturbative triples, CCSD(T).16-19 In 2009 Klopper et al.20 determined a zero point inclusive atomization energy for 1,1-difluoroethylene of 562.3 kcal/mol based on frozen core (FC) CCSD(T) with a cc-pCVQZ basis set. The frozen core approximation excludes the carbon and fluorine 1s pairs of electrons from the correlation treatment. The considerable basis set truncation error present with a quadruple-ζ basis set was addressed by means of explicitly correlated MP2-F12 calculations. The latter contribution was empirically scaled to fit a Received: December 7, 2010 Revised: January 10, 2011 Published: February 9, 2011 1440

dx.doi.org/10.1021/jp111644h | J. Phys. Chem. A 2011, 115, 1440–1451

The Journal of Physical Chemistry A

Figure 1. 1,1-Difluoroethylene (a), cis-1,2-difluoroethylene (b), trans1,2-difluoroethylene (c), tetrafluoroethylene (d), and difluoroethyne (e).

set of Active Thermochemical Table (ATcT)21,22 data on small molecules containing H, C, N, O, and F. In a subsequent paper appearing in 2010, Klopper et al. replaced the scaled MP2 correction with an unscaled CCSD-F12 correction.23 For the same set of 73 molecules, the 95% confidence limit rose slightly from 0.57 kcal/mol (scaled) to 0.81 kcal/mol (unscaled) and the maximum errors rose from 0.98 kcal/mol (scaled) to 1.48 kcal/ mol (unscaled). The atomization energy of 1,1-difluoroethylene increased by 0.24 kcal/mol, but other fluorinated hydrocarbons (such as CF4) changed by more than 0.7 kcal/mol. Two 1999 theoretical papers reported heats of formation for C2F4. The first, by Dixon et al.,24 incorporated some of the same components as the present study, although with smaller basis sets. CCSD(T) calculations with basis sets up through aug-ccpVQZ were performed at Møller-Plesset perturbation theory (MP2) geometries using double and triple-ζ basis sets. The zero point vibrational energy was based on the average of the CCSD(T)/aug-cc-pVDZ harmonic frequencies and the experimental fundamentals. Core/valence (CV) and scalar relativistic corrections were considered, but higher order correlation effects were unavailable at the time. The authors noted that the experimental 0 K heat of formation of C2F4 listed in the NIST-JANAF tables25 (-156.6 ( 0.7 kcal/mol) was too positive by more than 3 kcal/ mol. The calculated value would correspond to a 298 K heat of formation of -160.6 ( 1.5 kcal/mol. The second paper, by Bauschlicher and Ricca,26 called into question the error bars of the earlier theoretical paper. Using a similar approach to that of Dixon et al., they arrived at a final ΔHf0(298) = -160.5 ( 1.5 kcal/ mol, nearly identical to the earlier value. Thus, the later work supported both the value and the error bars of the work by Dixon. Furthermore, they suggested that the error was more likely to favor a more negative heat of formation. The 2009 work of Klopper at al. listed an atomization energy for C2F4 of 574.1 kcal/mol.20 This value would correspond to 0 and 298 K heats of formation of -160.3 and -161.1 kcal/mol, in good agreement with the much earlier values of Dixon et al. and Bauschlicher and Ricca. The purpose of the current work is to accurately predict the heats of formation for (1A1) cis-1,2-difluoroethylene, (1Ag) trans1,2-difluoroethylene, (1A1) 1,1-difluoroethylene, (1Ag) difluoroethyne, and (1Ag) tetrafluoroethylene (Figure 1) using a composite theoretical procedure10 closely related to the one used for obtaining the equilibrium structures.1 Experimental heats of formation for the cis and trans isomers of 1,2-C2H2F2 are not well established and carry error bars in excess of (1 kcal/ mol. Although the 1,1-C2H2F2 isomer has been assigned a smaller set of experimental error bars ((0.2 kcal/mol), the claimed level of accuracy may be unrealistic in view of the results obtained in this study. ATcT atomization energies have recently been published for C2F2 and C2F4.20 Our current approach is expected to be capable of achieving significantly better than (1 kcal/mol accuracy based on prior

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performance.8,10,27,28 A statistical comparison involving the heats of formation of 109 molecules whose experimental uncertainties were reported to be e1 kcal/mol showed a mean absolute deviation (εMAD) of 0.17 kcal/mol (root-mean-square error = 0.25 kcal/mol, maximum error = 0.98 kcal/mol for OCS 1Σþ) using the Computational Results Database (CRDB).29 The CRDB currently holds atomization energies for 242 molecules, including many of the 105 systems included in the study of Bakowies30 and later used by Klopper et al.20 If the statistical comparison was limited to molecules with experimental error bars of less than (0.2 kcal/mol, εMAD for our approach dropped to 0.10 kcal/mol (58 comparisons), with a maximum error of -0.42 kcal/mol for CF. Given the inherent uncertainty in the experimental values, this represents a very high level of agreement. In particular, we have shown good agreement between theory and experiment for the 298 K heats of formation of CF (-58.9 ( 0.2 theory vs -58.9 ( 0.15 expt), CF2 (-46.4 ( 0.3 theory vs -45.7 ( 0.3 expt), and CF3 (-112.1 ( 0.4 theory vs -112.4 ( 1.0 expt), with values expressed in kcal/mol.10 One (of two) leading sources of error in electronic structure calculations is due to the truncation of the 1-particle basis set expansion. To address this issue, two independent strategies will be employed for reliably establishing the CBS limit, where the 1-particle error vanishes. The first approach will involve standard coupled cluster theory calculations performed at the CCSD(T) level with large Gaussian basis sets. The second, complementary approach will replace CCSD(T) with explicitly correlated CCSD(T)-F12b calculations. By including nonlinear terms in the interelectronic distance, rij, the latter method converges to the CBS limit much more rapidly than standard CCSD(T).31-35 Sources of uncertainty in the results will be discussed to the extent they limit the achievable accuracy.

’ COMPUTATIONAL PROCEDURE Geometries. CCSD(T)(FC) geometry optimizations were performed on all species with the sequence of diffuse function augmented basis sets developed by Dunning and coworkers.36-44 The first six correlation consistent sets are conventionally labeled aug-cc-pVnZ (n = D, T, Q, 5 - 7). In this paper we sometimes abbreviate the names as aVnZ. The largest of the basis sets, aug-cc-pV7Z, corresponds to a [9s,8p,7d,6f,5g,4h,3i,2k] contraction on carbon and fluorine. It was only possible to use the full aV7Z basis set on C2F2. The geometry optimization for this molecule was performed without the presence of k functions. The k function contribution to the total energy was estimated by exploiting the uniformity in the correlation energy convergence pattern as a function of lmax, the largest angular momentum present in the basis set, as described elsewhere.8,45-47 Tests of this approximation against k functioninclusive calculations on a series of first row atoms and diatomic molecules established the accuracy of this estimate as (10-5 Eh. Errors in energy differences were e0.05 kcal/mol. By convention, the correlation consistent basis sets use only the spherical components of the Cartesian Gaussian basis functions, i.e., 5d, 7f, 9g, etc. CCSD(T)-F12b optimizations were performed with the ccpVnZ-F12 (n = D, T, Q) orbital basis sets48 combined with OptRI auxiliary basis sets,49 which are needed for the complementary auxiliary basis set (CABS) resolution of the identity step.50 Density fitting of the Fock and exchange matrices was accomplished with the cc-pVnZ/JKFIT basis sets of Weigend.51 1441

dx.doi.org/10.1021/jp111644h |J. Phys. Chem. A 2011, 115, 1440–1451

The Journal of Physical Chemistry A The geminal Slater exponent (β) values were taken from the recommendations in Table 5 of Hill et al.,52 namely 0.9 a0-1 (VDZ-F12), 1.0 a0-1 (VTZ-F12), and 1.0 a0-1 (VQZ-F12). The diagonal, fixed amplitude ansatz was used. The MP2 density fitting basis sets, used for all conventional 2-electron integrals needed in F12 except the integrals involving the Coulomb and exchange basis sets, were taken from the work of H€attig.53 Within the CCSD(T)-F12b method, the quasi-perturbative treatment of the triple excitations include no explicit correlation. Consequently, this piece of the F12 calculation converges at essentially the same rate as the triples portion of standard CCSD(T). All standard CCSD(T) and CCSD(T)-F12 calculations were performed with MOLPRO 2009.1.54,55 CBS estimates for CCSD(T)(FC) bond lengths were based on an exponential formula. We have recently discussed direct extrapolation of bond lengths versus geometry optimization on the CBS potential energy surface.55 The former simplification was found to yield predictions within 0.0005 Å of the latter for C-C distances when used with geometries up through aug-ccpV6Z. The agreement was within 0.0003 Å if aug-cc-pV7Z basis set geometries were available for the extrapolation. Atomization Energies. Vibrationless atomization energies, denoted here as ΣDe, that were obtained from standard CCSD(T) calculations extrapolated to the CBS limit using five simple formulas taken from the chemical literature. These include an exponential, a mixed Gaussian/exponential, 1/(lmax þ 0.5)4, 1/(lmax)3, and Schwenke’s collection of formulas.56-62 The first two formulas require three consecutive energies, e.g., aVDZ, aVTZ, and aVQZ, and the others require just two energies. Experience has shown that the 2-point formulas involving lmax work best when the smallest basis set is of aug-cc-pVQZ quality or better, which more closely match the saturated l space condition assumed in the 1/Z perturbation theory expansion of 2-electron, He-like systems examined by Schwartz which motivated these expressions.63,64 In this study, all of the extrapolation formulas, except the Schwenke expressions, were used with the total energies because, with basis sets of the quality used here, changes in the correlation energy heavily dominate changes in the Hartree-Fock energy. We have adopted the average of the formulas as our best estimate. The spread in CBS estimates serves as a rough estimate of the uncertainty in the average. Tests on a large number of molecules indicate that this estimate of the uncertainty for CBS(456) atomization energies may be overly conservative. For example, in the case of C2F2 we obtain an estimated uncertainty of (0.33 kcal/mol using half the spread in the five CBS(456) extrapolations. However, the CBS(567) value differs by only 0.01 kcal/ mol from the smaller basis set result. The F12b energies were extrapolated with the Schwenke-style formulas recommended by Hill et al.52 who developed extrapolation coefficients by fitting to a set of 14 reference molecules. For purposes of computing atomization energies, atomic open shell calculations on carbon and fluorine were based on the R/ UCCSD(T) method. It begins with restricted open-shell Hartree-Fock (ROHF) orbitals but allows a small amount of spin contamination in the solution of the CCSD equations.79-82 Orbital symmetry equivalencing was imposed. To improve upon the frozen core CCSD(T) results, it is necessary to account for a number of smaller effects that contribute to the residual error. A final composite estimate, which assumes additivity in the individual corrections, encompasses all physically important factors. The uncertainty in the

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final estimate is based on the uncertainties in all of the major components. This somewhat complex strategy is necessitated by the practical need to avoid a single all-inclusive calculation that would likely prove intractable. The procedure adopted for this study has been demonstrated to yield accurate results for many small molecules.8,10,27,65 Relaxing the frozen core approximation requires the use of special basis sets capable of describing inner shell correlation effects associated with the 1s electrons on carbon and fluorine. For this purpose we used the weighted core/valence sequence of basis sets, cc-pwCVnZ, n = D, T, Q.40 The CV correction is defined as the CV - FC difference in the properties computed at their respective optimal geometries. For the systems examined in this study, the CV correction is the largest of the minor corrections needed to create the final composite values. As with the frozen core component, the CV corrections were extrapolated to the CBS limit with the same five formulas. The F12b method can also be used for determining the CV correction with the cc-pCVnZ-F12 basis sets.66 Scalar relativistic (SR) corrections were obtained from Douglas-Kroll-Hess (DKH) CCSD(T)(FC) calculations67,68 using the recontracted cc-pVTZ-DK basis sets of de Jong et al.69 In first row compounds this small correction is relatively insensitive to the quality of the basis set or the level of theory. We include it here for the sake of consistency with our general approach that seeks to describe molecules composed of elements from throughout the entire periodic table. The relativistic correction is defined as the difference between the DKH value of a property and the corresponding CCSD(T)(FC)/cc-pVTZ nonrelativistic property, both computed at their respective optimal geometries. While CCSD(T) recovers a large fraction of the exact full configuration interaction (FCI) correlation energy, higher order (HO) recovery methods are still required to achieve the very highest levels of accuracy. For this purpose, CCSDT(FC) and CCSDT(Q)(FC) calculations were performed with the MRCC program of Kallay interfaced to MOLPRO.70a,70b Although it would have been preferable to include quadruple excitations in a fully iterative fashion, the number of determinants (∼1011 in the largest case) made geometry optimizations prohibitively expensive. Contributions to the atomization energy from triple and quadruple excitations are often of opposite signs and may individually be much larger than their sum. Consequently, it is important that they be balanced in terms of basis set convergence as much as possible. Experience has shown that combining an (n þ 1)-ζ basis set for the triples with an n-ζ basis set for the quadruples leads to a reasonably balanced combination. This choice reflects the inherently slower basis set convergence rate of the triples. The present results employed a [CCSDT/VTZ þ CCSDT(Q)/VDZ] combination for all systems except C2F2, where it was possible to perform [CCSDT/VQZ þ CCSDT(Q)/VTZ] calculations. On the basis of C2F2 and our experience with other small molecules, the T/VTZ þ (Q)/VDZ combination is expected to overestimate this piece of the HO correction in the other fluorinated systems. For C2F2 the [CCSDT/VTZ þ CCSDT(Q)/VDZ] yields a correction of 0.23 kcal/mol, whereas the [CCSDT/VTZ þ CCSDT(Q)/VDZ] combination obtained with the larger basis sets produces a value of -0.13 kcal/mol. To adjust for this deficiency in the smaller basis set value, we scaled the individual T/VTZ and (Q)/VDZ corrections by the ratio of the T(VQZ/VTZ) and (Q)/(VTZ/VDZ) corrections in C2F2. Specifically, we scaled up the (T) f T 1442


The Journal of Physical Chemistry A correction, which is negative, by 1.205. The T f (Q) correction, which is positive, was scaled down by 0.826. As with the other corrections, the geometries were optimized at each level of theory. We approximated the uncertainties in our HO corrections on the basis of [CCSDT/VTZ þ CCSDT(Q)/VDZ] f [CCSDT/VQZ þ CCSDT(Q)/VTZ] differences for C2H4, CF2, and C2F2. The residual difference between CCSDT(Q) and FCI was gauged via a continued fraction (cf) approximant. This technique was originally formulated by Goodson in terms of Hartree-Fock (HF), CCSD, and CCSD(T) energies.71 Tests of the accuracy of Goodson’s original sequence of energies showed rather disappointing results when compared to explicit FCI results.72 In the present study we used CCSD, CCSDT, and CCSDT(Q) energies. While this approximation is still far from perfect, benchmark calculations showed that in 42 of 49 cases it produced closer agreement with FCI compared to raw CCSDTQ results.10 We expect the same to be true of CCSDT(Q). Contributions from the continued fraction technique were included in our atomization energies, but not in the predicted structures because the effects are very small for the single configuration dominant systems examined here. Values of the T1 diagnostic, which Lee and co-workers have proposed as a measure of the extent of “nondynamical” correlation, were ∼0.012.73,74 In addition to the frozen core HO correction for structures and atomization energies, there is a smaller HO correction associated with core/valence effects. For example, in the case of C2 (1Σgþ), the HO(FC) correction to De is ∼0.5 kcal/mol and the HO(CV) correction is ∼0.2 kcal/mol. While it is feasible to obtain HO CV corrections for diatomics and some triatomics, they rapidly become computationally intractable for larger systems even with a medium size cc-pwCVTZ basis set. For example, in the case of 1 A1 CF2 the number of determinants (through quadruple excitations) approaches 1011. Use of the smaller cc-pwCVDZ basis set produced a correction that was less than half the ccpwCVTZ value. To account in some fashion for this effect, we performed CCSDT(Q)(CV)/cc-pwCVDZ calculations and scaled up the magnitude of the correction by 2.54 based on the cc-pwCVTZ/cc-pwCVDZ ratio for CF2. This ratio is similar to what has been observed in other molecules. The CCSD(T)(CV)/cc-pwCVDZ and CCSD(T)(FC)/cc-pwCVDZ optimal geometries were used for the (Q) calculations. Corrections were also included for the spin-orbit splittings in the isolated atoms because typical electronic structure calculations would not otherwise include these effects. We have chosen to use the tabulated values of Moore, namely 0.083 (C) and 0.385 (F) kcal/mol.83 Diagonal Born-Oppenheimer Correction. A first-order correction to the electronic energy arising from the breakdown of the Born-Oppenheimer approximation was computed at the CCSD(FC)/aug-cc-pVTZ level of theory with CCSD(T)(FC)/ aug-cc-pVTZ optimized geometries using the CFOUR program package.84a,84b Open shell atoms were described with the unrestricted CCSD method. The diagonal Born-Oppenheimer correction (DBOC) to the atomization energy is not expected to be large for any of the systems in this study. We include it here for the sake of consistency with our general approach. This correction is only required for very high accuracy in molecules containing light atoms. It was initially included in our composite approach adopted for a high accuracy study of the heats of formation of OH and H2O where the correction was ∼0.10 kcal/ mol.85,86 The DBOC is generally insensitive to basis set quality,

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the level of correlation recovery or the exact geometry at which it is evaluated. In the case of 1,1-difluoroethylene the RHF value differed by just 0.05 kcal/mol from the CCSD(FC) value. The DBOC was only applied to the atomization energies. Zero Point Energies and Thermal Corrections. Vibrational zero point vibration energies (ZPEs) for all of the systems except C2F2 were obtained from harmonic frequencies at the CCSD(T)(FC)/aug-cc-pVTZ level of theory. These were combined with anharmonic corrections calculated at the MP2(FC)/aug-ccpVDZ level of theory using the expression:75 X ZPE ¼ 0:5ðZPEH þ ZPEF Þ þ χ0 - 0:25ð χii Þ ð1Þ where ZPEH = 0.5(Σωi) and ZPEF = 0.5(Σνi); ωi are the harmonic frequencies, νi are the anharmonic fundamentals, χii are the anharmonicity constants, and the χ0 term is defined in the reference cited. The anharmonic corrections were obtained from Gaussian 03, which includes a second-order perturbative treatment of such effects using finite difference evaluations of third and semidiagonal fourth derivatives.76 Anharmonic corrections are considerably less sensitive to the level of theory than the harmonic frequencies.10 Comparisons with CCSD(T)(FC)/ aug-cc-pVDZ frequencies provided insight into the degree of convergence in the aug-cc-pVTZ results. The harmonic frequencies were determined with a central differencing algorithm that uses step sizes of (0.01 bohrs. We estimated the uncertainty in the ZPE values as half the difference between the aug-cc-pVDZ and aug-cc-pVTZ results. This is based on our experience with smaller molecules where it was possible to obtain harmonic ZPEs with the aug-cc-pVQZ basis set. Core/valence correlation effects were found to have a small impact (0.05-0.07 kcal/mol) on the ZPEs. All of the anharmonic ZPEs reported here include a correction obtained from CCSD(T)(CV)/cc-pwCVTZ calculations. The more difficult-to-determine effect of higher order CV effects on the ZPEs is currently beyond our means to compute. The approach for computing anharmonic ZPEs described above cannot be used on linear molecules. In the case of C2F2, an anharmonic vibrational energy was obtained by combining CCSD(T)(FC)/aug-cc-pVQZ harmonic frequencies with an anharmonic correction obtained from a vibrational configuration interaction (VCI) calculation at the CCSD(T)(FC)/aug-ccpVTZ level of theory. The method is based on the approach of Rauhut77 as implemented in MOLPRO. A many-mode, gridbased expansion was used for the representation of the anharmonic potential, which was truncated at three-mode contributions. The resulting VCI calculations involved up to quadruple excitations and included the Watson correction term.78 Use of aug-cc-pVTZ harmonic frequencies yields a ZPE that is 0.02 kcal/mol smaller. As mentioned previously, we adopted half the difference between the aug-cc-pVDZ and aug-cc-pVTZ values as a rough measure of the uncertainty in the ZPE. For C2F2 this value is 0.17 kcal/mol. Since the aVQZ value differed by only 0.02 kcal/mol, we conclude that this measure of the uncertainty in the ZPE for the other molecules is likely to be conservative, at least when it comes to the frozen core contribution to the property. The 0 to 298.15 K temperature conversion factors were based on standard thermodynamic and statistical mechanics expressions within the rigid-rotor/harmonic oscillator approximation87 and the CCSD(T)(FC) harmonic frequencies and geometries. Temperature corrections for the atoms were taken from Curtiss 1443



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Table 1. Frozen Core CCSD(T) and CCSD(T)-F12b Structures and Electronic Vibrationless Atomization Energies for C2H2F2, C2F2, and C2F4a method

ΣDe

r(CH)

R(CCF)

R(CCH)

CCSD(T)(FC)/CBS(aVQ56)b,c

583.95

1,1-Difluoroethylene 1.32051 (1.32035)d 1.31717 (1.31710)d

CCSD(T)-F12b/VDZ-F12 CCSD(T)-F12b/VTZ-F12

1.07645 (1.07641)d

125.15

119.36

577.97 582.36

1.32212 1.32071

1.31834 1.31744

1.07693 1.07674

125.13 125.15

119.36 119.36

CCSD(T)-F12b/VQZ-F12

583.67

1.32026

1.31694

1.07651

125.15

119.36

CCSD(T)-F12b/CBS(DT)

583.98

1.32034

1.31723

1.07674

125.15

119.36

CCSD(T)-F12b/CBS(TQ)

584.27

1.32008

1.31681

1.07648

125.15

119.36


574.51

1.32677

1.07862

122.18

122.74

CCSD(T)-F12b/VDZ-F12

568.52

1.32827

1.33622

1.07906

122.16

122.74

CCSD(T)-F12b/VTZ-F12

572.92

1.32695

1.33539

1.07879

122.17

122.74

CCSD(T)-F12b/VQZ-F12 CCSD(T)-F12b/CBS(DT)

574.21 574.54

1.32650

1.33487

1.07859

122.18

122.74


574.80


573.48

1.32606

1.07903

119.73

125.38


567.49

1.32752

1.34207

1.07963

119.74

125.37


571.89

1.32618

1.34135

1.07934

119.72

125.37


573.19

1.32572

1.34089

1.07913

119.72

125.38

CCSD(T)-F12b/CBS(DT) CCSD(T)-F12b/CBS(TQ)

573.52 573.79

CCSD(T)(FC)/CBS(aV567)b

382.99

1.18819

Difluoroethyne 1.28482

r(CC)

r(CF)

cis-1,2-Difluoroethylene 1.33508

trans-1,2-Difluoroethylene 1.34118


377.56

1.18969

1.28618


381.56

1.18843

1.28506


382.75

1.18815

1.28464


382.99


383.30

CCSD(T)(FC)/CBS(aVQ56)b

587.23

1.32423


579.85

1.32608

1.31434

123.29


585.23

1.32438

1.31315

123.30


586.94

1.32397

1.31262

123.31


587.18


587.76

Tetrafluoroethylene 1.31287

123.31

a

Atomization energies are given in kcal/mol. Bond lengths are in angstroms and bond angles are in degrees. b The CBS atomization estimates for the standard CCSD(T) method were based on an average of five formulas, as described in the text. Uncertainties in ΣDe are C2H2F2 and C2F4 = (0.4, C2F2 = (0.2 kcal/mol. The CBS bond lengths were estimated by performing an exponential extrapolation of the aug-cc-pVQZ through aug-cc-pV6Z internal coordinates for all molecules except C2F2 where the aug-cc-pV7Z basis set was used. c Feller et al.11 d Includes an adjustment to r(CC) of -0.00016 Å and r(CF) of -0.00007 Å based on CCSD(T)/aug-cc-pV7Z calculations on C2F2. An adjustment to r(CH) of -0.00004 Å was based on CCSD(T)/aug-ccpV7Z calculations on C2H4.

et al.88 The theoretical heats of formation at 0 K depend upon the 0 K heats of formation of the constituent elements. These were taken from the NIST-JANAF Tables, with H = 51.634 ( 0.0014, C = 169.98 ( 0.11, and F = 18.47 ( 0.07 kcal/mol.25

’ RESULTS AND DISCUSSION Structures, Zero Point Energies, and Heats of Formation. Table 1 contains the total atomization energies and optimized structures obtained from frozen core CCSD(T) and CCSD(T)F12b calculations. Systematic improvements in the one-particle

expansion via the progression of correlation consistent basis sets produces a monotonic increase in the total atomization energies accompanied by a corresponding monotonic decrease in the bond lengths with both the standard and explicitly correlated methods. In general, the CCSD(T)-F12b/cc-pVTZ-F12 properties are comparable in quality to the standard method with the much larger aug-cc-pV5Z basis set. As an illustration of the relative computational costs, the 668 basis function aug-cc-pV5Z calculation on 1,1-difluoroethylene required 4097 CPU sec on 12 2.0 GHz Opteron processors compared to just 325 CPU sec for the 248 basis function cc-pVTZ-F12 calculation. Similarly, the 1444


The Journal of Physical Chemistry A CCSD(T)-F12b/cc-pVQZ-F12 level of theory produces results of similar quality to the CCSD(T)/aug-cc-pV6Z values. Note that the F12 basis sets include more functions than their standard method counterparts with the same ζ notation. For example, the cc-pVDZ-F12 basis set consists of a (11s,6p,2d) primitive set contracted to [5s,5p,2d], whereas the standard cc-pVDZ basis set is a (9s,4p,1d) f [3s,2p,1d] contraction. Thus, a cc-pVDZ-F12 calculation on C2F4 would contain the same number of orbital basis functions as a cc-pVTZ calculation, but with a different mixture of angular momentum types. The sizes and angular momentum composition of the cc-pVnZ-F12 series of basis sets is closer to the standard aug-cc-pVnZ sequence. At the CBS limit, the standard method atomization energies in Table 1 are slightly closer to the CCSD(T)-F12b/CBS(DT) values than the F12b/CBS(TQ) values. Most of the standard method values carry an estimated uncertainty of (0.4 kcal/mol. Thus, the slightly larger differences with respect to the F12b/ CBS(TQ) estimates for the three C2H2F2 isomers may not be significant. However, even in the case of C2F2, where the uncertainty in the standard method falls to (0.2 kcal/mol, the F12b/CBS(DT) is fortuitously in exact agreement with it and the CBS(TQ) estimate is 0.3 kcal/mol larger. For C2F4 the F12b/ CBS(TQ) estimate lies slightly outside the CCSD(T) error bars. Although a CCSD(T)(FC)/aug-cc-pV7Z calculation on C2F4 would have allowed us to cut the uncertainty in the atomization energy in half, with more than 1600 basis functions, such a calculation was beyond our current computational capabilities. In place of a single very large calculation, several smaller aV7Z calculations were performed on prototype molecules, including ethylene (C2H4) and 1A1 difluorocarbene (CF2). These results suggest that the CBS(aVQ56) atomization energies for C2H2F2 and C2F4 should be within (0.03 kcal/mol of the prohibitively expensive CBS(aV567) values. It also proved possible to obtain a CBS(aV678) extrapolated atomization energy for 1A1 CF2. The 8-ζ basis set produced a value within 0.01 kcal/mol of the CBS(aV567) value and suggests that the CBS(aVQ56) estimates in Table 1 may be accurate to better than (0.1 kcal/mol. However, for analysis purposes we retain our original, more conservative estimates of the uncertainties. The current findings differ somewhat from the findings of our recent calibration study of hydrocarbons up through C6H14, where the CCSD(T)-F12b/CBS(DT) and CCSD(T)-F12b/ CBS(TQ) extrapolated atomization energies were in close agreement with each other.55 The average deviation between the F12b/CBS(DT) and F12b/CBS(TQ) ΣDe’s for 16 hydrocarbon molecules was 0.10 kcal/mol, with a maximum difference of 0.31 kcal/mol. Because the F12b/CBS(DT) and F12b/CBS(TQ) estimates were so close, comparisons with large basis set standard CCSD(T) results showed no statistical preference for one over the other. In the current case the average deviation is larger (0.35 kcal/mol) and the maximum deviation is almost 0.6 kcal/mol (for C2F4). In addition, the F12b(TQ) estimates are consistently larger than the smaller basis set (DT) values. For the five fluorinated hydrocarbons examined here, the standard method CBS atomization energies more closely match the (DT) values than do the larger basis set (TQ) values. In separate calculations on F2 with basis sets as large as aug-cc-pV8Z, we noticed a slight tendency for F12b to overshoot the CCSD correlation energy. The CCSD(FC)/CBS(aV78) extrapolation of the correlation energy yields a value 0.0004 Eh smaller in magnitude than the saturated reference basis set CCSD-F12b result of Hill et al.52 It may be possible that we are observing a

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similar effect in the current atomization energies, but the difference is small and the difficulty in pinning down the standard method complete basis set limit with sufficient precision makes it difficult to reach an unambiguous conclusion. On the basis of the level of agreement between the CCSD(T) and CCSD(T)-F12b atomization energies, we conclude that the CBS error bars adopted here and in our previous work8,10,24,47,65 provide a meaningful measure of the uncertainty in the CBS extrapolation. If anything, the CBS(aV567) and CBS(aV678) results on C2H4 and CF2 suggest that the error bars may be overly conservative. Although not evident from the results in Table 1, the error bars generated by CBS extrapolations with smaller basis sets in general encompass the values obtained with larger basis sets. We adopt the CBS(aV456), or CBS(aV567) for C2F2, values as our best estimates and will use them as the foundation for the molecular heats of formation. Structural data in Table 1 reveal good agreement between the standard and the F12b methods, although the latter method leads to somewhat shorter bond lengths. The VQZ-F12 C-C and CF bond lengths are roughly 0.0003 Å shorter than the CBS(aVQ56) limit values obtained with the standard method. For C2F2, where aug-cc-pV7Z(no k) bond lengths are available, the CBS(aV567) C-C and C-F bond lengths are 0.00004 and 0.00018 Å longer than the F12/VQZ values, respectively. In an effort to adjust the 1,1-difluoroethylene bond lengths, we report a second set of values (given in parentheses) that have been adjusted by the differences in CBS(aVQ56) and CBS(aV567) values for C2F2 and C2H4. Even with this adjustment, the C-C and C-F bond lengths obtained from the standard CCSD(T) method remain ∼0.0003 Å longer than the corresponding F12b/ CBS(TQ) values. Without explicit aV7Z geometry optimizations of the C2H2F2 and C2F4 compounds, it is difficult to know if the differences between the CBS(aVQ56) bond lengths and the F12b/VQZ-F12 values is significant or just an artifact of the remaining uncertainty in our extrapolated values. Our best equilibrium structures for C2F4 (rCC = 1.3221 Å, rCF = 1.3113 Å, CCF = 123.3°) and C2F2 (rCC = 1.1863 Å, rCF = 1.2834 Å) were obtained from the composite approach used to determine the atomization energies but without the continued fraction estimated FCI and the DBOC contributions. For C2F4, there is semiquantitative agreement with the vibrationally averaged, electron-diffraction structure (rCC = 1.313 ( 0.010 Å, rCF = 1.313 ( 0.035 Å, CCF = 123 ( 2°),89 and the theoretical bond lengths fall within the experimental error bars. Comparison with another electron-diffraction structure with larger uncertainties (rCC = 1.30 ( 0.02 Å, rCF = 1.33 ( 0.06 Å, CCF = 123 ( 3°) shows approximately the same level of agreement.90 For C2F2, the experimental structure listed in the compilation of Kuchitsu (rCC = 1.187 Å, rCF = 1.283 Å) is in good agreement with our present structure.91 Harmonic zero point vibrational energies obtained at the CCSD(T)(FC)/aug-cc-pVTZ and CCSD(T)-F12b(FC)/ccpVTZ-F12 levels of theory are presented in Table 2 together with the gas phase experimental fundamentals.92-94 The F12b results for 1,1-C2H2F2 are seen to be in agreement with the standard method and experimental values. As is generally the case, the harmonic ZPEs increase in size as the quality of the basis set improves, consistent with the decreases in the calculated bond lengths. Although we could not afford to compute CCSD(T)/ aug-cc-pVQZ ZPEs for the systems studied here, it appears likely that the CCSD(T)F12b(FC)/cc-pVTZ-F12 ZPEs would be similar in quality to aug-cc-pVQZ or aug-cc-pV5Z values 1445



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Table 2. CCSD(T) Harmonic Zero Point Vibrational Energies (kcal/mol) and Normal Mode Frequencies (cm-1)a system

basis

1,1-C2H2F2

ZPE

b2

a1

b1

a2

b1

a1

b2

b2

a1

a1

a1

b2

aVTZ

22.93

433.0

548.3

612.5

715.1

810.0

934.0

967.0

1328.9

1412.4

1767.4

3201.9

3308.0

VDZ-F12

22.99

436.2

553.6

616.8

724.2

804.2

939.2

969.6

1334.6

1415.5

1768.3

3204.4

3314.9

438

550

611

714

803

955

962

1302

1360

1728

3070

3154

exptb system

basis

ZPE

a1

a2

b2

b1

a2

a1

b2

a1

b2

a1

b2

a1

cis-1,2-C2H2F2

aVTZ

23.12

232.8

498.1

769.2

770.6

829.8

1018.3

1147.6

1278.9

1394.6

1751.3

3226.6

3252.2

255

482

756

768

866

1014

1127

1266

1376

1715

3135

3135

exptc system

basis

ZPE

bu

au

ag

bg

au

ag

bu

bu

ag

ag

bu

ag

trans-1,2-C2H2F2

aVTZ exptd

22.88

311.8 341

332.8 329

552.0 548

780.9 788

894.3 875

1150.6 1123

1178.7 1159

1290.9 1274

1303.3 1286

1735.2 1694

3232.5 3114

3241.9 3111

system

basis

ZPE

C2F2

aVQZ

8.22

expte system

basis

ZPE

C2F4

aVTZ

13.43

exptc

au

b2u

ag

σgþ

πg

πu

σuþ

σgþ

269.4

272.0

786.2

1365.8

2515.1

268

270

787

1349

2436

b1u

b2g

b3u

b1g

ag

b3u

b2u

b1g

ag

199.2

206.5

396.2

409.8

498.9

553.2

553.9

791.8

1196.3

1360.8

1363.1

1912.6

190

218

394

406

508

558

551

778

1186

1337

1340

1872

a AVTZ = CCSD(T)(FC)/aug-cc-pVTZ. aVQZ = CCSD(T)(FC)/aug-cc-pVQZ. VDZ-F12 = CCSD(T)-F12b/cc-pVDZ-F12. b Experimental IR gas phase fundamentals reported by Kagel et al.92 c Experimental IR gas phase fundamentals reported by Shimanouchi.93 d Experimental IR gas phase fundamentals reported by Craig et al.95 e Experimental IR gas phase fundamentals reported by McNaughton and Elmes.94

determined with the standard method. Because different modes converge at different rates, it is difficult to make a blanket statement regarding F12 and standard method frequencies. Rauhut et al. reported that F12b/VTZ-F12 harmonic frequencies were comparable in quality to CCSD(T)/aV5Z frequencies for five small molecules.35 Their analysis relied on a comparison with CCSD(T)-F12b/aV5Z frequencies. Theoretical total atomization energies are listed in Table 3. The corresponding heats of formation at 0 and 298.15 K are contained in Table 4 together with the available experimental values. The sum of the uncertainties in the various component pieces is used for the final theoretical error bars, i.e., cancellation of error is not assumed. As previously stated, these error bars may be overly conservative. Nonetheless, the current best theoretical values fall significantly outside the experimental error bars for the available values for 1,1-difluoroethylene. For cis- and trans-1,2difluoroethylene, the larger experimental error bars encompass the theoretical values. Compared to the two 1999 theoretical values for ΔHf0(C2H4 298K),24,26 the present value is more negative by a little more than 1 kcal/mol, placing it inside the (1.5 kcal/mol error bars of both earlier works. For C2F2 and C2F4 the present ΔHf0(298K) values are in good agreement with the ATcT atomization energies cited by Klopper et al.20 The current value for 1,1-difluoroethylene (562.61 ( 0.64) is close to the 2009 value of Klopper et al. (562.31 kcal/mol)20 and even closer to the 2010 value (562.55 kcal/mol) by some of the same authors.23 This is despite the differences in our approaches for estimating the frozen core and core/valence CCSD(T) components and the lack of DBOC, estimated FCI corrections and the higher order CV contributions in the approach of Klopper et al. All of these are of the same sign. Similar levels of agreement were found for C2F2 and C2F4. In the case of the latter molecule, the current value (574.38 ( 0.79 kcal/mol) is bracketed by the 2009 (574.14) and 2010 (574.81) values of Klopper et al.

After extrapolating to the CBS limit and correcting for CV, SR, and HO effects, we predict the cis-1,2-difluoroethylene isomer to lie 9.44 kcal/mol above the 1,1-isomer and the trans isomer to be another 1.01 kcal/mol higher in energy. It is difficult to assign error bars to these numbers because significant cancellation of error can be expected because of the similarity of the compounds. The use of nonextrapolated aug-cc-pV6Z and cc-pwCVQZ energies produces essentially the same energy differences. The experimental value for the cis/trans-difluoroethylene electronic energy difference is 1.086 ( 0.022 kcal/mol, as inferred from infrared spectroscopy measurements.95 The current calculations are in good agreement with this value. There is only a small correlation correction to the cis/trans energy difference for 1,2C2H2F2 as the Hartree-Fock value with the aug-cc-pV5Z basis set is 1.11 kcal/mol. For 1,1-difluoroethylene, our calculated heat of formation falls within the error bars of the Cox and Pilcher adjusted heat of formation (-82.2 ( 2.4 kcal/mol) based on the combustion reaction of Kolosev et al.96,97 However, it does not fall within the tabulated, more precisely defined, average value (-80.5 ( 0.2 kcal/mol) given by Cox and Pilcher. Alternative Estimates for ΔHf(C2F4). There have been a variety of experimental results for the heat of formation of C2F4, reported prior to the new ATcT value98-101 The following reactions have been used to obtain this value: ð2Þ C2 F4 þ 4K f 2C þ 4KF

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C2 F4 þ 4Na f 2C þ 4NaF

ð3Þ

C2 F4 þ 2H2 f 2C þ 4HFðaqÞ

ð4Þ

C2 F4 þ O2 f CO2 þ CF4

ð5Þ

C2 F4 þ O2 f 2COF2

ð6Þ



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Table 3. Total Atomization Energies (kcal/mol) for C2H2F2, C2F4, and C2F2 C2H2F2 C2F4 1,1-difluoroethylene

cis-1,2-difluoroethylene

trans-1,2-difluoroethylene

tetrafluoroethylene

583.95 ( 0.35

574.51 ( 0.34

573.48 ( 0.35

587.23 ( 0.44

2.54 ( 0.05

2.50 ( 0.05

2.50 ( 0.05

2.69 ( 0.07

ΔSR CCSD(T)-DKH/cc-pVQZ-DK

-0.68 ( 0.01

-0.70 ( 0.01

-0.69 ( 0.01

-1.06 ( 0.01

ΔHO(1) CCSDT(FC)/(cc-pVQZ)a

-1.14 ( 0.09

contribution ΣDe CCSD(T)(FC)/CBS(aVQ56) ΔCV CCSD(T)(CV)CBS(wCVDTQ)

-0.78 ( 0.08

-0.70 ( 0.08

-0.72 ( 0.08

ΔHO(2) CCSDT(Q)(FC)/(cc-pVTZ)b

0.60 ( 0.05

0.60 ( 0.05

0.61 ( 0.05

0.95 ( 0.05

ΔHO(3) cf. est FCI(FC)/cc-pVDZ

0.08

0.08

0.09

0.15

ΔHO(4) est CCSDT(Q)(CV)/cc-pwCVTZc ΔDBOC CCSD(FC)/aug-cc-pVTZ

0.46 0.08 ( < 0.01

0.46 0.07 ( < 0.01

0.46 0.07 ( < 0.01

0.68 0.07 ( < 0.01

-0.94

atomic SO anharmonic ZPE (CCSD(T)/aVTZ)d best composite ΣD0

-0.94

-0.94

-1.71

-22.70 ( 0.10

-22.95 ( 0.14

-22.66 ( 0.10

-13.48 ( 0.13

562.61 ( 0.64

552.93 ( 0.60

552.20 ( 0.61

574.38 ( 0.79

C2F2 contribution

difluoroethyne

ΣDe CCSD(T)(FC)/CBS(aV567)

382.99 ( 0.18

ΔCV CCSD(T)(CV)CBS(wCVDTQ)

2.94 ( 0.05

ΔSR CCSD(T)-DKH/cc-pVQZ-DK

-0.72 ( 0.01

ΔHO(1) CCSDT(FC)/cc-pVQZ

-1.02 ( 0.07

ΔHO(2) CCSDT(Q)(FC)/cc-pVTZ

0.89 ( 0.02

ΔHO(3) cf est. FCI(FC)/cc-pVDZ

0.16

ΔHO(4) est. CCSDT(Q)(CV)/cc-pwCVTZc ΔDBOC CCSD(FC)/aug-cc-pVTZ

0.46 0.06 ( < 0.01 -0.94

atomic SO anharmonic ZPE (CCSD(T)/aVQZ)d

-8.20

best composite ΣD0

376.62 ( 0.33

a Estimated cc-pVQZ value obtained by scaling the cc-pVTZ value by the VQZ/VTZ CCSDT correction ratio for C2F2. b Estimated cc-pVTZ value obtained by scaling the cc-pVDZ value by the VTZ/VDZ CCSDT(Q) correction ratio for C2F2. c Estimated higher order core/valence contribution based on a CCSDTQ(CV)/cc-pwCVDZ correction scaled by 2.54. The scale factor is based on the cc-pwCVTZ/cc-pwCVDZ ration for CF2 (1A1). d Anharmonic ZPEs including a CCSD(T)(CV)/cc-pwCVTZ correction for core core/valence effects. e Estimated value taken as the sum of the C2F2 and CF2 corrections.

All of these reactions suffer from experimental difficulties and from the use of different values for the known heats of formation, such as the well-known issues with ΔHf0(COF2).102 For example, the heats of formation of KF and NaF used in the above reactions have changed by more than 1 kcal/mol from the original respective values of -134.5 (KF)98 and -136.17 ( 0.3 (NaF)101 kcal/mol. The newest values are -135.9 ( 0.1 (KF) and -137.5 ( 0.2 (NaF) kcal/mol. This change has significant consequences for the experimental values because four salt molecules are produced as products, as has been noted in the literature.25,96 In addition, the form of the carbon that is produced is not well-established. In some cases, it has been assumed to be crystalline, weakly crystalline, and amorphous. Values of 0.0,99 1.5,103 2.5,100 and 3.98101 kcal/mol have all been used for the carbon heat of formation. There are also issues with the purity of the carbon that is formed in terms of the potential production of an alkali carbide and the purity of the starting alkali in terms of an oxide surface. All of these experiments led to values of -164 ( 2 (reaction 2),98 -162 ( 1 (reaction 2),100 -151.8 ( 1.1 (reaction 3),101 -151.7 ( 1.1 (reaction 4),103 and -151.3 (reaction 4)99 kcal/mol. We can correct the values for reactions 2 and 3 by the new ΔHf0 values for KF and NaF as has been done in

the tabulated values to obtain -169.6 (reaction 2), -167.6 (reaction 2), and -156.4 (reaction 3) kcal/mol. If the correction for the heat of formation of the produced carbon is actually closer to 2 kcal/mol rather than to the 4 kcal/mol used for reaction 3,101 then the new 0 K heat of formation from reaction 3 is -160.4 kcal/mol. Perhaps the best experiment for shedding light on the heat of formation of C2F4 corresponds to the heat of reaction (7): ð7Þ C2 F4 f C þ CF4 It allows us to combine the known heat of formation of CF4 and the published heats of reaction of -61.4399 and -63.5 ( 0.4103 kcal/mol. For many years the heat of formation of CF4 was not well-established, with early values such as -212.799 and 217.1103 kcal/mol. However, in recent years the situation has improved. The NIST-JANAF tables list ΔHf0(298K) for CF4 as -223.04 ( 0.31 kcal/mol,25 while Gurvich quotes the same values but reduces the error bars to (0.18 kcal/mol.104 Burcat and Ruscic provide a nearly identical value, but with still smaller error bars, ΔHf0(298K) = -223.09 ( 0.13 kcal/mol.105 A very recent Active Thermochemical Table (ATcT) zero-point inclusive atomization energy of 465.56 ( 0.14 kcal/mol implies a 1447



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Table 4. Heats of Formation (kcal/mol) for C2H2F2, C2F2, and C2F4 molecule 1,1-difluoroethylene

ΔHf(0K) calc

ΔHf(298.15K) calc

-82.4 ( 0.6

-84.1 ( 0.6

ΔHf(0K) expt

ΔHf(298.15K) expt -80.5 ( 0.2a -79.9 ( 0.8b -82.2 ( 2.4c

cis-1,2-difluoroethylene

-72.8 ( 0.6

-74.4 ( 0.6

d

-78.8 ( 1.0

-80.4 ( 1.0d

e

-71.7 ( 1.2

-73.3 ( 1.2e

-70.8 ( 3.1

f

trans-1,2-difluoroethylene difluoroethyne tetrafluoroethylene

-72.0 ( 0.6

-73.5 ( 0.6

0.3 ( 0.3

1.0 ( 0.3

-160.5 ( 0.8

-161.3 ( 0.8

-71.0 ( 1.2e

-72.6 ( 1.2e

-0.01 ( 0.41 4.6 ( 5h

0.67 ( 0.41g 5.0 ( 5h

g

-161.0 ( 0.2g

-161.8 ( 0.2g

h

-156.6 ( 0.7

-157.4 ( 0.7h

d

-156.8 ( 0.6

-157.6 ( 0.6d -156.2 ( 1.3i -158.8 ( 1.1j -169.6 ( 2k -167.6 ( 2l -161.6m -159.5n

Cox and Pilcher average selected value.96 b Cox and Pilcher96 adjusted value from the original value of -77.5 ( 0.8 kcal/mol reported by Neugebauer and Margrave.103 c Cox and Pilcher96 adjusted value from the original value of -79.6 ( 2.4 kcal/mol from Kolosev et al.97 d Gurvich et al.104 e Reference 105. f Jochims et al.111 g ATcT value reported by Klopper et al.20 h NIST/JANAF.25 i Original value is -151.7 ( 1.1 kcal/mol from Neugebauer and Margrave103 and adjusted by NIST/JANAF.25 j Original value is -151.1 ( 1.1 kcal/mol from Kolosev et al.101 and adjusted by Cox and Pilcher. 96 k Original value is -164 ( 2 kcal/mol from Wartenberg and Schiefer98 adjusted for the heat of formation of KF. l Original value is -164 ( 2 kcal/mol from Kirkbride and Davidson100 adjusted for the heat of formation of KF. m Using heat of reaction for reaction 7 from Duus99 and the modern value for ΔHf0(CF4), assuming the heat of formation of the product C to be 0.0. n Using heat of reaction for reaction 7 from Neugebauer and Margrave103 and the modern value for ΔHf0(CF4), assuming the heat of formation of the product C to be 0.0 kcal/mol. a

ΔHf0(298K) of -223.12 ( 0.14 kcal/mol using the ATcT atomic heats of formation.20-22 When we apply our composite theoretical approach through aug-cc-pV6Z to CF4, we find a ΔHf0(298K) of -223.2 ( 0.5 kcal/mol, in excellent agreement with the ATcT value. Other theoretical work on CF4 includes a W4lite zero-point inclusive atomization energy of 465.72 kcal/ mol and an estimated W4.3 value of 465.41 kcal/mol.106 The latter would imply a ΔHf0(298K) of -223.0 kcal/mol. More recently, Klopper et al. reported a zero-point inclusive atomization energy for CF4 of 465.94 kcal/mol.20 This implies a ΔHf0(298K) of -223.5 kcal/mol. Thus, all of the more recent theoretical and experimental heats of formation for CF4 are in good agreement. Use of the newer values for the heat of formation of CF4 together with the two reaction energies for reaction 7 gives ΔHf0(C2F4 298K) = -161.6 and -159.5 kcal/mol with error bars on the order of (1 kcal/mol or larger. The first value is in close agreement with our prediction and the second lies just outside the combined experimental and computational error bars. The first value is also in close agreement with the ATcT value of -161.8 ( 0.2 kcal/mol.20 Bond Dissociation Energies. Adiabatic and diabatic CC bond dissociation energies (BDEs) are presented in Table 5. Diabatic BDEs are defined with respect to the product configurations most closely resembling the bonding configuration in the reactant and a adiabatic BDE is dissociation to the ground state of the separated species.107 An adiabatic BDE will always be equal to or less than the corresponding diabatic BDE. For our systems, the ground states of the products are 3B1 (CH2), 1A0 (CHF), 1A1 (CF2), 2Π (CH), and 2Π (CF). Thus, for CH2 dCH 2 , the adiabatic and diabatic BDEs are the same

Table 5. CdC and CtC Adiabatic and Diabatic Bond Dissociation Energies in kcal/mol at 298.15 Ka molecule

C-C adiabatic

C-C diabatic

CH2dCH2

174.2

174.2

CH2dCF2

130.7

187.2

CHFdCHF(cis)

145.7

174.9

CF2dCF2

68.9

181.9

HCtCH

230.4

265.4

FCtCF

117.8

280.9

The heats of formation of the 3B1 state of CF2 and the 1A0 and 3A00 states for CHF are taken from Vasiliu et al.112 The heats of formation were calculated using a similar approach to the one used here with CCSD(T) calculations up through aug-cc-pVQZ. The differences in the calculated heats of formation for 3B1 CH2 and 1A1 CF2 from the higher level calculations in Feller et al.10 are only 0.5 kcal/mol. Consequently, the ground state for CHF and the excited states for CF2 and CHF were corrected by this amount. a

because the electron configuration in the two 3B1 CH2 products resembles the electron distribution needed to form a σ and π bond in the reactant. Because the ground states of CF2 and CHF are singlets, they do not have the bonding configuration to form the σ and π bond in the reactant and the diabatic and adiabatic CdC BDEs are different. The calculated CdC BDEs show interesting features. The adiabatic CdC BDE for CH2dCF2 is weaker than that for either cis- or trans-CHFdCHF even though CH2dCF2 is more stable. This is related to the fact that the singlet-triplet splitting in CF2 is more than twice that in CHF. Comparing the diabatic BDEs, we find that CH2CH2 has the smallest diabatic CdC BDE and CH2CF2 has the largest. The 1448


The Journal of Physical Chemistry A diabatic BDE for CF2CH2 is larger than that for the cis- and transCHFCHF isomers consistent with the increased stability of the 1,1-isomer. The diabatic CdC BDEs in cis- and transCHFdCHF are essentially the same as in CH2dCH2. The diabatic CdC BDE in CF2CF2 is 7.7 kcal/mol larger than that in CH2CH2 consistent with previous lower level calculations.108,109 We also note that the low value for the adiabatic CdC BDE in CF2CF2 is consistent with its higher reactivity. The fact that the diabatic CdC BDE is largest for CH2dCF2 is consistent with the fact that this molecule is the only one with polarized (ionic) σ and π bonds. The BDEs corresponding to triply bonded carbons behave in the same way. The ground state of both CH and CF are the 2 Π with a 4Σ- excited state that is lower lying in CH than in CF. The adiabatic CtC BDE in FCCF is much less than that in HCCH. The adiabatic CtC BDE in C2H2 is greater than the NtN BDE of N2, 225.9 kcal/mol,25 by about 4 kcal/mol. The diabatic BDE in HCCH corresponds to dissociation to the 4Σstate of CH with three unpaired electrons. We calculated the splitting between the 2Π and 4Σ- states of CH at the R/ UCCSD(T)(FC)/aug-cc-pV6Z level of theory and obtained a splitting of 17.5 kcal/mol. Combining this value with the calculated heat of formation of CH at 298.15 K (142.5 ( 0.1 kcal/mol)10 yields a diabatic BDE for the triply bonded carbons of 265.4 kcal/mol, which is substantially above the diabatic NtN BDE in N2. In the case of N2 the adiabatic and diabatic values are identical because the lowest state for the product N is the 4S state. We calculated the splitting between the 2Π and 4Σ- states for CF and obtained 81.5 kcal/mol. Combining this splitting with the calculated heat of formation of CF at 298 K (58.9 ( 0.2 kcal/ mol)10 where we note that there is a sign error in ref 10) gives a diabatic CtC BDE for FCCF of 280.9 kcal/mol. Thus, the diabatic CtC BDE in FCCF is more than twice the adiabatic value. The larger diabatic CtC BDE for FCCF is consistent with the fact that r(CtC) is shorter in this molecule than in HCCH (1.2024 Å).110

’ CONCLUSIONS A composite theoretical procedure has been used to determine the structures of C2F2 and C2F4 and the heats of formation of all five fluorinated hydrocarbons that are the subject of this study. Despite the application of theoretical methods that produce excellent agreement with experimental heats of formation for over 100 molecules, including CFn, n = 1 - 4, C2F2, and C2F4, the present findings are in only fair agreement with the available experimental heats of formation for 1,1-difluoroethylene. The larger experimental error bars ((1.2 kcal/mol) for the two 1,2-difluoroethylene encompass the theoretical values. Given the documented accuracy of the theoretical approach, a reexamination of the experimental findings for 1,1-difluoroethylene is needed. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT We thank the U.S. Department of Energy, Office of Basic Energy Sciences, for partial support. D.A.D. thanks the Robert

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Ramsay Fund at the University of Alabama for partial support. K. A.P. acknowledges the support of the National Science Foundation under grant CHE-0723997.

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Ab Initio Coupled Cluster Determination of the Heats of Formation of C

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