Article pubs.acs.org/JPCA
Evaluation of the Heats of Formation of Corannulene and C60 by Means of Inexpensive Theoretical Procedures Frank J. Dobek, Duminda S. Ranasinghe, Kyle Throssell, and George A. Petersson* Hall-Atwater Laboratories of Chemistry, Wesleyan University, Middletown, Connecticut 06459-0180, United States ABSTRACT: Inexpensive ab initio procedures that employ homologous sequences of isodesmic reactions for the calculation of enthalpies of formation of moderate-sized organic molecules were tested with benzene, naphthalene, phenanthrene, and triphenylene. Two size-consistent adjustable parameters were found to bring the calculated values within the uncertainty of the experimental values. These procedures were then applied to C20H10 (corannulene) and C60 (buckminsterfullerene). The results, specifically, ΔfH0298(C20H10) = 484 ± 4 kJ mol−1 and ΔfH0298(C60) = 2531 ± 15 kJ mol−1, are in excellent agreement with both the recent definitive W1h calculations of Karton et al. for corannulene [ΔfH0298(C20H10) = 485.2 ± 7.9 kJ mol−1] and their estimated value for buckminsterfullerene [ΔfH0298(C60) = 2521.6 ± 13.6 kJ mol−1] (J. Phys. Chem. A 2013, 117, 1834−1842). We support their conclusion that the experimental values should be reexamined.
1. INTRODUCTION The recent article1 by Karton et al. presents a tour de force on the definitive calculation of enthalpies of formation. Beyond their goal of providing more accurate energies for corannulene (C20H10) and buckminsterfullerene (C60), they also provided a series of benchmark W1h energies1,2 that we employed in this work to calibrate less costly methods. We restricted ourselves to calculations that could easily be applied to C60 with no more than an inexpensive dual-processor workstation. It was crucial to our procedure that these calculations could be applied without compromise to an entire homologous sequence of reactions. One might expect uncertainties in experimental heats of formation to increase linearly with molecular weight, as they are generally determined by measured heats of combustion,3 which would tend to have consistent errors per gram of sample. However, the estimated uncertainties for the species in Table 1 deviate from the anticipated linear increase (Figure 1) and instead grow approximately as the number of carbon atoms raised to the power of 2.5 for the larger species that are solids at room temperature. It has been suggested1 that this rapid increase in uncertainty results from a combination of difficulties in both the purification of nonpolar species and the measurement of heats of sublimation for these species. We are not aware of any fundamental reason for the exponent of 2.5. It is simply an empirical observation about literature estimates of these experimental uncertainties for the particular species in Table 1, but this observation implies that, beyond a certain molecular size, calculated heats of formation might be more reliable than experimental values. Any properly size-consistent computational method will necessarily have errors that grow linearly with molecular size for a homologous sequence of chemical reactions (Figure 2). Note that the magnitude of these errors varies by a factor of 2 © 2013 American Chemical Society
Table 1. Experimental Gas-Phase Enthalpies of Formation species C(atom) hydrogen methane ethane propane ethylene benzene naphthalene phenanthrene triphenylene corannulene buckminsterfullerene a
C(g) H2 CH4 C2H6 C3H8 C2H4 C6H6 C10H8 C14H10 C18H12 C20H10 C60
ΔfH0298 (kJ/mol)
ref
716.87 ± 0.06 0.00a −74.53 ± 0.06 −83.76 ± 0.17 −104.70 ± 0.50 52.4 ± 0.2 83.2 ± 0.3 150.6 ± 1.5 202.2 ± 2.3 270.1 ± 4.4 458.5 ± 9.2 2560 ± 100
3 4 5 5 5 3 6 6 6 6 5
By definition.
between alkanes and alkenes, with intermediate errors for the aromatic species of current interest. Thus, it was important to employ homologous series of reactions when we applied this size-consistent behavior of computational errors to correct the inexpensive CBS-4M and APF-D/3Za1P computational models.7−9 The spirit of these corrections is similar to that of the bond additivity corrections10 of Melius and Allendorf, but our selective focus on a specific homologous series of reactions provides improved accuracy within our narrowly focused application. Received: April 27, 2013 Revised: May 14, 2013 Published: May 15, 2013 4726
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time to 1 day for C60. Both models employed the HF/3-21G*optimized geometries, vibrational zero-point energies, and thermal corrections from the CBS-4M output. We repeated the APF-D/3Za1P single-point energies at the B3LYP/631G(2df,p)-optimized geometries from the Supporting Information of ref 1, to test the veracity of the HF/3-21G*optimized geometries. The energy of each of the molecules in Table 2 decreased at the B3LYP geometry. The increments varied from only 0.2 kJ mol−1 for H2 to 20.2 kJ mol−1 for C60, but had little effect on the final results (vide infra). All calculations were performed with a modified version of Gaussian 09.11 We employed three homologous sequences of reactions. The first was a slight modification of the commonly used protocol based on the enthalpy of atomization CnH 2m(g) → nC(g) + mH 2(g)
(1)
Because the current methods cannot be applied to solid graphite, it was necessary to substitute the experimental enthalpy of formation for gas-phase carbon atoms.3 However, we retained the calculated enthalpy of H2 rather than substitute the experimental enthalpy of a hydrogen atom. We also applied the first two sequences of isodesmic reactions from Karton et al.,1 namely
Figure 1. Reported uncertainty in experimental enthalpies of formation increases linearly with molecular size for small molecules, but increases more rapidly for species larger than benzene.
C2nHm(g) + (4n − m)CH4(g) → nC2H4(g) + [(4n − m)/2]C2H6(g)
(2)
and CnHm(g) + [(n − m)/2]C2H4(g) → [(2n − m)/6]C6H6(g)
(3)
The first sequence preserves the numbers of carbon−carbon single and double bonds, but not the hybridization. These are the bond separation reactions introduced12 by Hehre et al. in 1970. The second sequence includes only sp2 hybridized carbons. We did not employ isodesmic reactions involving naphthalene and larger reference species, because the experimental uncertainties exceed 0.1 kJ mol−1 per carbon atom for the solids (Table 1). Each of the three homologous sequences of reactions includes two independent variables, n and m. The errors in the enthalpies of formation calculated from these reactions will necessarily vary linearly with these variables.
3. RESULTS The calculated enthalpies of formation obtained from the homologous sequence of “atomization reactions” are reported in Table 2, with the empirical size-consistent correction parameters given as footnotes. Note that the correction varies with both the number of carbon atoms and the number of hydrogen atoms in the aromatic species. These parameters were (least-squares) fit to the W1h enthalpies of formation1 for C 6 H 6 , C 10 H 8 , C 14 H 10 , and C 18 H 12 . The values of ΔfH0298(C20H10) and ΔfH0298(C60) were then predicted from these fits. The improved B3LYP geometries reduce the predicted value of ΔfH0298(C60) by less than 5 kJ mol−1. Although these results are consistent with the experimental data, the magnitude of the corrections (the APF-D correction is almost 700 kJ mol−1for C60) might make one skeptical about the use of these atomization reactions.
Figure 2. Errors in APF-D/3Za1P enthalpies of formation relative to the corresponding W1h values increase linearly with molecular size within each homologous sequence.
2. COMPUTATIONAL METHODS The CBS-4M model7 required 1 week to complete the calculation for C60 using our dual quad-core workstation and thus represents about the upper limit of complexity that is readily available without access to large-scale high-performance computing. We also applied the new APF-D density functional8 with a modest 3Za1P basis set,9 which reduced the computer 4727
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Table 2. Calculated Enthalpies of Formation from Eq 1, Dissociation to C(g) and H2(g) ΔfH0298 (kJ/mol) aromatic species benzene naphthalene phenanthrene triphenylene corannulene buckminsterfullerene
C6H6 C10H8 C14H10 C18H12 C20H10 C60
dissociation reaction
CBS-4M
C6H6 → 6C(atom) + 3H2 C10H8 → 10C(atom) + 4H2 C14H10 → 14C(atom) + 5H2 C18H12 → 18C(atom) + 6H2 C20H10 → 20C(atom) + 5H2 C60 → 60C(atom)
81.8 151.4 207.5 267.1 482.0 2529.2
a
APF-Db
APF-Dc
expt
84.0 149.0 205.8 269.0 484.8 2527.4
84.4 148.6 205.5 269.3 482.4 2522.8
83.2 ± 0.3 150.6 ± 1.5 202.2 ± 2.3 270.1 ± 4.4 459.5 ± 9.2 2560 ± 100
CBS-4M + 8.17 × (number of C atoms) − 4.00 × (number of H atoms). bAPF-D/3Za1P//HF/3-21G* + 11.24 × (number of C atoms) + 0.59 × (number of H atoms). cAPF-D/3Za1P//B3LYP/6-31G(2df,p) + 11.36 × (number of C atoms) + 0.87 × (number of H atoms). a
Table 3. Calculated Enthalpies of Formation from Isodesmic Reactions with Methane, Eq 2a ΔfH0298 (kJ/mol) aromatic species benzene naphthalene phenanthrene triphenylene corannulene buckminsterfullerene
C6H6 C10H8 C14H10 C18H12 C20H10 C60
1a 1b 1c 1d
isodesmic reaction
CBS-4M
C6H6 + 6CH4 → 3C2H4 + 3C2H6 C10H8 + 12CH4 → 5C2H4 + 6C2H6 C14H10 + 18CH4 → 7C2H4 + 9C2H6 C18H12 + 24CH4 → 9C2H4 + 12C2H6 C20H10 + 30CH4 → 10C2H4 + 15C2H6 C60 + 120CH4 → 30C2H4 + 60C2H6
81.8 151.4 207.5 267.1 482.0 2529.2
b
c
APF-D
APF-Dd
W1he
expt
84.0 149.0 205.8 269.0 484.9 2528.2
84.4 148.6 205.5 269.3 482.4 2522.8
82.3f 153.1 208.9 271.2 488.9
83.2 ± 0.3 150.6 ± 1.5 202.2 ± 2.3 270.1 ± 4.4 459.5 ± 9.2 2560 ± 100
Sequence 1 in ref 1. bCBS-4M + 4.64 × (number of C atoms) − 2.07 × (number of H atoms). cAPF-D/3Za1P//HF/3-21G* + 0.71 × (number of C atoms) + 0.85 × (number of H atoms). dAPF-D/3Za1P// B3LYP/6-31G(2df,p) − 0.42 × (number of C atoms) + 1.48 × (number of H atoms). e Reference 1. fW1BD, ref 13. a
Table 4. Calculated Enthalpies of Formation from Isodesmic Reactions with Ethylene, Eq 3a ΔfH0298 (kJ/mol) aromatic species naphthalene phenanthrene triphenylene corannulene buckminsterfullerene
C10H8 C14H10 C18H12 C20H10 C60
2a 2b 2c 2d
isodesmic reaction
CBS-4M
C10H8 + C2H4 → 2C6H6 C14H10 + 2C2H4 → 3C6H6 C18H12 + 3C2H4 → 4C6H6 C20H10 + 5C2H4 → 5C6H6 C60 + 30C2H4 → 20C6H6
151.7 207.9 267.7 483.0 2534.7
b
c
APF-D
APF-Dd
W1he
expt
149.2 206.1 269.3 484.9 2533.2
148.5 205.9 270.2 484.6 2540.6
149.8 206.6 268.2 485.2
150.6 ± 1.5 202.2 ± 2.3 270.1 ± 4.4 459.5 ± 9.2 2560 ± 100
Sequence 2 in ref 1. bCBS-4M − 0.86 × (number of C atoms) + 0.64 × (number of H atoms). cAPF-D/3Za1P//HF/3-21G* − 2.18 × (number of C atoms) + 2.35 × (number of H atoms). dAPF-D/3Za1P// B3LYP/6-31G(2df,p) − 1.84 × (number of C atoms) + 1.96 × (number of H atoms). e Reference 1. a
Table 5. Best Estimates for Enthalpies of Formation ΔfH0298 (kJ/mol) a
aromatic species
avg (W1h)
benzene naphthalene phenanthrene triphenylene corannulene buckminsterfullerene
83.4 ± 1.1 149.7 ± 1.3 206.5 ± 0.9 268.8 ± 1.0 483.6 ± 1.2 2531.4 ± 5.6
W1h
b
82.3 149.8 206.6 268.2 485.2
avg (expt)c
expt
82.9 ± 1.1 149.0 ± 0.9 205.8 ± 0.7 268.2 ± 0.7 484.1 ± 3.8 2542.2 ± 44.2
83.2 ± 0.3 150.6 ± 1.5 202.2 ± 2.3 270.1 ± 4.4 459.5 ± 9.2 2560 ± 100
From Tables 3 and 4 fits to W1h energies. bReference 1. cFrom corresponding fits to experimental ΔfH0298 values for C6H6, C10H8, C14H10, and C18H12.
a
sequence of reactions. Unfortunately, agreement this close also indicates that the data are not independent. It is apparent that ΔfH0298(Catom) was adjusted to be consistent with the values for methane, ethane, and ethylene.3 We shall omit the results in Table 2 from our final results to avoid redundancy. The calculated enthalpies of formation obtained from the homologous sequence of isodesmic reactions described by eq 3 are reported in Table 4, with the empirical size-consistent correction parameters again given as footnotes. The improved B3LYP geometries reduce the energy of the products more
The calculated enthalpies of formation obtained from the homologous sequence of isodesmic reactions described by eq 2 are reported in Table 3, with the empirical size-consistent correction parameters again given as footnotes. The APF-D corrections are almost an order-of-magnitude smaller than those in Table 2, lending more credibility to these predictions. However, both the APF-D and CBS-4M results in Table 3 are virtually identical to those in Table 2, demonstrating the effectiveness of size-consistent empirical corrections in removing the dependence on the choice of the homologous 4728
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ACKNOWLEDGMENTS The authors gratefully acknowledge support from Gaussian, Inc., and Wesleyan University for computational facilities supported by National Science Foundation Grant CNS0959856.
than that of the reactants for this sequence and, thus, increase the predicted value of ΔfH0298(C60) by 7.4 kJ mol−1. The close agreement with the values in Tables 2 and 3 is striking. In fact, the variation in the APF-D or CBS-4M enthalpies between the two isodesmic sequences is no greater than the variation in the W1h values between the two sequences (Tables 3 and 4). The inexpensive APF-D and CBS-4M calculations with sizeconsistent empirical corrections mimic the powerful W1h calculations within the error limits for these high-level calculations.
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REFERENCES
(1) Karton, A.; Chan, B.; Raghavachari, K.; Radom, L. Evaluation of the Heats of Formation of Corannulene and C60 by Means of HighLevel Theoretical Procedures. J. Phys. Chem. A 2013, 117, 1834−1842. (2) Martin, J. M. L.; Oliveira, G. Towards standard methods for benchmark quality ab initio thermochemistryW1 and W2 theory. J. Chem. Phys. 1999, 111, 1843−1856. (3) Stevens, W. R.; Ruscic, B.; Baer, T. Heats of Formation of C6H5•, C 6H5+ , and C6 H5 NO by Threshold Photoelectron Photoion Coincidence and Active Thermochemical Tables Analysis. J. Phys. Chem. A 2010, 114, 13134−13145. (4) Ruscic, B.; Pinzon, R. E.; Morton, M. L.; von Laszewski, G.; Bittner, S.; Nijsure, S. G.; Amin, K. A.; Minkoff, M.; Wagner, A. F. Introduction to Active Thermochemical Tables: Several “Key” Enthalpies of Formation Revisited. J. Phys. Chem. A 2004, 108, 9979−9997. Active Thermochemical Tables (ATcT), version alpha 1.110; Argonne National Laboratory: Lemont, IL, 2010; see also http://atct.anl.gov/Thermochemical%20Data/version%20Alpha%201. 110/index.html (accessed April, 2013). (5) Pittam, D. A.; Pilcher, G. Measurements of Heats of Combustion by Flame Calorimetry. Part 8.Methane, Ethane, Propane, n-Butane and 2-Methylpropane. J. Chem. Soc., Faraday Trans. 1 1972, 68, 2224− 2229. Afeefy, H. Y.; Liebman, J. F.; Stein, S. E. Neutral Thermochemical Data. In NIST Chemistry WebBook; NIST Standard Reference Database Number 69; Linstrom, P. J., Mallard, W. G., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, 2005; see http://webbook.nist.gov (accessed April, 2013). (6) Roux, M. V.; Temprado, M.; Chickos, J. S.; Nagano, Y. Critically Evaluated Thermochemical Properties of Polycyclic Aromatic Hydrocarbons. J. Phys. Chem. Ref. Data 2008, 37, 1855−1996. (7) Ochterski, J. W.; Petersson, G. A.; Montgomery, J. A., Jr. A Complete Basis Set Model Chemistry. V. Extensions to Six or More Heavy Atoms. J. Chem. Phys. 1996, 104, 2598−2619. Montgomery, J. A., Jr.; Frisch, M. J.; Ochterski, J. W.; Petersson, G. A. A Complete Basis Set Model Chemistry. VII. Use of the Minimum Population Localization Method. J. Chem. Phys. 2000, 112, 6532−6542. (8) Austin, A.; Petersson, G. A.; Frisch, M. J.; Dobek, F. J.; Scalmani, G.; Throssell, K. A Density Functional with Spherical Atom Dispersion Terms. J. Chem. Theory Comput. 2012, 8, 4989−5007. (9) Petersson, G. A.; Zhong, S.; Montgomery, J. A., Jr.; Frisch, M. J. On the Optimization of Gaussian Basis Sets. J. Chem. Phys. 2003, 118, 1101−1109. Zhong, S.; Ericka, C.; Barnes, E. C.; George, A.; Petersson, G. A. Uniformly Convergent n-Tuple-ζ Augmented Polarized (nZaP) Basis Sets for Complete Basis Set Extrapolations. I. Self-Consistent Field Energies. J. Chem. Phys. 2008, 129, 184116− 184127. Barnes, E. C.; Petersson, G. A. MP2/CBS Atomic and Molecular Benchmarks for H through Ar. J. Chem. Phys. 2010, 132, 114111−114119. Ranasinghe, D. S.; Petersson, G. A. CCSD(T)/CBS Atomic and Molecular Benchmarks for H through Ar. J. Chem. Phys. 2013, 138, 144104−144114. (10) Melius, C. F.; Allendorf, M. D. Bond Additivity Corrections for Quantum Chemistry Methods. J. Phys. Chem. A 2000, 104, 2168− 2177. Melius, C. F. In Chemistry and Physics of Energetic Materials; Bulusu, S. N., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1990; p 21. (11) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin,
4. CONCLUSIONS The APF-D and CBS-4M calculations with size-consistent empirical corrections applied to two different homologous sequences of isodesmic reactions gave six separate estimates for the enthalpy of formation for each of the polycyclic aromatic hydrocarbons. The average values are reported in Table 5, along with the variance of the six estimates for each species and the W1h and experimental values for comparison. The small variance among our six estimates and the small rms deviation from the available W1h values (0.8 kJ mol−1) suggests that the estimated W1h value for buckminsterfullerene (2531 ± 6 kJ mol−1) is within the accuracy estimated1 for such a calculation (±7.7 kJ mol−1) if it were possible. The reliability of the APF-D or CBS-4M enthalpies with size-consistent empirical corrections rests not on the absolute accuracy of the inexpensive APFD or CBS-4M calculations, but on the consistency of their errors relative to high accuracy calculations. The size-consistent empirical corrections must be adapted to each unique homologous sequence for a specific computational model and applied only to that sequence with that model. We must also account for the uncertainty in the experimental data (Table 1) employed in the three reaction schemes (±2.5 kJ mol−1 for C20H10 and ±9.5 kJ mol−1 for C60). Our final estimates are thus ΔfH0298(C20H10) = 484 ± 4 kJ mol−1 and ΔfH0298(C60) = 2531 ± 15 kJ mol−1. These values are in excellent agreement with both the recent definitive W1h calculations of Karton et al.1 for corannulene [ΔfH0298(C20H10) = 485.2 ± 7.9 kJ mol−1] and their estimated value for buckminsterfullerene [ΔfH0298(C60) = 2521.6 ± 13.6 kJ mol−1]. We therefore also agree with their conclusion that the experimental values should be reexamined. Finally, we consider how we would have fared if the very accurate and rigorously size-consistent W1 energies had not been available. The second set of average values in Table 5 were obtained through identical fits to the experimental values of 0 for C6H6, C10H8, C14H10, and C18H12 and thus ΔfH298 employed no results from calculations more sophisticated than the CBS-4M and APF-D/3Za1P models. The small random errors associated with these experiments increase the variance of the predictions far more than they influence the values. This is encouraging for applications of size-consistent empirical corrections to other homologous sequences.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest. 4729
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K. N.; Staroverov, V. N.; Keith, T.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö .; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, revision A.02, 01; Gaussian, Inc.: Wallingford, CT, 2009. (12) Hehre, W. J.; Ditchfield, R.; Radom, L.; Pople, J. A. Molecular Orbital Theory of the Electronic Structure of Organic Compounds. V. Molecular Theory of Bond Separation. J. Am. Chem. Soc. 1970, 92, 4796−4801. (13) Barnes, E. C.; Petersson, G. A.; Frisch, M. J.; Montgomery, J. M., Jr.; Martin, J. M. Spin Unrestricted Coupled-Cluster and BruecknerDoubles Variations of W1 Theory. J. Chem. Theory Comput. 2009, 5, 2687−2693.
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