J. Phys. Chem. C 2007, 111, 18503-18506
18503
Estimation of Standard Heats of Formation of Fullerenes Using Pentagon-Centered Motifs Cheng Hua Sun,†,‡ Ya Nan Chen,†,§ Gao Qing Lu,*,‡ and Hui Ming Cheng*,† Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China, ARC Center of Excellence for Functional Nanomaterials, UniVersity of Queensland, Brisbane, QLD 4072, Australia, and Northeast Yucai School, Shenyang, China ReceiVed: June 13, 2007; In Final Form: September 23, 2007
Standard heats of formation (∆H0f ) calculated at B3LYP/6-31G(d) for 114 isolated pentagon rule fullerenes were reproduced within 4.2 kcal/mol on the basis of 24 pentagon-centered motifs. With the use of a C30capped (5,5) single-walled carbon nanotube as test molecule, it is found that (i) both hexagon- and pentagoncentered motifs can be employed to estimate the ∆H0f of fullerenes, while the latter gives better results for large fullerenes; and (ii) with the consideration of nonplanar strain energy, the estimated results of ∆H0f can be further improved. The estimation scheme of pentagon-centered motifs can serve as a rapid prescreening tool for viable isomers of large fullerenes and is expected to be equally valuable for carbon nanotubes.
As a basic thermodynamic parameter, the standard heat of formation (∆H0f ) of fullerenes is fundamentally important for both experimental and theoretical studies. Experimentally, accurate values of ∆H0f can be obtained through calorimetric measurements if pure samples are available. Unfortunately, due to their diminishing solubility in common solvents and the complexity of allotropic forms, there are considerable challenges in the isolation, purification, and characterization of fullerenes, which undoubtedly underline the need to develop theoretical approaches to obtaining ∆H0f for fullerenes.1 However, due to the complexity of their allotropic forms, even semiempirical methods are too expensive as prescreening tools for fullerenes with hundreds of atoms. Thereby, it is necessary to develop simple rules and estimation schemes to predict the thermodynamic stability of large fullerenes. Topologically, the isolated pentagon rule (IPR)2 and hexagonneighbor rule (HNR)3 have shown excellent predictive power. The idea behind IPR is that in a stable fullerene pentagons prefer to be isolated from one another, which is favored on both π-electronic and steric grounds.4 A typical example for this rule is C60 itself: out of the 1812 possible isomers of 60 carbon atoms, nature selects icosahedral C60,5 and, in the case of C116, imposition of IPR reduces the set of candidates from 1207119 to 6063 structural isomers.4 Different from IPR, HNR focuses on the distribution of hexagons and suggests that hexagons should have similar environments so that the steric strain can be spread evenly,3 which has been successfully applied to C84,3 C116,4 and C118.6 Thereby, both IPR and HNR can simplify the searching of stable isomers for fullerenes. However, such simplification is not sufficient since the set of isomers satisfying IPR and HNR increases remarkably with the cluster size, which stimulates chemists to develop quantitative estimation schemes to predict the thermodynamic properties of large fullerenes without time-consuming calculations.
Accurate values of ∆H0f of 115 fullerenes with 60-180 carbon atoms were calculated at the level of B3LYP/631G(d).1 Those values have been employed as the benchmark for the development of theoretical estimation schemes. Based on 30 distinct structural motifs with a hexagon center, the calculated values of ∆H0f were reproduced within 3.0 kcal/ mol.1 The above success can be summarized in two points. First, values of ∆H0f of a set of fullerenes which can represent the essential characteristics of arbitrary fullerenes are calculated using reliable methods of quantum chemistry. Second, an approximate formula of ∆H0f with undetermined coefficients is proposed, and the coefficients are further obtained through fitting the calculated values of ∆H0f . Inspired by Cioslowski et al.,1 our group studied the nonplanar strain energies (Enp) of fullerenes at the level of B3LYP/6-31G(d), and the calculated values of Enp for 115 fullerenes were reproduced with a rootmean-square (rms) error of 6.8 kcal/mol only on the basis of 8 motifs.7 Different from the motifs developed by Cioslowski et al.,1 the motifs employed in our work are pentagon-centered. Thereby, the total number of motifs for any fullerene is 12 according to the Euler rule, which can simplify the description of arbitrary fullerenes and the estimation of thermodynamic properties. What about the reliability of estimating ∆H0f using pentagon-centered motifs? Although the validity of pentagon-centered motifs for the estimation of Enp has been illustrated in our previous work,7 their applicability to the estimation of ∆H0f of fullerenes is unknown. In the present work, an estimation scheme based upon pentagon-centered motifs is developed and the values of ∆H0f of 114 IPR fullerenes with 60-102 atoms calculated at the B3LYP/6-31G(d) level are reproduced within 4.2 kcal/mol. Using C30-capped (5,5) single-walled carbon nanotubes (SWNTs) as test molecules, the merits and limitations of pentagon-centered motifs are discussed and compared with hexagon-centered motifs.
* Corresponding authors. E-mail:
[email protected] (H. M. Cheng);
[email protected] (G. Q. Lu). † Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences. ‡ University of Queensland. § Northeast Yucai School.
2. Motifs and Calculation Methods From the previous work,7 eight pentagon-centered motifs are obtained and shown in Figure 1. Due to the IPR,2 the central pentagons (red atoms) are surrounded by five hexagons. Hence, the essential difference among parts A-H of Figure 1 is the
1. Introduction
10.1021/jp0745900 CCC: $37.00 © 2007 American Chemical Society Published on Web 11/29/2007
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Sun et al.
Figure 1. Possible motifs of IPR fullerenes with a consideration of the first and second neighborhoods of the central pentagon (red spheres). Motifs A-H were initially proposed in ref 7.
Figure 2. Possible motifs of IPR fullerenes with a consideration of the first, second, and third neighborhoods of the central pentagon.
third neighbor; for parts A-H, only the first and second neighbors of the central pentagon are considered. Theoretically, there are 64 possible configurations if the third neighbor is taken into account. However, only 24 among these configurations satisfy the IPR, as shown in Figure 2. Motifs are named according to the first and second neighborhoods of the central pentagon. On the basis of the motifs shown in Figures 1 and 2, two approximate formulas of ∆H0f are proposed:
∑I NII + sN (I ) A, B, ..., H)
(1.1)
∑I NII + sN (I ) A1, A2, ..., H8)
(1.2)
∆Hf0 est1 ) ∆Hf0 est2 )
where I represents the contribution of motif I, and NI is the number of motif I contained in the object fullerene. N is the total number of carbon atoms, and s is the contribution of one atom to ∆H0f . According to previous studies,1 remarkable improvement on the estimation of ∆H0f can result from adding a term proportional (N-constant)-1 to the sum of the contributions of motifs. Unfortunately, ∆H0f est is barely improved with such modification in this work. Therefore, we define two coefficients of I and s that are determined through fitting based
on eqs 1.1 and 1.2. Fullerenes are described as the combination of the motifs shown in Figures 1 and 2 and coefficients used in eqs 1.1 and 1.2 can be found in Supporting Information (PART I and PART II). The error is defined as the difference between the estimated and calculated results by
error ) ∆H0f est - ∆H0f cal (I ) A, B, ..., H)
(2)
To compare the results with the results estimated by hexagoncentered motifs,1 the same calculated values of ∆H0f are employed as standard data in the following fittings. The fitting is based on the least-square theory, rather than singular value decomposition formalism8 used in ref 1. 3. Results and Discussion Calculated and estimated values of ∆H0f are listed in the Supporting Information (PART III). As shown in the previous work,7 each fullerene is assigned a sequence number ranging from 1 to 114 for the sake of clarity. The values of errors defined by eq 2 for 114 fullerenes are shown in Figure 3, parts a and b, as a function of the sequence number. Results estimated on the basis of 30 hexagon-centered motifs obtained from ref 1 are also employed for reference, as shown in Figure 3c. The quality
Standard Heats of Formation of Fullerenes
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Figure 3. Errors of ∆H0f estimated on the basis of pentagon-centered motifs (Figures 1 and 2) and hexagon-centered motifs (see ref 1).
Figure 4. (a) Schematic framework of the atomic structure at the end of C30-capped (5,5) SWNTs where a, b, c, d, and e indicate different hexagons around the central pentagon. (b) Estimated ∆H0f based on hexagon-centered motifs and pentagon-centered motifs (see Figure 2) are shown with a comparison to the calculated results at the B3LYP/6-311G(d) and B3LYP/6-31G(d) levels. (c) Schematic frameworks of hexagoncentered motifs 135, 13/66, 6666, 666655, and 666666 (initially proposed in ref 1).
of the estimation is evaluated by the mean absolute deviation (MAD) and the rms error, which are given as the insets of Figure 3. Accordingly, it can be summarized that (i) ∆H0f est2 is much better than ∆H0f est1 with rms decreasing from 13.9 to 7.7 kcal/ mol, which can be equally attributed to the use of more motifs that allow for more flexible and comprehensive description of local structural characteristics; and (ii) ∆H0f est2 is close to the estimated results using 30 hexagon-centered motifs, although the formula of ∆H0f est used in the two works are different. So far, both hexagon-centered motifs and pentagon-centered motifs can be employed to estimate ∆H0f . As shown in Figure 3b, however, values of ∆H0f est for C90 (D5h) and C96 (D3h) show errors of 38.4 and 31.1 kcal/mol, respectively. Such estimations can be improved through the correction for nonplanar strain energy as discussed in the following analysis. The purpose of developing motifs is to estimate the properties of fullerenes with hundreds of atoms for which calculations based on quantum chemistry are too time-consuming and expensive. Therefore, it is necessary to check the validity of the motifs before they are applied to large fullerenes. Testing based on model molecules is a good choice according to our previous experience.7 Previously, it has been suggested that isolable fullerenes containing 100 or more atoms prefer to be cylindrical,9 which is inspired by the fact of the successful synthesis of SWNTs with large molecular weight,10 indicating that tubular fullerenes present excellent stability. Thereby C30capped (5,5) SWNTs with N g100 are employed as model
molecules, and the schematic framework of the atomic structures at the end is shown in Figure 4a, where a, b, c, d, and e indicate different hexagons around the central pentagon. Under the scheme of hexagon-centered motifs,1 a, b, c, and d correspond to motifs 135, 13/66, 6666, and 666655, with other hexagons being motif 666666. Schematic frameworks of these motifs have been shown in Figure 4c. And ∆H0f of C30-capped (5,5) SWNTs can be estimated by
∆Hf0 est ) 1067.610 + (n - 40) × 30.336 - 8050.751(N 30.050)-1 (3.1) where n is the number of hexagons and can be determined by n ) (N - 20)/2 based on the Euler rule. Similarly, under the scheme of pentagon-centered motifs shown in Figure 2, C30capped (5,5) SWNTs can be described as the combination of 2 motifs of G1 and 10 motifs of E1. On the basis of eq 1.2 and coefficients determined above, ∆H0f can be estimated by
∆Hf0 est ) 58.630 + 8.260N
(3.2)
The detailed derivations of eqs 3.1 and 3.2 can be found in Supporting Information (PART IV). The estimated values of ∆ H0f by eqs 3.1 and 3.2 are shown in Figure 4b as a function of N. For the sake of comparison, the calculated values of ∆H0f at the level of B3LYP/6-311G(d) are also shown as benchmarks.11 Considering that the coefficients used in eqs 3.1 and 3.2 are
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Sun et al. results based upon eq 4 are better than those obtained from eq 3.2. Thereby, the estimation of ∆H0f can be improved with the consideration of Enp. 4. Conclusions The values of ∆H0f cal (B3LYP/6-31G(d)) of 114 IPR fullerenes with 60-102 atoms are reproduced within 4.2 kcal/ mol on the basis of 24 pentagon-centered motifs, which is low enough to make the present scheme a viable alternative to semiempirical approaches. Using a C30-capped (5,5) SWNT as a test molecule, the applicability of the developed scheme to large fullerenes is confirmed and compared with that of the hexagon-centered motifs. It is found that (i) both hexagoncentered motifs and pentagon-centered motifs can be employed to estimate ∆H0f , while the latter can give better results for large fullerenes; and (ii) with the consideration of Enp, the estimated results of ∆H0f can be improved remarkably. The estimation schemes presented in this work can serve as a rapid prescreening tool for viable isomers of large fullerenes and is expected to be equally valuable for carbon nanotubes.
Figure 5. (a) Estimated ∆H0f with an Enp correction based on eq 4; (b) estimated ∆Hf0 based on eqs 3.2 and 4 with a comparison to the results calculated at the B3LYP/6-311G(d) level.
based on the results calculated at the B3LYP/6-31G(d) level, approximate values of ∆H0f cal at B3LYP/6-31G(d) are obtained through linear fitting to the values of ∆H0f cal of C60 (Ih), C70 (D5h), C80 (D5d), and C90 (D5h),1 which can be also viewed as C30-capped (5,5) SWNTs. Clearly, fitted ∆H0f values at the B3LYP/6-31G(d) level present only a slight difference from the results calculated at the B3LYP/6-311G(d); thereby, these data can serve as the benchmarks for the evaluation of results estimated from eqs 3.1 and 3.2. From Figure 4b, it is clear that eq 3.2 gives a better estimation than eq 3.1. From Figure 4b, it is found that both hexagon- and pentagoncentered motifs present overestimation probably because that the contributions of all motifs are obtained from fullerenes with C60-C102 whose strain energy per atom is higher than that of large ones. Thereby, the estimation of ∆H0f may be improved with the addition of a term of nonplanar strain energy Enp by
∆Hf0 est3 )
∑I NII + sN + kEnp (I ) A1, A2, ..., H8)
Acknowledgment. We thank Prof. J. Cioslowski for kindly offering the coordinates and total energies of IPR fullerenes used in this work. The financial support from NSFC grants (90606008 and 90406012) and MOST grants (2006CB932703), and from the ARC Centre of Excellence for Functional Nanomaterials Australia for this work are acknowledged. We also acknowledge the Center of Computational Molecular Sciences, University of Queensland, for allowing the access to the high-performance computing clusters for some calculations. Supporting Information Available: The description of fullerenes using pentagon-centered motifs and values of ∆H0f are listed in Tables S1 (PART I) and S2 (PART III). The coefficients used in eqs 1.1, 1.2, 3.1, 3.2, and 4 and the detailed derivations of eqs 3.1 and 3.2 are also presented as PART II and PART IV. Values of Enp of C100-C210 are presented as PART V. These materials are available free of charge via the Internet at http://pubs.acs.org. References and Notes
(4)
Values of the coefficients s, I, and k are listed in the Supporting Information (PART II). Enp has been calculated on the basis of the theory of π-Orbital Axis Vector,12-15 and values for Enp of fullerenes with C60-C102 have been published in ref 7. More details regarding the calculation of Enp can be found in our previous study of supershort SWNTs.16 For C30-capped (5,5) SWNTs, Enp has been obtained through two : (i) values of Enp of C70 (D5h), C80 (D5d), and C90 (D5h) have been calculated on the basis of the structures optimized at B3LYP/6-31G(d), and (ii) through the linear fitting, an approximate expression has been constructed, Enp ) 2.455N + 322.667, based on which values of Enp of C100-C210 are estimated and presented in the Support Information (PART V). Estimated results of ∆H0f of C60-C102 and C30-capped (5,5) SWNTs are shown in Figure 5parts a and b, respectively. For the former, MAD has decreased from 5.4 to 4.2 kcal/mol and rms from 7.7 to 5.6 kcal/mol. Moreover, the estimated results for both C90 (D5h) and C96 (D3h) have been improved remarkably, as demonstrated in Figure 5a. Such an improvement is expected for large fullerenes, as shown in Figure 5b. Although the estimation of Enp is pretty rough,
(1) Cioslowski, J.; Rao, N.; Moncrieff, D. J. Am. Chem. Soc. 2000, 122, 8265. (2) Kroto, H. W. Nature 1987, 329, 529. (3) Raghavachari, K. Chem. Phys. Lett. 1992, 190, 397. (4) Achiba, Y.; Fowler, P. W.; Mitchell, D.; Zerbetto, F. J. Phys. Chem. A 1998, 102, 6835. (5) Kroto, H. W.; Heath, J. R.; O’Braen, S. C.; Curl, R. F.; Smalley, R. E. Nature 1985, 318, 162. (6) Schmalz, T. G.; Seitz, W. A.; Klein, D. J.; Hite, G. E. J. Am. Chem. Soc. 1988, 101, 1113. (7) Sun, C. H.; Lu, G. Q.; Cheng, H. M. J. Phys. Chem. B 2006, 110, 218. (8) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes; Cambridge University Press: New York, 1992; Chapter 2. (9) Achiba, Y.; Kikuchi, K.; Aihara, Y.; Wakabayashi, T.; Miyake, Y.; Kainosho, M. The Chemical Physics of the Fullerenes 10 (and 5) Years Later; Andreoni, W., Ed.; Kluwer Academic Press: Dordrecht, 1996; p 139. (10) Iijima, S.; Ichihashi, T. Nature 1993, 363, 603. (11) Cioslowski, J.; Rao, N.; Moncrieff, D. J. Am. Chem. Soc. 2002, 124, 8485. (12) Haddon, R. C.; Scott, L. T. Pure Appl. Chem. 1986, 58, 137. (13) Haddon, R. C. J. Phys. Chem. 1987, 91, 3719. (14) Haddon, R. C. J. Am. Chem. Soc. 1986, 108, 2837. (15) Haddon, R. C. Science 1993, 261, 1545. (16) Sun, C. H.; Finnerty, J. J.; Lu, G. Q.; Cheng, H. M. J. Phys. Chem. B 2005, 109, 12406.
