C38 to C80 and C112 to C120 - American Chemical Society

May 22, 2007 - to search for the lowest-energy structures of higher fullerenes C110 to C120. In addition, a systematic study of low-lying structures o...
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J. Phys. Chem. C 2007, 111, 17671-17677

17671

ARTICLES Search for Lowest-Energy Fullerenes 2: C38 to C80 and C112 to C120† Nan Shao, Yi Gao, and Xiao Cheng Zeng* Department of Chemistry, UniVersity of NebraskasLincoln, Lincoln, Nebraska 68588 ReceiVed: January 6, 2007; In Final Form: April 10, 2007

An efficient computational approach that combines semiempirical density-functional based tight-binding method (DFTB) for geometry optimization and density-functional theory for single-point energy calculation is employed to search for the lowest-energy structures of higher fullerenes C110 to C120. In addition, a systematic study of low-lying structures of lower fullerenes C38 to C80 is undertaken. For the latter study, the targeted isomers amount to 131 164, including 17 IPR (isolated pentagon rule) isomers and all non-IPR isomers. Non-IPR isomers dominate the low-lying population of C72 but are gradually phased out of the low-lying population when the fullerene size increases toward C80. An unexpected manner of pentagonal adjacency was observed, that is, for fullerenes containing an adjacent-pentagon chain with less than five pentagons, the longer chain incurs less energy penalty than the shorter chain when the top-two lowest-energy fullerene cages (for all C38- C70) have the same number of adjacent pentagons. For higher fullerenes C112 to C120, a full set of total 32 795 IPR isomers were optimized using the DFTB method. An energy cutoff value of 6.3 kcal/mol was used to collect low-lying candidate isomers for the second-stage single-point energy calculation at the DFT level. Multiple candidates for the lowest-energy structure were identified for C112, C118, and C120. Among them, C112:3299 and C118:7308 exhibit a large HOMO-LUMO gap. For C116, a sole candidate for the lowestenergy structure was identified, namely, C116:6061 which has a high Th symmetry and a large HOMOLUMO gap and is 12.5 kcal/mol lower in energy than the second lowest-energy isomer. Thus, C116: 6061 is most likely to be isolated first in the laboratory among the five large fullerenes. 13C NMR spectra of the ten lowest-energy isomers of C112 to C120 were calculated for comparison with the experiment.

I. Introduction Since the discovery of the prototypical fullerene “buckminsterfullerene” C60 by Smalley and co-workers,1a fullerene chemistry has been developed into a mature field. To date, isolation of higher fullerenes up to C96 has been reported.1b-f Most of these higher fullerenes have been proven to follow the so-called isolated pentagon rule (IPR).2 In fact, C60 is the smallest fullerene having only one IPR isomer, followed by C70 and those larger than C70. Recently, increasing theoretical efforts have been devoted to the exploration of larger IPR fullerenes such as C98-C1203-5 as well as endohedral metallofullerenes and exohedral adducts with non-IPR fullerene cages at lower size (smaller than C80).6 Experimental studies have shown that non-IPR structures of lower fullerenes are generally unstable. However, many endohedral or exohedral derivatives of non-IPR fullerenes are stable, for example, the newly isolated metal-carbide endofullerene Sc2C2@C687a and [email protected],7b Another example is the non-IPR fullerene derivative C64H4, which has been synthesized by Wang et al.8 through introducing methane in the fullerene productions (the Kratschmer-Huffman method). This non-IPR fullerene has been viewed as a bridging structure between C60 and C70. Besides IPR and non-IPR fullerenes, nonclassical †

Part of the special issue “Richard E. Smalley Memorial Issue”. * To whom correspondence should be addressed. E-mail: [email protected].

fullerenes whose cages contain heptagon or tetrahedron also exist. For example, C58F18 and C58F17CF3 which contain a sevenmembered ring were recently isolated and confirmed by mass spectrometry and fluorine nuclear magnetic resonance spectroscopy.9 Theoretical studies of the stabilities of fullerenes require enumeration and construction of all fullerene isomers. If the non-IPR isomers were included, a complete set of fullerene isomers would entail a very large number for large fullerenes (e.g., 31 924 isomers for C80). Ab initio quantum-chemistry calculations are required to accurately determine the relative stabilities among the low-energy isomers. For higher fullerenes, the density functional theory (DFT) method has been proven to be a cost-effective choice to calculate structural parameters of fullerenes. DFT has comparable accuracy but less computational cost than the second-order Møller-Plesset perturbation method (MP2).10 To reduce computational costs, some mathematic methods, e.g., the IPR rule, hexagon-neighbor rule (HNR),11 or approximate standard enthalpy formula,12 have been developed as prescreening tools to reduce the number of candidate isomers, especially for higher fullerenes. Also, an efficient screening tool is usually needed to further reduce the number of low-energy candidates for ab initio calculations. Popular screening tools include the empirical force field method or semiempirical methods (e.g., PM3, TBMD, and SAM1).

10.1021/jp0701082 CCC: $37.00 © 2007 American Chemical Society Published on Web 05/22/2007

17672 J. Phys. Chem. C, Vol. 111, No. 48, 2007 In previous theoretical studies, structures of all fullerene isomers of C20-C36 as well as the IPR isomers of C60-C86 have been optimized using either AM1, noniterative density functional tight binding (NCC-DFTB), or self-consistent charge density functional tight binding (SCC-DFTB) methods.13 It has been shown that results of DFTB calculations correlate well with those of DFT calclulations.13b,14 Using IPR and PAPR rules (pentagon adjacency penalty rule: the most stable isomer is expected to have the lowest number of adjacent pentagons), Martin et al. selected the best isomers of fullerene from C20 to C70 and calculated their ionization potentials as well as C2 and C2+ dissociation energies at the B3LYP/6-31G* level of theory.15 Several fullerenes have been fully investigated using ab initio methods. For example, all 40 isomers of C40 and 271 isomers of C50 have been fully optimized using both DFT and Hartree-Fork (HF) theory.16,17 The top 20 low-energy isomers of C72-C78 have been sorted out using TB Monte Carlo, PM3 semiempirical, and HF/3-21G optimization, respectively.18 For fullerenes larger than C110, C116, C118, and C120 have been studied at the AM1, DFTB, or QCFF/PI (quantum-consistent force field) semiempirical level.3,4 However, calculations at the ab initio level have not been reported for large fullerenes beyond C110. Very recently, top 1000 IPR isomers of higher fullerenes from C132 to C160 were determined based on a topological scheme (approximate standard enthalpy formula12) followed with a screening using the PM3 and TB methods.19 The top 10 lowlying isomers obtained from this scheme were considered as candidates for the lowest-energy structures. DFT calculations at the B3LYP/6-31G* level were then performed for these candidate isomers. Note that the standard enthalpy formula was derived on the basis of 115 IPR fullerenes of C60-C180.12 If the typical error of the formula is less than the energy cutoff value for the screening, the obtained lowest-energy structures are expected to be reliable. In our previous paper (paper 1),5 we reported the best candidates for the lowest-energy structure of fullerenes C82C110. We used the DFTB method as a screening tool and predicted the lowest-energy structures based on DFT calculation. The aim of this paper is 2-fold: (1) To systematically search and study the low-lying structures of lower fullerenes C38C80, considering all possible isomers without employing any topological or mathematical pre-screening tools to reduce the number of candidate isomers and (2) to predict the best candidates for the lowest-energy structures of higher fullerene C112-C120 because of the growing experimental interests for this size range of higher fullerenes. II. Computational Details We used the same computational approach introduced in paper 1.5 All isomers were optimized using the DFTB method, whereas the energy ranking of the low-lying isomers were determined by using DFT single-point energy calculation. The first series of calculations was carried out for lower fullerenes C38-C80, with targeted isomers amount to 131 164. Here, both non-IPR and IPR isomers were considered. In the second series of calculations, for higher fullerene C112-C120, a total of 32 795 IPR isomers were considered. At first, the topological structures of 32 795 IPR isomers were generated by CaGe software20 and then optimized using the DFTB method. Guided by the energy error bar tested in our previous work (paper 1),5 an energy cutoff criterion of 6.3 kcal/mol for the DFTB relative energy was employed to provide isomer candidates for the next-stage DFT calculations. For lower fullerenes C38-C80, a much larger energy cutoff value of 18.0 kcal/mol (75.3 kJ/mol) was adopted. Albertazzi

Shao et al. et al. have shown that the penalty of pentagonal adjacency is typically 80∼100 kJ/mol.16 Thus, the selection of 18.0 kcal/ mol (as cutoff value) allows us to examine isomers containing one more pairs of pentagon-pentagon contact than the lowestenergy structures. In both series of calculations, all isomers that met the energy cutoff criterion were subject to the single-point energy calculation using the hybrid functional PBE1PBE21 with a mid-size basis set 6-311G*. This level of theory and the basis set appear to be sufficiently accurate to simulate the interconversion of small carbon clusters22 or to determine energy ranking of C20 isomers.23 Finally, 13C NMR chemical shifts were calculated for the top ten isomers of higher fullerenes C112C120. Structures of these low-lying isomers were reoptimized using the PBE1PBE/6-31G* method. Since the DFT relative energy of these low-lying isomers is typically less than 1 kcal/ mol, the calculated NMR chemical shifts provide an additional characteristic for these candidate isomers to compare with future experiments. All DFT calculations were performed using Gaussian 03.24 III. Results and Discussions A. C38-C80 Non-IPR and IPR Isomers. A total of 131 164 geometries of fullerene isomers of C38-C80 were fully optimized using the DFTB method. Relative energies of low-energy isomers (those met the energy cutoff criterion 18.0 kcal/mol with respect to the lowest-lying DFTB isomer) are listed in the Table 1 and in the Supporting Information, Table S1, together with their PBE1PBE/6-311G* relative energies and HOMOLUMO gaps. All isomers are labeled according to the Flower and Manolopoulos scheme,25 along with their point-group symmetries. The predicted low-lying structures of C38-C52 and C72-C78 (Table 1 and Table S1) are readily compared with literature results. Figure 1 shows the relative energies of lowlying structures calculated at PBE1PBE/6-311G*//DFTB level of theory, to compare with relative energies calculated at full ab initio levels: B3LYP/6-31G* 26a and B3LYP/6-31G*//HF/ 3-21G*.18 The two lines in Figure 1 nearly overlap, indicating that almost all low-lying isomers identified from the present study exhibit the same energy ranking as previous studies. Only two isomers were missed in this study, C74:5991 and C76:10000, because the two did not meet the energy cutoff criterion. At the B3LYP/6-31G*//HF/3-21G level,18 the calculated relative energies of the two isomers are 15.4 and 17.5 kcal/mol, respectively, indicating that the semiempirical DFTB method entails slightly larger errors in structural optimization than the ab initio HF/3-21G method. We also noticed that a very recent paper26b reported the same top five isomers of C52 as ours. However, a major difference is the energy ranking of C52:437 (with a symmetry T), which is the third lowest energy isomer predicted at the B3LYP/6-31G* level but has a much larger relative energy in our single-point energy calculation. The smaller relative energies for the last isomer of the selected fullerenes (C40, C46, C48, and C50) shown in Figure 1 further demonstrate that structure optimization at the B3LYP/6-31G* level is more accurate than at the DFTB level. Nevertheless, geometry optimization with the DFTB method is computationally about 2 orders of magnitude faster than with the HF/321G method. In fact, ab initio methods are generally impractical for geometry optimization of tens of thousands of higher fullerenes. Note that, in Table 1, the predicted lowest-energy isomers (in boldface) C54:540, C58:1205, C64:1988, C66:1789, and C68:3981 and two nearly isoenergetic lowest-energy isomers C56:864 and C56:916 are consistent with the previous prediction based on the B3LYP/6-31G* optimization.15,6c

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J. Phys. Chem. C, Vol. 111, No. 48, 2007 17673

TABLE 1: Relative Electronic Energies ∆E (kcal/mol), Point-Group Symmetries, and HOMO-LUMO Gaps (eV) for Low-Lying Isomers of C38-C80a PBE1PBE/ 6-311G*//DFTB b

CN

C38(17) C40 (40) C42(45) C44 (89)

C46 (116)

C48 (199)

C50 (271) C52 (437)

C54 (580) C56 (924) C58 (1205)

C60(1812) C62 (2385)

label

c

17 38 39 31 29 45 75 89 72 69 55 109 108 114 116 103 107 99 67 199 171 196 197 139 149 138 165 163 271 270 266 263 422 365 424 437 378 540 537 541 916 864 843 913 1205 1198 1196 1195 1078 1812d 1994 1993 1824

PBE1PBE/ 6-311G*//DFTB

DFTB

sym.

∆E

Egap

∆E

C2 D2 D5d Cs C2 D3 D2 D2 D3h C1 C2V C2 Cs C1 C2 C1 Cs Cs C1 C2 C2 C1 Cs C1 C1 C2V C1 C2 D5h D3 Cs C2 C2 C1 C1 T C2 C2W C1 C1 D2 Cs C2 C2V C3W C1 Cs C2 C2 Ih C2 C1 C1

0.000 0.000 10.173 16.396 22.189 0.000 0.000 0.696 8.024 20.308 182.974 0.000 2.075 6.697 7.708 5.914 5.953 6.984 18.277 2.901 0.000 3.532 5.537 11.625 16.214 13.186 15.864 13.515 2.921 0.000 8.295 11.157 0.000 11.180 15.095 23.171 16.515 0.000 13.332 15.444 0.081 0.000 4.042 8.711 0.000 9.779 11.032 11.091 7.471 0.000 0.000 0.534 2.375

1.878 2.014 2.204 1.633 1.252 2.095 1.905 1.959 2.259 1.633 1.905 1.633 1.823 1.415 1.361 1.469 2.095 1.660 1.878 1.252 1.633 1.388 1.714 1.769 1.388 1.769 1.742 2.204 1.361 2.422 1.905 2.204 1.361 1.333 1.497 0.680 1.714 1.361 1.524 1.415 1.742 1.959 1.714 1.388 0.626 1.333 1.061 1.197 1.415 2.884 1.333 1.197 1.306

0.000 0.000 12.897 14.682 17.875 0.000 0.000 0.073 7.848 17.224 17.672 0.000 2.747 3.559 5.045 5.572 6.788 8.016 17.075 0.000 1.473 1.824 3.264 11.690 12.291 13.227 15.587 15.633 0.000 2.811 9.861 12.984 0.000 10.766 16.010 17.534 17.815 0.000 12.636 15.414 0.000 0.275 4.194 5.252 0.000 11.520 12.689 13.677 15.272 0.000 0.000 0.007 5.200

b

CN

C64 (3465) C66 (4478)

C68 (6332)

C70(8149) C72 (11190) C74(14246) C76 (19151) C78 (24109) C80 (31924)

label

c

1956 1812 1998 1814 1995 1560 1972 1988 1989 1994 1789 1913 3995 1676 1503 3981 4000 1923 3893 1969 1909 3966 3914 3895 3992 3896 6000d 5999 11190d 5779 6000d 9999d 10000d 7665 7679 10000d 22000d 21999d 1999d 30000d 7999d 20000d 13999d 19999d 14000d 29988 8000d 7632 7634 11996 13988 13938 29997

DFTB

sym.

∆E

Egap

∆E

C2 C1 C1 C1 C1 C2 C1 D2 Cs C2 Cs C2V C2 C1 C1 C2 C2 Cs C2 C1 C2V C1 C2 C1 C1 C2 D5h C2W D6d C2 D3h D2 Td C2V C1 C2W D3h C2V D3 C2V D5h D2 C2V D5d D3 C1 Ih C1 C1 C1 C1 C2 C1

11.012 11.259 16.673 14.613 16.965 13.238 19.125 0.000 7.239 13.517 0.000 5.954 10.044 17.067 17.983 0.000 1.555 17.242 15.777 13.333 19.757 10.966 14.744 14.800 16.977 21.518 0.000 0.000 11.008 18.887 0.000 0.000 21.797 24.934 20.752 0.000 5.435 6.295 9.860 10.344 12.399 0.000 4.076 0.255 9.409 12.114 34.921 14.573 18.640 19.482 18.946 17.214 25.515

1.524 1.687 1.361 1.415 1.469 1.361 1.170 2.313 2.145 1.932 2.068 1.197 1.442 1.714 1.687 2.449 1.959 0.708 0.871 1.388 1.143 2.095 1.170 1.170 1.850 1.170 2.857 1.633 2.640 1.197 0.735 2.095 0.708 1.007 1.551 1.714 1.551 2.150 1.714 0.743

11.385 13.603 15.318 15.447 16.072 16.908 17.026 0.000 6.481 12.211 0.000 2.634 6.729 15.040 15.461 0.000 1.145 6.482 7.445 9.435 9.509 10.567 11.498 11.502 14.790 17.571 0.000 0.000 15.156 17.891 0.000 0.000 10.474 14.447 16.387 0.000 2.701 8.573 10.825 0.000 0.483 0.998 1.938 2.276 4.133 11.810 12.288 12.436 13.914 15.038 15.986 16.365 17.744

1.391 0.946 1.148 0.83 1.660 0.680 1.578 1.361 1.170 1.252 1.687 1.088

a An energy cutoff value of 18.0 kcal/mol was used to obtain all the leading candidates for the lowest-energy isomers. The boldface denotes the best candidates for the lowest-energy isomer. b The number of isomers is given in parentheses. c The labels are according to Flowler and Manolopoulos (ref 25). d IPR structure.

The fullerene structures considered in refs 15 and 6c show a minimum number of adjacent pentagons (APs), suggesting that the PAPR is robust. The number and combination of APs are listed in Table 2 for the lowest-energy and the second lowestenergy structures of non-IPR fullerenes C40-C68. C50 is the only fullerene that disobeys the minimum-pentagon-adjacent rule. Interestingly, all of the lowest-energy isomers possess longer adjacent pentagonal chains than the second lowest-energy isomers when the two isomers have the same number of APs. This can be seen in Table 2 for C44, C48, C56, and C66 (the top two isomers of C56 can also be viewed as isoenergetic due to a

small energy difference of 0.081 kcal/mol). This result is somewhat unexpected because the AP chains have a tendency to be short.15 When an AP chain contains more than five pentagons, the longer chain causes more energy penalty as expected, such as in the case of C40 and C46. In this case, the penalty of APs correlates with both the number of AP pairs and the way of their combination. Fullerene C60 and C70 have 1812 and 8149 isomers, respectively. Their well-known (and only) IPR structure, with Ih and D5h symmetry, respectively, has the lowest energy. Fullerene C62 has two candidate isomers for the lowest-energy structure,

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Shao et al.

Figure 1. Relative energy of low-lying isomers for fullerene from C38 to C80 calculated at three levels of theory: PBE1PBE/6-311G*//DFTB, B3LYP/6-31G*//B3LYP/6-31G* (ref 26), and B3LYP/6-31G*//HG/3-21G* (ref 17). The energy cutoff value is 18 kcal/mol.

TABLE 2: Label of Structure and the Number of Adjacent Pentagons (NAP) of the Lowest-Energy and Second Lowest-Energy Isomers of Non-IPR Fullerene C40-C68 lowest-energy structure

second lowest-energy structure

CN

label

structurea

NAP

label

structure

NAP

40 44 46 48 50 52 54 56 58 62 64 66 68

38 75 109 171 270 422 540 864 1205 1994 1988 1789 3981

2 × 65 (2 × 43)+(2 × 21) (2 × 43)+(2 × 21) (1 × 43)+(2 × 32)+(2 × 10) 6 × 21 (2 × 32)+(1 × 21)+(4 × 10) (2 × 32)+(6 × 10) (1 × 32)+(2 × 21)+(5 × 10) (3 × 21)+(6 × 10) (3 × 21)+(6 × 10) (2 × 21)+(8 × 10 (1 × 32)+(9 × 10) (2 × 21)+(8 × 10)

10 8 8 7 6 5 4 4 3 3 2 2 2

39 89 108 199 271 365 537 916 1078 1993 1989 1913 4000

1010+(2 × 10) 4 × 32 (1 × 54)+(2 × 32)+(1 × 10) (2 × 32)+(3 × 21) (5 × 21)+(2 × 10) (1 × 32)+(4 × 21)+(1 × 10) (1 × 32)+(3 × 21)+(3 × 10) (4 × 21)+(4 × 10) (2 × 32)+(6 × 10) (3 × 21)+(6 × 10) (2 × 21)+(8 × 10) (2 × 21)+(8 × 10) (2 × 21)+(8 × 10)

10 8 8 7 5 6 5 4 4 3 2 2 2

a The label of the structure follows the nomenclature of (n × Fm) by Jiao H. J. et. al (ref 32), in which F is the number of the fused pentagons, m is the shared C-C bonds, and n is the number of the Fm combination. The sum of n × m is equal to the number of adjacent pentagons (NAP) and the sum of n × F is equal to 12.

namely C62:1994 and C62:1993, because the relative energies between the two are within 1 kcal/mol. Interestingly, a nonclassical fullerene isomer of C62 containing a heptagon has been isolated experimentally.27 This nonclassical fullerene isomer is actually ∼9 kcal/mol lower in energy than the best classical fullerene isomer (based on DFT calculation).27 Another nonclassical fullerene isomer of C62 containing a four-membered ring was also isolated by adding and rearranging tetrazines on C60.28 Because of the overwhelming stability of C60 and C70, nonclassical fullerenes containing either heptagon or tetrahedron are expected to be very stable at sizes of C58, C62, C68, and C72. These fullerenes can be viewed via adding or removing two carbon atoms from highly stable fullerenes C60 and C70. Fullerene C80 has only seven IPR isomers, but the total number of isomers of C80 amounts to 31 924. Two IPR isomers of C80, C80:20000 and C80:19999 with D2 and D5d symmetry respectively, have been isolated in the laboratory.2d,29 Also, the seven IPR isomers of C80 have been fully studied theoretically.30 A recent ab initio calculation at the MP2(FC)/6-31G*//B3LYP/

6-31G* level30a shows that C80:20000 with D2 symmetry is the lowest-energy structure. Our calculation confirms this finding. Our calculation also shows that the second lowest-energy isomer is C80:19999 (with symmetry D5d) and is only 1.1 kcal/mol higher in energy than the lowest-energy isomer (see Table S2). The energy difference between the two is less than 2.2-2.9 kcal/mol obtained from B3LYP/6-31G*12,30b,30c and MP2(FC)/ 6-31G*//B3LYP/6-31G*30a calculations. As shown in Table 1, C80:29988 is the best non-IPR isomer with a relative energy of 12.1 kcal/mol. Due to its large HOMO-LUMO gap of 1.66 eV, it is possible that C80:29988 is a chemically stable isomer. Last, C80:8000 is a very special IPR isomer since it has Ih symmetry as C60. The Jahn-Teller distortion reduces its symmetry to D2h.30d The pristine C80:8000 cage is unlikely to be stable due to its large relative energy 34.9 kcal/mol. However, this cage can make a stable endohedral metallofullerene.31 The transition from non-IPR to IPR structures occurs in the size range of C70 to C80. Besides C60 and C70, C72 is the smallest fullerene that has an IPR isomer. In Table 1, the only IPR isomer

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J. Phys. Chem. C, Vol. 111, No. 48, 2007 17675

TABLE 3: Relative Electronic Energies ∆E (kcal/mol), Symmetries, Ring Spiral Codes, and HOMO-LUMO Gaps (eV) of the Best Candidates for Lowest-energy Structures of C112-C120a PBE1PBE/6-311G*//DFTB CNb C112 (3342)

C114 (4468) C116 (6063) C118 (8148) C120 (10774)

DFTB

labelc

ring spiral code

symmetry

∆E

Egap

∆E

3299 3336 3189 3168 1848 1842 4462 2244 4073 6061 7924 7308 7933 5878 10253 4811 10243 8781 10764 10268

1 7 10 24 26 28 31 33 36 52 55 57 1 7 14 20 22 25 31 41 46 48 51 54 1 7 10 18 24 27 35 41 45 48 50 54 1 7 10 18 23 27 37 41 45 47 50 54 1 7 9 22 24 27 39 42 45 47 49 54 1 7 9 22 24 27 33 39 42 44 48 58 1 7 11 25 27 31 33 35 37 43 51 59 1 7 9 22 25 33 36 39 41 43 45 59 1 7 10 14 30 32 35 37 41 50 53 56 1 7 20 23 25 28 32 35 41 44 50 60 1 7 10 24 27 31 35 40 43 47 50 61 1 7 10 19 23 26 40 43 47 50 56 60 1 7 10 24 27 32 35 40 43 50 55 58 1 7 10 13 26 31 36 45 50 52 54 57 1 7 10 24 27 32 35 40 43 50 57 62 1 7 9 23 26 31 34 38 42 55 58 60 1 7 10 24 27 31 36 42 47 52 56 59 1 7 10 18 23 26 37 41 45 54 57 59 1 7 20 22 25 28 32 43 48 50 53 62 1 7 10 24 27 32 36 42 47 52 56 60

D2 D2 C2 C1 C1 C1 Cs C1 C1 Th C1 C2W C2 C1 C2 Cs C1 C1 D5 C2

0.0000 0.6620 0.7670 1.2720 1.4490 2.5940 0.0000 1.1000 2.6820 0.0000 0.0000 0.0510 1.5850 2.6420 0.0000 0.0890 0.4200 1.2010 1.9250 2.0080

1.524 1.197 0.925 1.116 1.143 1.034 0.952 1.197 1.170 1.742 1.415 1.823 1.225 1.306 1.061 1.116 1.279 1.197 0.680 1.116

4.131 0.000 1.649 4.144 5.344 4.677 5.871 5.890 6.159 0.000 2.364 4.722 0.000 3.579 0.000 4.308 2.078 5.269 1.495 2.608

a The energies are calculated at PBE1PBE/6-311G* level of theory and based on the DFTB optimized geometries. The boldface denotes the top candidates for the lowest-energy isomer. b The number of IPR isomers is given in parentheses. c The labels are according to Flowler and Manolopoulos.

C72:11190 is 11.0 kcal/mol higher in energy than the lowestenergy (non-IPR) isomer C72:5999. For C74, the best non-IPR isomer C74:5991 is 21.9 kcal/mol higher in DFTB energy than the lowest-energy (IPR) isomer C74:6000. The energy difference is larger than the cutoff value 18.0 kcal/mol. However, C74: 5991 has a relative energy of 14.7 kcal/mol at PBE1PBE/6311G*//DFTB level (see Table S1) and a much larger HOMOLOMO gap (2.23 eV) than that (0.74 eV) of C74:6000. The two IPR isomers of C76, C76:9999 and C76:10000, are the first and third lowest-energy isomers. In particular, C76:9999 is notably lower in energy (20.8 kcal/mol) than the second lowest-energy isomer C76:7679. In addition, its large HOMO-LOMO gap (2.10 eV) renders C76:9999 chemically stable. The best nonIPR isomer C76:7679 with C1 symmetry is only 1.0 kcal/mol lower in energy than the IPR isomer C76:10000, indicating an overlap in energy between the least stable IPR cage and the best non-IPR cage.3 This overlap in energy can be also seen for C78. Four of the five IPR isomers of C78 occupy the topfour positions in the energy ranking, except the last IPR isomer: C78:2000 (with symmetry D3h) which is the ninth in the energy ranking. In summary, non-IPR isomers dominate the low-lying population of C72 but are gradually phased out of the low-lying population toward C80. Knowing the stabilities of nonIPR isomers can also offer good candidates for fullerene derivatives. It appears that the energy cutoff value 18.0 kcal/ mol is reasonable to obtain cage candidates for fullerene derivatives. For example, the endohedral fullerene Sc2C2@C68 discovered by Wang et al.7a contains a non-IPR fullerene cage C68:1909. This fullerene cage is 9.5 kcal/mol higher in DFTB energy than the lowest-energy isomer (see Table 1). Last, it is known that fullerenes with a large HOMO-LUMO gap and high symmetry are not necessarily the lowest-energy structure.4 Among the lowest-energy structures, C42:45, C50: 270, C56:864, C64:1988, C66:1789, C68:3981, and C76:9999, as well as C60 and C70, all exhibit the largest HOMO-LUMO gap in their peer group. Although the IPR isomer C72:11190 is the second lowest-energy isomer, it has a very large theoretical HOMO-LUMO gap of 2.64 eV (only smaller than 2.88 eV of C60 and 2.86 eV of C70). More experiments on C72 fullerene are needed to identify which of the isomers (among C72:11190, C72:5999 and non-classic fullerenes containing heptagon or tetragon) is the most stable structure. On the other hand, the

Figure 2. Best candidates for the lowest-energy structure of higher fullerene C112-C120.

lowest-energy structures C58:1205 and C74:6000 have a very small HOMO-LUMO gap (0.63 and 0.74 eV) and thus are expected to be relatively less stable.

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Shao et al.

Figure 3. Calculated 13C NMR spectra for the 10 candidate lowest-energy structures of C112-C120. C116:6061 Th is predicted to be more easily isolated experimentally than other higher fullerenes.

B. Lowest-Energy IPR Isomers of C112-C120. All 32 795 IPR isomers of C112-C120 were geometrically optimized using the DFTB method. Those isomers meeting the energy cutoff criterion (6.3 kcal/mol) were collected as low-lying isomers. Subsequent DFT calculations at the PBE1PBE/6-311G* level were carried out to determine the energy ranking as well as the best candidates for the lowest-energy structure. DFTB energies and relative energies of low-lying isomers are listed in Table S3, together with the PBE1PBE/6-311G* electronic energies. Those isomers with relative energy (DFT calculation) less than 3.0 kcal/mol were identified as the best candidates and presented in Table 3. Their corresponding spiral codes and HOMOLUMO gaps are also presented in Table 3. In particular, isomers with relative energies less than 1.0 kcal/mol are highlighted in boldface in Table 3 and shown in Figure 2. From Table 3 and Table S3, one can see that the lowest-energy isomers of C116 and C120 are identical to the lowest DFTB isomers, while those of C112, C114, and C118 are different. For C114, for example, the predicted best three isomerss4462, 2244, and 4073sall have a modest relative energy (5.9-6.2 kcal/mol) in DFTB calculation. In paper 1,5 we predicted that C102:603 and C108:1771 are the sole candidates for the lowest-energy structure of C102 and C108, respectively. Thus, both structures are expected to be isolated more easily than the other three fullerenes (C104, C106, and C110). Here, C116:6061 has high Th symmetry and a large HOMO-LUMO gap of 1.74 eV and is 12.5 kcal/mol lower in energy than the second lowest-energy isomer C116:6060 (Table S3). Therefore, we expect that C116:6061 is more easily isolated and characterized than the other four fullerenes (C112, C114, C118, and C120). In Table 3, one can also see that three isomers of C112, two of C118, and three of C120 have relative energies within 1.0 kcal/mol. However, due to their large HOMO-LUMO gap, C112: 3299 and C118:7308 may be more likely to be isolated than their peer candidates. In Figure 2, C118:7308 shows an interesting heart-like shape with C2V symmetry, whereas C116: 6061 shows a quasioctahedral structure. As pointed out in paper

1, the HOMO-LUMO gap of fullerene tends to decrease with increasing the fullerene size. For the best candidate isomers of C112-C120, the averaged HOMO-LUMO gap is 1.2 eV which is lower than 1.8 eV, the averaged gap for C38-C80. Finally, the simulated 13C NMR spectra for the top 10 isomers of C112-C120 are displayed in Figure 3. Structures of the 10 best isomers were reoptimized at the PBE1PBE/6-31G* level. Multiple isoenergetic isomers are shown in Figure 3 for C112, C118, and C120 because their relative energies are within 1.0 kcal/ mol. The calculated NMR data are given in Table S4, which are readily to be compared with experimental data. It is worthy to note that the 13C NMR spectrum of C116:6061 is quite unique in that it has only seven lines separated in two regions, 119153 ppm and ∼222 ppm. IV. Conclusions We have employed an efficient computational approach which combines the semiempirical density-functional based tightbinding method (DFTB) for geometry optimization and densityfunctional theory (PBE1PBE/6-311G*) for single-point energy calculation to search for the lowest-energy structures of higher fullerenes C110-C120. We have also performed a systematic study of low-lying structures of lower fullerenes C38-C80. For the latter, the targeted isomers amount to 131 164, which include both IPR and non-IPR isomers. The predicted lowest-energy isomers based on PBE1PBE/6-311G*//DFTB calculation are consistent with those previously predicted based on full ab initio calculations at B3LYP/6-31G* and B3LYP/6-31G*//HF/3-21G* levels. Non-IPR isomers dominate the low-lying population of C72 but are gradually phased out of the low-lying population when the size of fullerenes increases toward C80. An unexpected manner of pentagonal adjacency was observed; that is, for fullerenes containing an adjacent-pentagon chain with less than five pentagons, the longer chain incurs less energy penalty than the shorter chain when the top-two lowest-energy fullerene cages (from C38 to C70) have the same number of adjacent pentagons.

Search for Lowest-Energy Fullerenes For higher fullerenes C112-C120, 20 isomers have been identified as the best candidates for the lowest-energy structures. Their relative energies, spiral codes, and HOMO-LUMO gaps are also presented. Among the 20 isomers, 10 isomers with relative energies less than 1.0 kcal/mol are highlighted in Figure 2, including three isomers of C112, two of C118, and three of C120. Isomers C112:3299 and C118:7308 are possibly to be isolated prior to their peer counterparts due to their large HOMOLUMO gaps. 13C NMR spectra of the 10 lowest-energy structures were also calculated. The spectra serve as additional characteristic information to be compared with experimental measurements. Finally, it is worth singling out fullerene C116, because it is a sole candidate for the lowest-energy structure, namely, C116:6061. The latter has a high Th symmetry and a large HOMO-LUMO gap and is 12.5 kcal/mol lower in energy than the second lowest-energy isomer. Moreover, it has a unique 13C NMR spectrum with only seven lines. We thus predict that C116:6061 is more easily isolated and characterized in the laboratory than other higher fullerenes from C112 to C120. Acknowledgment. This research was supported by grants from DOE (DE-FG02-04ER46164), the Nebraska Research Initiative, and the John Simon Guggenheim Foundation (X.C.Z.) and by the Research Computing Facility at University of Nebraska-Lincoln. Supporting Information Available: Complete reference 24, total electronic energies, relative energies, and HOMO-LUMO gaps for fullerenes C38-C80 and C112-C120 and 13C NMR chemical shifts for C112-C120 are collected. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) (a) Kroto, H. W.; Heatch, J. R.; O’Brien, S. C.; Curl, R. F.; Smalley, R. E. Nature 1985, 318, 162-163. (b) Achiba, Y.; Kikuchi, K.; Aihara, Y.; Wakabayashi, T.; Miyake, Y.; Kainosho, M. In The Chemical Physics of Fullerenes 10 (And 5) Years Later; Andreoni, W., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1996; pp 139-147. (c) Minami, T.; Miyake, Y.; Kikuchi, K.; Achiba, Y. In The 18th Fullerene General Symposium; Osawa, E., Ed.; Fullerene Research Association of Japan: Okazaki, Japan, 2000; 1B02, p 42. (d) Wang, G. W.; Saunders, M.; Khong, A.; Cross, R. J. J. Am. Chem. Soc. 2000, 122, 3216-3217. (e) Hennrich, F. H.; Michel, R. H.; Fischer, A.; Richard, S. S.; Gilb, S.; Kappes, M. M.; Fuchs, D.; Burk, M.; Kobayashi, K.; Nagase, S. Angew. Chem., Int. Ed. Engl. 1996, 35, 1732-1734. (f) Tagmatarchis, N.; Arcon, D.; Prato, M.; Shinohara, H. Chem. Commun. 2002, 24, 2992-2993. (2) Kroto, H. W. Nature 1987, 329, 529. (3) Fowler, P. W.; Heine, T.; Zerbetto, F. J. Phys. Chem. A 2000, 104, 9625-9629. (4) Achiba, Y.; Fowler, P. W.; Mitchell, D.; Zerbetto, F. J. Phys. Chem. A 1998, 102, 6835-6841. (5) Shao, N.; Gao, Y.; Yoo, S.; An, W.; Zeng, X. C. J. Phys. Chem. A 2006, 110, 7672-7676. (6) (a)Wang, C. R.; Kai, T.; Tomiyama, T.; Kobayashi, Y.; Nishibori, E.; Takata, M.; Sakata, M.; Shinohara, H. Nature 2000, 408, 426-427. (b) Slanina, Z.; Chen, Z. F.; Schleyer, P. v. R.; Uhlik, F.; Lu, X.; Nagase, S. J. Phys. Chem. A 2006, 110, 2231-2234. (c) Dı´az-Tendero, S.; Alcamı´, S.

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