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Size dependence of ionization potentials and dissociation energies for neutral and singly-charged Cn fullerenes (n = 40–70)
Chemical Physics Letters 416 (2005) 14–17 www.elsevier.com/locate/cplett
Size dependence of ionization potentials and dissociation energies for neutral and singly-charged Cn fullerenes (n = 40–70) Goar Sa´nchez, Sergio Dı´az-Tendero, Manuel Alcamı´, Fernando Martı´n
*
Departamento de Quı´mica, C-9, Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain Received 12 July 2005; in final form 1 September 2005 Available online 3 October 2005
Abstract Dissociation energies and ionization potentials of neutral and singly-charged fullerenes have been obtained from density functional theory calculations for sizes ranging from 40 to 70 atoms. Good agreement with available experimental data has been obtained. Our results confirm that magic number fullerenes with n = 50, 60 and 70 present the largest ionization potentials and dissociation energies. We have found that the most stable isomer for n = 62 is a non-classical structure with a chain of four adjacent pentagons surrounding a heptagon, and for n = 50 is a structure of D3 symmetry that violates the pentagon adjacency penalty rule. Both unusual structures lead to the best agreement with experiment. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction In the last decade a large number of collision experiments have been performed to study the fragmentation properties of fullerenes (see, e.g., [1,2] and references therein). It is now well established that moderately excited neutral, singly- and doubly-charged fullerenes preferentially decay by emission of a neutral C2 molecule [3,4]. Thus, the knowledge of the corresponding dissociation energy is crucial to interpret the mass spectra that arise from those experiments. In particular, the determination of the dissociation energy of neutral and singly-charged C60 has received considerable attention and has been the subject of a long standing controversy that has only been settled down very recently (see, e.g., the reviews [5,6] and references therein). In recent experiments, the dissociation energy of singly-charged fullerenes, þ Cþ n ! Cn2 þ C2
ð1Þ
have been measured as a function of fullerene size n for n = 40–70 [7–9]. These experiments show that the dissocia*
Corresponding author. E-mail address: [email protected] (F. Martı´n).
0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.09.033
tion energy varies significantly with n. Such a variation is not yet fully understood. In contrast with the large number of theoretical works devoted to determine the binding energies of C60 and Cþ 60 (see, e.g., [10]), there have been much less attempts to determine these quantities in smaller or larger fullerenes. The most complete calculation of dissociation energies for fullerenes other than C60 have been performed by Zhang et al. [11], but only for neutral systems. These authors have determined dissociation energies for fullerenes as small as C20 by using a tight-binding model combined with a scheme to generate energetically favorable structures. Only classical structures made of pentagonal and hexagonal rings were considered. Ionization potentials have been evaluated by Martin et al. [12] for n = 52–60. Recently, we have reported high level theoretical calculations of dissociation energies and ionization potentials in the range n = 50–60 for neutral, singly- and doubly-charged fullerenes [13]. In this work, we have extended the calculations reported in [13] to provide dissociation energies for neutral and singly-charged species as well as ionization potentials for the former for sizes ranging from 40 to 70 atoms. In Section 2, we briefly describe the theoretical methods used in this work. Our results on dissociation energies and ionization
G. Sa´nchez et al. / Chemical Physics Letters 416 (2005) 14–17
potentials are given and discussed in Section 3. Comparison with the available experimental measurements is also shown. We end with some conclusions in Section 4. 2. Computational details We have employed in our calculations the density functional theory (DFT) with the B3LYP functional for exchange and correlation. This functional combines the BeckeÕs three parameter nonlocal hybrid exchange potential with the nonlocal correlation functional of Lee, Yang and Parr. The geometries of all the structures have been optimized by using the 6-31G(d) basis set. The B3LYP functional has been proved to be good a choice for the description of carbon clusters [13]. In the case of small carbon clusters, the calculated geometries and the vibrational frequencies are very close to those obtained at higher levels of theory [14,15]. The calculations have been carried out with the GAUSSIAN98 program [16]. The starting geometry of the classical structures investigated have been obtained with the help of the CAGE program [17]. 3. Results and discussion Classical fullerenes are cages made of twelve pentagons and a variable number of hexagons. For a given size, (i.e., for a given number of hexagons), there are many possible arrangements of pentagons and hexagons [18]. Thus, the selection of the appropriate arrangement is actually the first difficulty one meets in the search for the most stable isomer. More precisely, the relative position of the pentagonal faces determines the ring strain in the cage and, therefore, its stability. In general, the larger the number of adjacent pentagons (AP), the larger the strain. Thus, the most stable fullerenes have all pentagons isolated. This rule, known as the Isolated Pentagon Rule (IPR), was first proposed by Kroto [19]. Only C60 and fullerenes with a number of atoms equal or greater than 70 present isomers that follow the IPR. For other fullerenes, it is impossible to have all pentagons isolated. In this case, the most stable isomer is expected to have the lowest number of APs [20,21]. This second rule is known as the Pentagon Adjacency Penalty Rule (PAPR) or Minimum Pentagon-Adjacency Rule [19,21–23]. In previous works, a few exceptions to the PAPR have been found for both neutral and cationic fullerenes [24– 28]. In particular, we have shown [24] using MP2 theory that C50 does not follow the PAPR: the isomer C50(D3) containing 6 APs is 21.4 kcal/mol more stable than the isomer C50(D5h) containing 5 APs. At the B3LYP/6-31G(d) level of theory used in this work, C50(D3) is more stable by 2.3 kcal/mol, in agreement with [25,26,28]. The situation is the opposite for cationic species: C50(D5h) is 9.5 kcal/mol more stable than C50(D3) and, therefore, the PAPR is fulfilled. In the present work, we have also considered a few nonclassical structures containing heptagons and/or squares
15
that result from extraction or addition of C2 units from the most stable isomers of neighboring size. For C62 we have found that two non-classical structures are more stable than the classical ones. In particular, an isomer of C2v symmetry containing a square ring is 10.4 kcal/mol more stable than the most stable classical isomer. This structure has been proposed in [29] and synthesized in [30]. We have also found that a second non-classical isomer proposed by Ayuela et al. [22] containing a heptagon is even more stable: it has Cs symmetry and its energy is 13.5 kcal/mol lower than that of the most stable classical fullerene. In the case of the cationic species, these isomers are even more stable: 10.2 and 17.7 kcal/mol, respectively. Direct C2 extraction from the non-classical structures containing either the square or the heptagon leads the well-known icosahedral isomer of C60. We have summarized in Table 1 the symmetry, the relevant ring characteristics (apart from hexagons and isolated pentagons) and the number of adjacent pentagons for the most stable isomers found at the B3LYP/6-31G(d) level of theory. These properties are common to both neutral and singly-charged species, except for n = 50, 66 and 68. A more detailed analysis of the structure of the different isomers for a given cluster size will be presented elsewhere. We have used the absolute electronic energies of the most stable isomers to evaluate adiabatic ionization potentials of neutral Cn fullerenes and C2 dissociation energies of neutral and singly-charged Cþ n fullerenes. The results are also given in Table 1. For the calculation of C2 dissociation energies, we have used the absolute energy of C2 obtained at the same level of theory corrected as explained in [13]. Dissociation energies are very similar for neutral and cationic species except in the size interval n = 58–64. Thus, similar qualitative trends are observed concerning their variation with cluster size. Our results for neutral fullerenes agree reasonably well with those of Zhang et al. [11]. Fig. 1 shows the dissociation energy for C2 emission from singly-charged Cþ n (Eq. 1) as a function of cluster size, n. Experimental values [7,9,31,32] are also shown for comparison. The data of Barran et al. [31] were obtained in arbitrary units; they have been renormalized to the calculated value for Cþ 54 . Our results show the same trends as the experimental values, in particular the pronounced þ þ peaks observed in the vicinity of Cþ 50 , C60 and C70 . The maximum dissociation energies correspond to maþ þ gic number fullerenes: Cþ 50 , C60 and C70 . The great stability þ þ of C60 and C70 is due to the absence of adjacent pentagons in the cage, which implies low ring strain. In addition, Cþ 60 is nearly spherical (Ih symmetry), which provides a surþ plus of stability. Fullerenes between Cþ 60 and C70 always þ have APs. C50 is the smallest fullerene in which pentagons can be arranged in pairs of APs (smaller fullerenes always have chains of three or more adjacent pentagons). Furthermore, in a spherical electronic model of fullerenes, its neutral C50 analogue has a closed electronic shell with 2(l + 1)2 = 2(4 + 1)2p electrons, which results in spherical aromaticity and, therefore, in additional stability [33,24].
G. Sa´nchez et al. / Chemical Physics Letters 416 (2005) 14–17
16
Table 1 Symmetry, structure (namely ring structures different from hexagons and isolated pentagons), number of adjacent pentagons (NAP), first ionization potentials (IP) in eV, and C2 dissociation energies in eV for neutral (DE) and singly-charged (DE+) fullerenes for the most stable isomers at the B3LYP/631G(d) level of theory Cn
Symmetry
Structure
NAP
IP (eV)
DE (eV)
DE+ (eV)
C70 C68 C66 C64 C62 C60 C58 C56 C54 C52 C50 C48 C46 C44 C42 C40
D5h C2 (C2) Cs (C2v) D2 Cs Ih Cs D2 C2v C2 D3 (D5h) C2 C2 D2 D3 D2
– 2-2AP (2-2AP) 1-C3AP (2-2AP) 2-2AP 1Hp + 1-C4AP – 3-2AP 4-2AP 2-C3AP 2-C3AP + 1-2AP 6-2AP (5-2AP) 1-C4AP + 2-C3AP 2-C4AP + 2-2AP 2-C4AP + 2-2AP 3-C4AP 2-C6AP
0 2 2 2 3 + Hp 0 3 4 4 5 5 7 8 8 9 10
7.02 6.87 6.86 6.78 6.36 7.14 6.41 6.65 6.61 6.54 7.26 6.85 7.04 7.15 7.13 7.01
10.36 8.36 7.88 8.99 5.53 11.37 8.11 8.47 8.44 7.79 9.63 8.60 7.79 8.61 8.50 –
10.11 8.29 7.97 8.57 6.31 10.64 8.35 8.43 8.37 8.10 9.63 8.79 7.90 8.59 8.38 –
Structures of neutral and singly-charged species are generally the same; when they differ, those of the singly-charged species are given within parenthesis. Notations: 2AP, two adjacent pentagons; C3AP, a chain of three adjacent pentagons; 1Hp, a heptagon; C4AP, a chain of four adjacent pentagons; C6AP, a chain of six adjacent pentagons
Fig. 2 shows our results for the first ionization potential of neutral Cn fullerenes as a function of cluster size. The ionization potential has been obtained by subtracting the absolute energies of the Cn and Cþ n in their optimum geometries. In all cases but three, Cn and Cþ n have the same isomeric form (see Table 1) and, therefore, there is no ambiguity. For C50, C66 and C68 the isomeric form of the neutral species differs from that of the cation. In this case, we have subtracted the energy of the most stable isomer of Cn and that of the same isomeric form of Cþ n . This implies that, apart from bond relaxation, no reorganization of the fullerene cage is allowed during the ionization process. The
12
Dissociation energy (eV)
11
+ Cn →
+ Cn-2 +
C2
10 9 8 7 6 5
44
48
52
56
60
64
68
Cluster size n Fig. 1. Comparison between calculated and experimental C2 dissociation energy as a function of cluster size. Circles: this work. Experimental values: Squares: Barran et al. [31] scaled to Cþ 54 ; triangles up: Laskin et al. [32]; triangles left: Tomita et al. [7]; triangles right: Concina et al. [9].
Ionization potential (eV)
These two reasons make C50 and its corresponding cationic partner singular cases in a wide range of sizes: n = 40–58. The lowest dissociation energies correspond to C62 and Cþ 62 . In this case, the products of dissociation are the icosahedral C60 and Cþ 60 fullerenes, respectively. Therefore, it is not surprising that the energy required is smaller than for the other fullerenes. If one takes the most stable classical isomer of Cþ 62 instead of that containing the heptagonal þ ring, one obtains dissociation energies for Cþ 64 and C62 that are, respectively, 1 eV larger and smaller than those shown in Fig. 1. This would deteriorate the agreement with experiment, which proves that the Cþ 62 fullerene obtained in the experiments is indeed a non-classical structure as that considered in the present work.
8
7
6
40
44
48
52
56
60
64
68
Cluster size n Fig. 2. Comparison between calculated and experimental first ionization potential as a function of cluster size. Circles: this work. Experimental values: Diamonds: Zimmerman et al. [34]; squares: McElvany et al. [35]. Dashed line: result obtained for a C50 fullerene of D5h symmetry (see text).
G. Sa´nchez et al. / Chemical Physics Letters 416 (2005) 14–17
calculated ionization potentials should be considered as adiabatic. Fig. 2 also includes the experimental data of [34,35]. Our results are in good agreement with the experimental data, except for a constant downward shift of 0.5 eV. The maximum values of the curve appear for C50, C60 and C70, which is due again to the large stability of these clusters. Nevertheless, the case of C50 must be analyzed with some caution. As mentioned above, the most stable C50 isomer has D3 symmetry and does not follow the PAPR. Therefore, the theoretical value given in Fig. 2 corresponds to the C50 ðD3 Þ ! Cþ 50 ðD3 Þ þ e ionization process. However, if we consider the ionization energy of the other isomer, i.e. C50 ðD5h Þ ! Cþ 50 ðD5h Þ þ e , one obtains a value of 6.75 eV (dashed line in Fig. 2), which is even farther from the experimental value of Zimmerman et al. [34]. This comparison with experiment indicates that the D3 isomer is indeed the one observed experimentally. 4. Conclusion C2 dissociation energies of neutral and singly-charged Cn fullerenes (n = 42–70) and first ionization potentials of neutral Cn fullerenes (n = 40–70) have been computed by means of DFT-B3LYP/6-31G(d). Good agreement with experiment has been obtained. We have found that magic number fullerenes such as C50, C60 and C70 (and the correþ þ sponding cationic species Cþ 50 , C60 and C70 ) present larger ionization potentials and dissociation energies than their neighbor fullerenes. We have also found that the most stable isomer of C62 and Cþ 62 is a non-classical fullerene with a chain of four adjacent pentagons surrounding a heptagon. This structure, originally proposed by Ayuela et al. [22], leads to dissociation energies in good agreement with experimental ones, whereas the use of a classical structure only made of hexagons and pentagons leads to results that þ significantly differ from experiment for both Cþ 62 and C64 . We have also found that a D3 isomer of C50 that violates the pentagon adjacency penalty rule is the one that gives the best agreement with the measured ionization potentials. Acknowledgements We thank the CCC-UAM and CIEMAT for allocation of computer time. Work partially supported by the DGI Project Nos. BFM2003-00194, BQU2003-00894 and CTQ2004-00039/BQU, and the CAM Project No. GR/ MAT/0083/2004. References [1] E.E.B. Campbell, Fullerene Collision Reactions, first ed., Kluwer Academic Publishers, Dordrecht, 2003.
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[2] R. Dendg, O. Echt, Int. J. Mass. Spectrom. 233 (2004) 1. [3] S. Martin, L. Chen, A. Denis, R. Bredy, J. Bernard, J. Desesquelles, Phys. Rev. A 62 (2000) 022707. [4] S. Matt, R. Parajuli, A. Stamatovic, P. Scheier, T.D. Ma¨rk, J. Laskin, C. Lifshitz, Eur. Mass Spectrom. 5 (1999) 477. [5] C. Lifshitz, Int. J. Mass Spectrom. 198 (2000) 1. [6] K. Hansen, E.E.B. Campbell, Int. J. Mass. Spectrom. 233 (2004) 215. [7] S. Tomita, J.U. Andersen, C. Gottrup, P. Hvelplund, U.V. Pedersen, Phys. Rev. Lett. 87 (2001) 073401. [8] K. Gluch, S. Matt-Leubner, O. Echt, B. Concina, P. Scheier, T.D. Ma¨rk, J. Chem. Phys. 121 (2004) 2137. [9] B. Concina, K. Gluch, S. Matt-Leubner, O. Echt, P. Scheier, T.D. Ma¨rk, Chem. Phys. Lett. 407 (2005) 464. [10] A.D. Boese, G.E. Scuseria, Chem. Phys. Lett. 294 (1998) 233. [11] B.L. Zhang, C.H. Xu, C.Z. Wang, C.T. Chan, K.M. Ho, Phys. Rev. B 46 (1992) 7333. [12] S. Martin, L. Chen, R. Bredy, J. Bernard, M.C. Buchet-Poulizac, A. Allouche, J. Desesquelles, Phys. Rev. A 66 (2002) 063201. [13] S. Dı´az-Tendero, M. Alcamı´, F. Martı´n, J. Chem. Phys. 119 (2003) 5545. [14] J.M.L. Martin, J. El-Yazal, J.P. Franc¸ois, Chem. Phys. Lett. 242 (1995) 570. [15] J.M.L. Martin, J. El-Yazal, J.P. Franc¸ois, Chem. Phys. Lett. 252 (1996) 9. [16] M.J. Frisch et al., GAUSSIAN98, Revision A.11, Gaussian, Inc., Pittsburgh, PA, 2001. [17] G. Brinkmann, O.D. Friedrichs, A. Dress, T. Harmuth, MatchCommun. Math. Comput. Chem. 36 (1997) 233. [18] P.W. Fowler, D.E. Manolopoulos, An Atlas of Fullerenes, Oxford University Press, Oxford, 1995. [19] H.W. Kroto, Nature 329 (1987) 529. [20] P.W. Fowler, T. Heine, D. Mitchell, G. Orlandi, R. Schmidt, G. Seifert, F. Zerbetto, J. Chem. Soc., Faraday Trans. 92 (1996) 2203. [21] E. Albertazzi, C. Domene, P.W. Fowler, T. Heine, G. Seifert, C.V. Alsenow, F. Zerbetto, Phys. Chem. Chem. Phys. 1 (1999) 2913. [22] A. Ayuela, P.W. Fowler, D. Mitchell, R. Schmidt, G. Seifert, F. Zerbetto, J. Phys. Chem. 100 (1996) 15634. [23] E.E.B. Campbell, P.W. Fowler, D. Mitchell, F. Zerbetto, Chem. Phys. Lett. 250 (1996) 544. [24] S. Dı´az-Tendero, M. Alcamı´, F. Martı´n, Chem. Phys. Lett. 407 (2005) 153. [25] L. Zhechkov, T. Heine, G. Seifert, J. Phys. Chem. A 108 (2004) 11733. [26] X. Lu, Z. Chen, W. Thiel, P. Schleyer, R. Huang, L. Zheng, J. Am. Chem. Soc. 126 (2004) 14871. [27] S. Dı´az-Tendero, F. Martı´n, M. Alcamı´, Chem. Phys. Chem. 6 (2005) 92. [28] X. Zhao, J. Phys. Chem. B 109 (2005) 5267. [29] W. Qian, M.D. Bartberger, S.J. Pastor, K.N. Houk, C.L. Wilkins, Y. Rubin, J. Am. Chem. Soc. 122 (2000) 8333. [30] W. Qian, S. Chuang, R.P. Amador, T. Jarrosson, M. Sander, S. Pieniazek, S.I. Khan, Y. Rubin, J. Am. Chem. Soc. 125 (2003) 2066. [31] P.E. Barran, S. Firth, A.J. Stace, H.W. Kroto, K. Hansen, E.E.B. Campbell, Int. J. Mass Spectrom. 167 (1997) 127. [32] J. Laskin, B. Hadas, T.D. Ma¨rk, C. Lifshitz, Int. J. Mass Spectrom. 177 (1998) L9. [33] M. Bu¨hl, A. Hirsch, Chem. Rev. 101 (2001) 1153. [34] J.A. Zimmerman, J.R. Eyler, S.B.H. Bach, S.W. McElvany, J. Chem. Phys. 94 (1991) 3556. [35] S.W. McElvany, S.B.H. Bach, in: Proceedings of the 39th ASMS Conference on Mass Spectrometry and Allied Topics, vol. 39, 1991, p. 422.
Size dependence of ionization potentials and dissociation energies for neutral and singly-charged Cn fullerenes (n = 40–70) Goar Sa´nchez, Sergio Dı´az-Tendero, Manuel Alcamı´, Fernando Martı´n
*
Departamento de Quı´mica, C-9, Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain Received 12 July 2005; in final form 1 September 2005 Available online 3 October 2005
Abstract Dissociation energies and ionization potentials of neutral and singly-charged fullerenes have been obtained from density functional theory calculations for sizes ranging from 40 to 70 atoms. Good agreement with available experimental data has been obtained. Our results confirm that magic number fullerenes with n = 50, 60 and 70 present the largest ionization potentials and dissociation energies. We have found that the most stable isomer for n = 62 is a non-classical structure with a chain of four adjacent pentagons surrounding a heptagon, and for n = 50 is a structure of D3 symmetry that violates the pentagon adjacency penalty rule. Both unusual structures lead to the best agreement with experiment. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction In the last decade a large number of collision experiments have been performed to study the fragmentation properties of fullerenes (see, e.g., [1,2] and references therein). It is now well established that moderately excited neutral, singly- and doubly-charged fullerenes preferentially decay by emission of a neutral C2 molecule [3,4]. Thus, the knowledge of the corresponding dissociation energy is crucial to interpret the mass spectra that arise from those experiments. In particular, the determination of the dissociation energy of neutral and singly-charged C60 has received considerable attention and has been the subject of a long standing controversy that has only been settled down very recently (see, e.g., the reviews [5,6] and references therein). In recent experiments, the dissociation energy of singly-charged fullerenes, þ Cþ n ! Cn2 þ C2
ð1Þ
have been measured as a function of fullerene size n for n = 40–70 [7–9]. These experiments show that the dissocia*
Corresponding author. E-mail address: [email protected] (F. Martı´n).
0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.09.033
tion energy varies significantly with n. Such a variation is not yet fully understood. In contrast with the large number of theoretical works devoted to determine the binding energies of C60 and Cþ 60 (see, e.g., [10]), there have been much less attempts to determine these quantities in smaller or larger fullerenes. The most complete calculation of dissociation energies for fullerenes other than C60 have been performed by Zhang et al. [11], but only for neutral systems. These authors have determined dissociation energies for fullerenes as small as C20 by using a tight-binding model combined with a scheme to generate energetically favorable structures. Only classical structures made of pentagonal and hexagonal rings were considered. Ionization potentials have been evaluated by Martin et al. [12] for n = 52–60. Recently, we have reported high level theoretical calculations of dissociation energies and ionization potentials in the range n = 50–60 for neutral, singly- and doubly-charged fullerenes [13]. In this work, we have extended the calculations reported in [13] to provide dissociation energies for neutral and singly-charged species as well as ionization potentials for the former for sizes ranging from 40 to 70 atoms. In Section 2, we briefly describe the theoretical methods used in this work. Our results on dissociation energies and ionization
G. Sa´nchez et al. / Chemical Physics Letters 416 (2005) 14–17
potentials are given and discussed in Section 3. Comparison with the available experimental measurements is also shown. We end with some conclusions in Section 4. 2. Computational details We have employed in our calculations the density functional theory (DFT) with the B3LYP functional for exchange and correlation. This functional combines the BeckeÕs three parameter nonlocal hybrid exchange potential with the nonlocal correlation functional of Lee, Yang and Parr. The geometries of all the structures have been optimized by using the 6-31G(d) basis set. The B3LYP functional has been proved to be good a choice for the description of carbon clusters [13]. In the case of small carbon clusters, the calculated geometries and the vibrational frequencies are very close to those obtained at higher levels of theory [14,15]. The calculations have been carried out with the GAUSSIAN98 program [16]. The starting geometry of the classical structures investigated have been obtained with the help of the CAGE program [17]. 3. Results and discussion Classical fullerenes are cages made of twelve pentagons and a variable number of hexagons. For a given size, (i.e., for a given number of hexagons), there are many possible arrangements of pentagons and hexagons [18]. Thus, the selection of the appropriate arrangement is actually the first difficulty one meets in the search for the most stable isomer. More precisely, the relative position of the pentagonal faces determines the ring strain in the cage and, therefore, its stability. In general, the larger the number of adjacent pentagons (AP), the larger the strain. Thus, the most stable fullerenes have all pentagons isolated. This rule, known as the Isolated Pentagon Rule (IPR), was first proposed by Kroto [19]. Only C60 and fullerenes with a number of atoms equal or greater than 70 present isomers that follow the IPR. For other fullerenes, it is impossible to have all pentagons isolated. In this case, the most stable isomer is expected to have the lowest number of APs [20,21]. This second rule is known as the Pentagon Adjacency Penalty Rule (PAPR) or Minimum Pentagon-Adjacency Rule [19,21–23]. In previous works, a few exceptions to the PAPR have been found for both neutral and cationic fullerenes [24– 28]. In particular, we have shown [24] using MP2 theory that C50 does not follow the PAPR: the isomer C50(D3) containing 6 APs is 21.4 kcal/mol more stable than the isomer C50(D5h) containing 5 APs. At the B3LYP/6-31G(d) level of theory used in this work, C50(D3) is more stable by 2.3 kcal/mol, in agreement with [25,26,28]. The situation is the opposite for cationic species: C50(D5h) is 9.5 kcal/mol more stable than C50(D3) and, therefore, the PAPR is fulfilled. In the present work, we have also considered a few nonclassical structures containing heptagons and/or squares
15
that result from extraction or addition of C2 units from the most stable isomers of neighboring size. For C62 we have found that two non-classical structures are more stable than the classical ones. In particular, an isomer of C2v symmetry containing a square ring is 10.4 kcal/mol more stable than the most stable classical isomer. This structure has been proposed in [29] and synthesized in [30]. We have also found that a second non-classical isomer proposed by Ayuela et al. [22] containing a heptagon is even more stable: it has Cs symmetry and its energy is 13.5 kcal/mol lower than that of the most stable classical fullerene. In the case of the cationic species, these isomers are even more stable: 10.2 and 17.7 kcal/mol, respectively. Direct C2 extraction from the non-classical structures containing either the square or the heptagon leads the well-known icosahedral isomer of C60. We have summarized in Table 1 the symmetry, the relevant ring characteristics (apart from hexagons and isolated pentagons) and the number of adjacent pentagons for the most stable isomers found at the B3LYP/6-31G(d) level of theory. These properties are common to both neutral and singly-charged species, except for n = 50, 66 and 68. A more detailed analysis of the structure of the different isomers for a given cluster size will be presented elsewhere. We have used the absolute electronic energies of the most stable isomers to evaluate adiabatic ionization potentials of neutral Cn fullerenes and C2 dissociation energies of neutral and singly-charged Cþ n fullerenes. The results are also given in Table 1. For the calculation of C2 dissociation energies, we have used the absolute energy of C2 obtained at the same level of theory corrected as explained in [13]. Dissociation energies are very similar for neutral and cationic species except in the size interval n = 58–64. Thus, similar qualitative trends are observed concerning their variation with cluster size. Our results for neutral fullerenes agree reasonably well with those of Zhang et al. [11]. Fig. 1 shows the dissociation energy for C2 emission from singly-charged Cþ n (Eq. 1) as a function of cluster size, n. Experimental values [7,9,31,32] are also shown for comparison. The data of Barran et al. [31] were obtained in arbitrary units; they have been renormalized to the calculated value for Cþ 54 . Our results show the same trends as the experimental values, in particular the pronounced þ þ peaks observed in the vicinity of Cþ 50 , C60 and C70 . The maximum dissociation energies correspond to maþ þ gic number fullerenes: Cþ 50 , C60 and C70 . The great stability þ þ of C60 and C70 is due to the absence of adjacent pentagons in the cage, which implies low ring strain. In addition, Cþ 60 is nearly spherical (Ih symmetry), which provides a surþ plus of stability. Fullerenes between Cþ 60 and C70 always þ have APs. C50 is the smallest fullerene in which pentagons can be arranged in pairs of APs (smaller fullerenes always have chains of three or more adjacent pentagons). Furthermore, in a spherical electronic model of fullerenes, its neutral C50 analogue has a closed electronic shell with 2(l + 1)2 = 2(4 + 1)2p electrons, which results in spherical aromaticity and, therefore, in additional stability [33,24].
G. Sa´nchez et al. / Chemical Physics Letters 416 (2005) 14–17
16
Table 1 Symmetry, structure (namely ring structures different from hexagons and isolated pentagons), number of adjacent pentagons (NAP), first ionization potentials (IP) in eV, and C2 dissociation energies in eV for neutral (DE) and singly-charged (DE+) fullerenes for the most stable isomers at the B3LYP/631G(d) level of theory Cn
Symmetry
Structure
NAP
IP (eV)
DE (eV)
DE+ (eV)
C70 C68 C66 C64 C62 C60 C58 C56 C54 C52 C50 C48 C46 C44 C42 C40
D5h C2 (C2) Cs (C2v) D2 Cs Ih Cs D2 C2v C2 D3 (D5h) C2 C2 D2 D3 D2
– 2-2AP (2-2AP) 1-C3AP (2-2AP) 2-2AP 1Hp + 1-C4AP – 3-2AP 4-2AP 2-C3AP 2-C3AP + 1-2AP 6-2AP (5-2AP) 1-C4AP + 2-C3AP 2-C4AP + 2-2AP 2-C4AP + 2-2AP 3-C4AP 2-C6AP
0 2 2 2 3 + Hp 0 3 4 4 5 5 7 8 8 9 10
7.02 6.87 6.86 6.78 6.36 7.14 6.41 6.65 6.61 6.54 7.26 6.85 7.04 7.15 7.13 7.01
10.36 8.36 7.88 8.99 5.53 11.37 8.11 8.47 8.44 7.79 9.63 8.60 7.79 8.61 8.50 –
10.11 8.29 7.97 8.57 6.31 10.64 8.35 8.43 8.37 8.10 9.63 8.79 7.90 8.59 8.38 –
Structures of neutral and singly-charged species are generally the same; when they differ, those of the singly-charged species are given within parenthesis. Notations: 2AP, two adjacent pentagons; C3AP, a chain of three adjacent pentagons; 1Hp, a heptagon; C4AP, a chain of four adjacent pentagons; C6AP, a chain of six adjacent pentagons
Fig. 2 shows our results for the first ionization potential of neutral Cn fullerenes as a function of cluster size. The ionization potential has been obtained by subtracting the absolute energies of the Cn and Cþ n in their optimum geometries. In all cases but three, Cn and Cþ n have the same isomeric form (see Table 1) and, therefore, there is no ambiguity. For C50, C66 and C68 the isomeric form of the neutral species differs from that of the cation. In this case, we have subtracted the energy of the most stable isomer of Cn and that of the same isomeric form of Cþ n . This implies that, apart from bond relaxation, no reorganization of the fullerene cage is allowed during the ionization process. The
12
Dissociation energy (eV)
11
+ Cn →
+ Cn-2 +
C2
10 9 8 7 6 5
44
48
52
56
60
64
68
Cluster size n Fig. 1. Comparison between calculated and experimental C2 dissociation energy as a function of cluster size. Circles: this work. Experimental values: Squares: Barran et al. [31] scaled to Cþ 54 ; triangles up: Laskin et al. [32]; triangles left: Tomita et al. [7]; triangles right: Concina et al. [9].
Ionization potential (eV)
These two reasons make C50 and its corresponding cationic partner singular cases in a wide range of sizes: n = 40–58. The lowest dissociation energies correspond to C62 and Cþ 62 . In this case, the products of dissociation are the icosahedral C60 and Cþ 60 fullerenes, respectively. Therefore, it is not surprising that the energy required is smaller than for the other fullerenes. If one takes the most stable classical isomer of Cþ 62 instead of that containing the heptagonal þ ring, one obtains dissociation energies for Cþ 64 and C62 that are, respectively, 1 eV larger and smaller than those shown in Fig. 1. This would deteriorate the agreement with experiment, which proves that the Cþ 62 fullerene obtained in the experiments is indeed a non-classical structure as that considered in the present work.
8
7
6
40
44
48
52
56
60
64
68
Cluster size n Fig. 2. Comparison between calculated and experimental first ionization potential as a function of cluster size. Circles: this work. Experimental values: Diamonds: Zimmerman et al. [34]; squares: McElvany et al. [35]. Dashed line: result obtained for a C50 fullerene of D5h symmetry (see text).
G. Sa´nchez et al. / Chemical Physics Letters 416 (2005) 14–17
calculated ionization potentials should be considered as adiabatic. Fig. 2 also includes the experimental data of [34,35]. Our results are in good agreement with the experimental data, except for a constant downward shift of 0.5 eV. The maximum values of the curve appear for C50, C60 and C70, which is due again to the large stability of these clusters. Nevertheless, the case of C50 must be analyzed with some caution. As mentioned above, the most stable C50 isomer has D3 symmetry and does not follow the PAPR. Therefore, the theoretical value given in Fig. 2 corresponds to the C50 ðD3 Þ ! Cþ 50 ðD3 Þ þ e ionization process. However, if we consider the ionization energy of the other isomer, i.e. C50 ðD5h Þ ! Cþ 50 ðD5h Þ þ e , one obtains a value of 6.75 eV (dashed line in Fig. 2), which is even farther from the experimental value of Zimmerman et al. [34]. This comparison with experiment indicates that the D3 isomer is indeed the one observed experimentally. 4. Conclusion C2 dissociation energies of neutral and singly-charged Cn fullerenes (n = 42–70) and first ionization potentials of neutral Cn fullerenes (n = 40–70) have been computed by means of DFT-B3LYP/6-31G(d). Good agreement with experiment has been obtained. We have found that magic number fullerenes such as C50, C60 and C70 (and the correþ þ sponding cationic species Cþ 50 , C60 and C70 ) present larger ionization potentials and dissociation energies than their neighbor fullerenes. We have also found that the most stable isomer of C62 and Cþ 62 is a non-classical fullerene with a chain of four adjacent pentagons surrounding a heptagon. This structure, originally proposed by Ayuela et al. [22], leads to dissociation energies in good agreement with experimental ones, whereas the use of a classical structure only made of hexagons and pentagons leads to results that þ significantly differ from experiment for both Cþ 62 and C64 . We have also found that a D3 isomer of C50 that violates the pentagon adjacency penalty rule is the one that gives the best agreement with the measured ionization potentials. Acknowledgements We thank the CCC-UAM and CIEMAT for allocation of computer time. Work partially supported by the DGI Project Nos. BFM2003-00194, BQU2003-00894 and CTQ2004-00039/BQU, and the CAM Project No. GR/ MAT/0083/2004. References [1] E.E.B. Campbell, Fullerene Collision Reactions, first ed., Kluwer Academic Publishers, Dordrecht, 2003.
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