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Structures and stabilities of C60-rings
2 March 2001
Chemical Physics Letters 335 (2001) 524±532
www.elsevier.nl/locate/cplett
Structures and stabilities of C60-rings Yuxue Li, Yuanhe Huang *, Shixuan Du, Ruozhuang Liu Department of Chemistry, Beijing Normal University, Beijing 100875, People's Republic of China Received 1 November 2000; in ®nal form 4 January 2001
Abstract A method to construct various C60 -rings is given. 36 C60 -rings with Dn h and Cn v symmetries have been investigated using self-consistent-®eld molecular orbital method. Their stabilization energies (E) are mainly aected by the distortion of the C60 balls (r), the number of double bonds introduced into pentagons per C60 (j) and the number of formed intermolecular bonds (k). Their electronic properties are discussed and compared with those of single C60 . Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Rings constructed of C60 balls have attracted interest due to their special structures. Theoretical studies on two rings composed of 3 and 4 C60 balls have been carried out [1,2]. Recently, based on good-quality experimental data, these two small closed rings were reported in the phototransformation of C60 at 50°C [3]. It is exciting to think that the experiment indicates the possibility of synthesis of a new kind of important C60 derivative, i.e., cyclic oligomers of C60 . C60 -ring structures can also be found in many 2D polymers that have been reported widely. For instance, 3-membered and 4-membered C60 -rings can be found in rhombohedral and tetragonal phases, respectively [4±6]. There are 4-membered singly bonded C60 -rings in the 2D structure of Na4 C60 [7±9]. Moreover, usually 1D C60 polymers can be treated as n-membered C60 -rings with n ! 1. Besides the C60 polymers, some other
*
Corresponding author. Fax: +86-010-62200567. E-mail address: [email protected] (Y. Huang).
fullerene polymers such as s2D or 3D C36 polymers which have been proposed [10±14], can also be regarded as the extension of the ring-based structures. 2D polymerized C74 is predicted to be stable [15], and to consist of 6-membered rings. These rings may become signi®cant in the further study of fullerene solids. Therefore, a systematic study of the C60 -rings is indicated. So far as we know, no such work has been performed though there are theoretical investigations of the two 3- and 4-membered C60 -rings [1,2]. In this Letter, a method for constructing various C60 -rings is given. The geometric structures and electronic properties of C60 -rings with Dn h and Cn v symmetries are investigated using the self-consistent-®eld molecular orbital method (SCF-MO). In particular, we propose a way to express the relationship between the structure and stability of these C60 -rings. 2. Construction of C60 -rings and computation details Consider a C60 -ring C60 n with Dn h symmetry. For convenience, we build a reference frame in the
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 0 6 4 - 1
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
Fig. 1. Reference frame for C60 -rings with Dn h symmetry.
C60 balls constructing the ring as shown in Fig. 1. The center of the C60 ball O is set as the origin point of the reference frame, the x-axis is points to the center of the ring and the z-axis is vertical to the ring plane. Based on the reference frame in Fig. 1, if the C60 balls rotate around the y-axis, the symmetry element rv remains but C2 disappears and the C60 -rings reduce to Cn v symmetry. Likewise, if the C60 balls rotate around z- or x-axes, Cn h or Dn symmetries will appear. Adjacent C60 balls in the ring will form bonds through carbon atoms situated on their adjacent faces and the connected area will move on the surface of the C60 balls with the change of the ring size so that the possibly bonding area on the C60 balls de®nes a `bondingbelt'. One symmetry group may be related to several bonding-belts, as shown in Fig. 2. The C60 -rings composed of n C60 balls here are all have a principal symmetry axis Cn .
525
In the present Letter, we focus on the structures built from the Dn h -A and Cn v -A bonding-belts for a preliminary study. In fact, the C60 -rings constructed through Dn h -B bonding-belt are also calculated, but only one stable D4h ring is obtained. Only bondingbelts resulting in stable C60 -rings are signi®cant. We describe the C60 -rings with the following notation: hsymmetry Dn h or Cn v i-h linking pattern described by indices of bonding atoms in bonding-beltsi. For instance, the ring D6h -cc0 dd0 is a D6h ring that consists of six C60 balls and the connection between neighbor C60 balls in the ring is through the four bonds of Cc ±Cc , Cc0 ±Cc0 , Cd ±Cd and Cd0 ±Cd0 . The superscript primes can be omitted, since bonds with and without primes are equivalent and the ring now is expressed by D6h -ccdd (see Fig. 5). Lots of linking patterns are possible. The geometrical and electronic structures of C60 -rings studied are calculated by using the semiempirical SCF-MO method at modi®ed neglect of diatomic overlap (MNDO) [16] level with the GA M E S S program package [17]. The rings are very large, the optimization work is hard and time consuming. Therefore, the semiempirical method is suitable. Here, the size of the C60 -rings is considered from 3 to 8. Symmetry constraints are applied in the calculations. 3. Results and discussion 3.1. Stability 3.1.1. C 60 -rings with Dn h symmetry The Dn h C60 -rings are constructed by forming covalent bonds on the Dn h -A bonding-belt
Fig. 2. Some bonding-belts as shown by bold line area on the C60 balls. The Greek alphabets denote atom indices.
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Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
Table 1 linking details, the number of double bonds introduced into pentagons per C60 ball j Energy E (kcal/mol), standard deviation r (A), of the Dn h C60 -rings and bond lengths (A) C60 -rings C60 Ih
E
r 0.000
Linking details
j
Bond lengths Ca ±Ca
Cb ±Cb
Cc ±Cc
Cd ±Cd
Ce ±Ce
±
±
±
±
1.601 1.605 1.577
1.576
1.581 1.657 1.748
1.563 1.531 1.528
±
±
0
± 1.620
D3h -a D3h -bb D3h -abb D3h -abbcc
11.499 21.163 )9.551 1.477
0.0950 0.1018 0.0904 0.0882
Singly (5±1; 3) (5±1; 2; 3) (5±1; 2; 3; 4; 5)
0 2 2 8
D3h -bbcc D4h -bbcc D5h -bbcc
11.317 58.544 160.066
0.0966 0.1739 0.2775
(5±1; 2; 3; 4)
8
D3h -cc D4h -cc D5h -cc D6h -cc
37.004 )11.099 )8.928 6.612
0.1216 0.0391 0.0435 0.0691
5=62 2
4
1.583 1.562 1.564 1.574
D4h -ccdd D5h -ccdd D6h -ccdd D7h -ccdd
82.965 )8.152 )36.116 )39.629
0.2578 0.1731 0.1294 0.1132
(6±1; 2; 3; 4)
0
1.523 1.527 1.536 1.548
1.728 1.653 1.622 1.604
D5h -ccddee D6h -ccddee D7h -ccddee D8h -ccddee
97.317 )2.368 )44.713 )59.639
0.2845 0.1908 0.1346 0.1087
(6±1; 2; 3; 4; 5; 6)
0
1.537 1.540 1.550 1.563
1.565 1.569 1.571 1.572
90.251 60.621 49.199
0.1924 0.1594 0.1461
(6±1; 4)
2
1.635 1.623 1.620
3.545
0.1284
(6±1; 2; 3; 4)
2
1.586
37.488
0.1586
6=62 2
0
D5h -dd D6h -dd D7h -dd D8h -ddee D8h -ee
between neighbor C60 balls. 24 stable C60 -rings have been obtained by full optimization from these starting structures, with symmetry constrains. The energy per C60 ball related to that of single C60 (EC60 ) with Ih symmetry is given by E E C60 n =n
EC60 ;
1
where E C60 n is the energy of the C60 -rings. The calculated results are listed in Table 1. In order to describe the linking patterns more clearly, the linking details are also given in Table 1. They represent the relative position of the bonding atoms in the 6- or 5-membered carbon ring on the C60 ball. For example, link (6±1; 4) indicates that two para-carbon atoms on the 6-membered carbon ring form the intermolecular bonds. Some
1.556 1.582
1.732 1.643 1.604 1.581
1.575 1.613
traditional notations are also used, such as 5=6 2 2. Table 1 shows that the bond lengths between neighbor C60 balls are in the range of 1.52±1.75 A. Since there is at least one bond with a bond length among the intermolecular bonds of 1.52±1.64 A for whichever of the rings, the rings can be considered as covalent C60 oligomers. Formation of covalent bonds between neighbor C60 balls will cause distortion of the C60 balls, which results in change of the strain energy. It was estimated that about 80% heat of formation of C60 is made up of strain energy [18]. We thus may suppose that strain energy in C60 -rings still plays an important role in thermodynamic stability. Generally speaking, the distortion of C60 balls in
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
the C60 -ring will lead to an increase of strain energy. For a certain linking pattern, it is reasonable to consider that the intermolecular bonds will determine the overall magnitude of the bonding energy for dierent ring sizes. Then, the strain energy due to distortion becomes a key factor in determining the stable ring size since dierent ring sizes are associated with dierent degrees of distortion. In the following, we estimate the change of strain energy for the C60 -rings studied in the present Letter. We chose the Ih C60 ball optimized at the MNDO level as a benchmark for the distortion of
Fig. 3. E r and E
8:5j
527
C60 balls in the C60 -rings. See Fig. 1, where an is the tangent plane of the C60 ball perpendicular to line OO0 ; Pn is the tangent point and T is the position of a possible bonding atom on the bonding-belt. TQ is the vertical distance from the possible bonding atom to an and concerned with the relative distance of possible bonding atom pair, such as T ; H . If there is an atom at Pn , it is more ready to bond than others. This situation causes the smallest distortion after bonding. At this point, TQ 0. It is reasonable to think that the longer TQ is, the bigger is the distortion introduced on
kEb r distribution for the C60 -rings, numbers (1)±(8) denote the ring size (n). (a) Dn h , (b) Cn v .
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Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
bonding, because the possible bonding atom pairs must become close enough to form the intermolecular bonds. We propose a parameter r, the standard deviation of TQ, to estimate the degree of distortion s P 2 TQ TQ r ; 2 m where m is the number of TQ including reference point Pn TQ 0 at Pn for a certain linking pattern, such as m 5 for `bbcc'. The calculation of r is dependent on the structure of a single undistorted C60 ball, the linking pattern and the ring are also listed in Table 1. size. The values of r (A) According to the data in Table 1, it is found that the energies of C60 -rings with a given linking pattern are nearly linear functions of their r values as shown in Fig. 3a-(1). The average slope is about 785 kcal/ but the intercepts are dierent for dierent mol/A, linking patterns. The smaller the r is, the lower the energy. This indicates that the distortion has a great in¯uence upon the stability. Therefore, the C60 -ring with the smallest r is predicted to be the most stable for a certain linking pattern. This is useful as r can be obtained through simple calculation.
For the comparison of relative stability for all rings with dierent sizes and linking patterns, in addition to the distortion of C60 balls, two factors must be considered: (a) the number of covalent bonds formed; (b) the number of double bonds introduced into pentagons on each C60 ball. We can see the trend in Fig. 3a-(1) that lines for the rings with more intermolecular bonds have small intercepts, indicating that more bonds cause the C60 -ring to be more stable. On the other hand, several previous calculations indicate that the heat of formation increases with the introduction of double bonds into pentagons on C60 balls for C60 H2 , C60 2 O and C60 2 [19,20]. The increment is about 8.5 kcal/mol per double bond. In the C60 rings, this rule still applies. As examples, Fig. 4 shows the changes of the bonds on the C60 balls for D4h -cc and D7h -ccdd rings from a view of valance bonding theory. It is found that the energies de®ned by Eq. (1) for C60 -rings can be expressed approximately as (in kcal/mol) E ar 8:5j
kEb b;
3
where j is the number of double bonds introduced into pentagons per C60 ball, k is the number of
Fig. 4. Changes of bonds on the C60 balls for C60 -rings D4h -cc and D7h -ccdd. The arrows indicate the shift direction of one half bond when the reaction takes place. A double bond forms when two half-bonds shift to the same single bond. The short straight line are also given. The double connected to the C60 ball with one end point is a half intermolecular bond. Calculated bond lengths (A) bonds accompanied by are those introduced into pentagons.
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
covalent bonds formed between two C60 balls, Eb can be considered as the average energy change for forming an intermolecular bond, a corresponds to the change of stain energies per unit of r, and b is from other contributions. Both a and b are constants for the rings constructed from the same bonding-belt. j and k are constants for rings with the same linking pattern. The increase of energies is thus proportional to r for the rings with the same linking pattern (Fig. 3a-(1)). Eq. (3) can also be written as E
8:5j
kEb ar b:
529
vary approximately linearly with r regardless of the linking patterns and ring sizes. The ®tted parameters are Eb 17:0 kcal/mol, a 784:7 kcal and b 52:3 kcal/mol. The most stable =mol/A C60 -rings obtained for dierent sizes are shown in Fig. 5. 3.1.2. C 60 -rings with C n v symmetry For the Cn v rings, the above discussion is also applicable. To take advantage of this method and to test its reliability, only two or three C60 -rings with smallest r are fully optimized by the same MNDO SCF-MO method for each linking pattern. As with Dn h structures, for a certain linking pattern, smaller r is associated with lower energy, as shown in Table 2 and Fig. 3b-(1). The sign `)' in
4
The distribution of E 8:5j kEb r is given in Fig. 3a-(2). All the data concentrate near a straight line, i.e., the values of E 8:5j kEb
Fig. 5. Structures of the most stable Dn h C60 -rings for each size (n 3±8). Table 2 Linking patterns, linking details, the number of double bonds introduced into pentagons per C60 ball j, the size of the ring n, standard and energy E (kcal/mol) for Cn v C60 -rings deviation r (A) Linking patterns
Linking details
j
n
a
Singly
5
r E
0.0493 31.741
0.2536 100.567
0.4522
0.6134
0.7416
0.8444
ab
6=62 2
0
r E
0.0398 )46.881
0.2018 7.301
0.3426
0.4521
0.5386
0.6081
bc
5=62 2
4
r E
0.2860
0.0946 )11.861
0.0542 )13.060
0.1110 4.308
0.1649
0.2098
abc
(6±1,2,3)
3
r E
0.2422 15.652
0.1664 )7.883
0.2935
0.4001
0.4815
0.5449
bcd
(5±1,2,3)
7
r E
0.5802
0.2963
0.1547 37.146
0.1099 21.404
0.1347 30.023
0.1753
3
4
5
6
)
7
8
530
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
Table 3 Energy gaps, HOMO and LUMO levels and symmetries of Dn h and Cn v C60 -rings (in eV) C60 rings
Eg
HOMO
C60
6.569
)9.132
(Hu )
)2.563
(T1u ) (A01 ) (A02 ) (A02 ) E00
LUMO
D3h -a D3h -bb D3h -abb D3h -abbcc
6.251 6.272 6.231 5.293
)8.909 )8.969 )8.925 )8.389
(A02 ) (A002 ) (A002 ) 0 E
)2.659 )2.697 )2.694 )3.097
D3h -bbcc D4h -bbcc D5h -bbcc
5.380 5.306 5.391
)8.444 )8.392 )8.411
E0 (B2g ) (E01 )
)3.064 )3.086 )3.020
E00 (B2u ) (A002 )
D3h -cc D4h -cc D5h -cc D6h -cc
5.668 5.627 5.665 5.646
)8.574 )8.555 )8.563 )8.553
(A001 ) (B1u ) (E002 ) (A1u )
)2.906 )2.928 )2.898 )2.906
E0 (B1g ) (E02 ) (A1g )
D4h -ccdd D5h -ccdd D6h -ccdd D7h -ccdd
6.191 6.324 6.335 6.370
)8.803 )8.901 )8.895 )8.915
(A1u ) (E002 ) (B1g ) (E2 )
)2.612 )2.577 )2.561 )2.544
(B2g ) (A02 ) (A2g ) (A2 )
D5h -ccddee D6h -ccddee D7h -ccddee D8h -ccddee
6.174 6.191 6.240 6.270
)8.795 )8.795 )8.830 )8.849
(E002 ) (B1g ) (E2 ) (A2u )
)2.620 )2.604 )2.591 )2.580
(A02 ) (A2g ) (A2 ) (B1g )
D5h -dd D6h -dd D7h -dd
5.807 5.984 6.098
)8.702 )8.759 )8.814
(A2 ) (A2g ) (A2 )
)2.895 )2.776 )2.716
(A002 ) (A2u ) (A1 )
D8h -ddee
6.286
)8.876
(B1g )
)2.591
(B1u )
D8h -ee
6.087
)8.754
(B2u )
)2.667
(B2g )
C3v -a C4v -a
5.097 4.447
)8.267 )7.900
E (B1 )
)3.170 )3.453
E (B2 )
C3v -ab C4v -ab
6.479 6.261
)9.029 )8.879
E (A2 )
)2.550 )2.618
E (A1 )
C4v -bc C5v -bc C6v -bc
5.937 5.946 5.970
)8.721 )8.746 )8.776
(A1 ) (A1 ) (A1 )
)2.784 )2.800 )2.806
(B1 ) (E2 ) (B1 )
C3v -abc C4v -abc
5.938 5.992
)8.724 )8.697
E (A1 )
)2.786 )2.705
E (A2 )
C5v -bcd C6v -bcd C7v -bcd
4.950 4.896 4.903
)8.218 )8.202 )8.212
(E2 ) (B2 ) (E2 )
)3.268 )3.306 )3.309
(E2 ) (B1 ) (E2 )
Table 2 denotes that no converged result was obtained in the molecular orbital calculations, a situation which occurs when r > 0:29. As with Dn h rings, the values of E 8:5j kEb (Eb 17:0 kcal/mol) for the Cn v
rings increase with r and are distributed about a straight line (see Fig. 3b-(2)). The constants a and and )27.4 kcal/mol, reb are 334.5 kcal/mol/A spectively. C3v -ab has been found by experimental work [3]. It has the smallest r and no double bond
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
is introduced into the pentagons on C60 balls, hence it is predicted to be the most stable of the Cn v rings in Table 2. If this C60 -ring is extended to a 2D structure, it will form a rhombohedral phase [4±6]. Since a Cn v < a Dn h , the in¯uence of the distortion of the C60 balls is greater in Dn h than in Cn v structures for the C60 -rings based on the de®nition of Eq. (2). We would like to point out that MNDO calculation overestimated the stabilities of fullerene dimers compared with more realistic ab initio DFT method [21,22], hence it may be worth doing a similar comparison in the future for the rings. 3.2. Electronic structures The calculated energy levels and symmetries of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), as well as the energy gaps (Eg ) between them, are listed in Table 3 for all the C60 -rings studied. All the HOMO levels and most of the LUMO levels of the C60 -rings are higher and lower respectively than that of the single C60 ball. Compared with single C60 , therefore, these C60 rings will both lose and accept electrons more easily. All the C60 -rings discussed here have smaller energy gaps than that of single C60 ball. Smaller energy gaps may lead to lower chemical stability, and thus they may more reactive than the single C60 ball. Moreover, large j results in small energy gap for the C60 -rings. Thus, the double bonds introduced into the pentagons possibly decrease both the thermodynamic and kinetic stability of the C60 -rings. From Table 3, it can also be seen that doubly degenerate HOMOs or LUMOs occur only for the C60 -rings with odd size (n 3; 5; 7) and Dn h or Cn v geometries. It should be pointed out that both singly and doubly bonded 1D C60 polymers have only C2 rotational symmetry and hence nondegenerate frontier bands [23,24]. 4. Conclusions The concept of the bonding-belt is proposed for building C60 -rings with various linking patterns
531
and symmetries. Two kinds of C60 -rings constructed through Dn h and Cn v bonding-belts are investigated using the semiempirical MNDO molecular orbital method. It is found that the stability of the C60 -rings is mainly dependent on the three factors: the change of strain energies due to the distortion of C60 balls, the number of double bonds introduced into the pentagons on C60 balls and the number of covalent bonds formed between neighbor C60 molecules. Distortion of the C60 balls always has a considerable eect on the stability of the C60 -rings. A strain-associated factor r can be used to predict the energy changes of the C60 -rings due to distortion. Since the LUMOs and HOMOs of all but two rings are bracketed by those of the single C60 molecule, the C60 -rings should have correspondingly greater tendencies to lose or accept electrons. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 29873007). References [1] D. Porezag, G. Jungnickel, Th. Frauenheim, G. Seifert, A. Ayuela, M.R. Pederson, Appl. Phys. A 64 (1997) 321. [2] D. Porezag, Th. Frauenheim, Carbon 37 (1999) 463. [3] T. Pusztai, G. Oszlanyi, G. Faigel, K. Kamaras, L. Granasy, S. Pekker, Solid State Commun. 111 (1999) 595. [4] F. Rachdi, C. Goze, L. Hajji, K.F. Thier, G. Zimmer, M. ~ ez-Regueiro, Appl. Phys. A 64 (1997) 295. Mehring, M. N un [5] A.M. Rao, P.C. Eklund, U.D. Venkateswaran, J. Tucker, M.A. Duncan, G.M. Bendele, P.W. Stephens, J.L. Hodeau, ~ ez-Regueiro, I.O. Bashkin, E.G. L. Marques, M. N un Ponyatovsky, A.P. Morovsky, Appl. Phys. A 64 (1997) 231. [6] Y. Iwasa, T. Furudate, T. Fukawa, T. Ozaki, T. Mitani, T. Yagi, T. Arima, Appl. Phys. A 64 (1997) 251. [7] G. Oszlanyi, G. Faigel, G. Baumgartner, L. Forr o, in: Molecular Nanostructures, World Scienti®c, Singapore, 1998, pp. 343. [8] S. Pekker, G. Oszlanyi, G. Faigel, in: Molecular Nanostructures, World Scienti®c, Singapore, 1998, p. 306. [9] M. Hjort, S. Stafstr om, Europhys. Lett. 46 (1999) 382. [10] P.G. Collins, J.C. Grossman, M. C^ ote, M. Ishigami, C. Piskoti, S.G. Louie, M.L. Cohen, A. Zettl, Phys. Rev. Lett. 82 (1999) 165.
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Chemical Physics Letters 335 (2001) 524±532
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Structures and stabilities of C60-rings Yuxue Li, Yuanhe Huang *, Shixuan Du, Ruozhuang Liu Department of Chemistry, Beijing Normal University, Beijing 100875, People's Republic of China Received 1 November 2000; in ®nal form 4 January 2001
Abstract A method to construct various C60 -rings is given. 36 C60 -rings with Dn h and Cn v symmetries have been investigated using self-consistent-®eld molecular orbital method. Their stabilization energies (E) are mainly aected by the distortion of the C60 balls (r), the number of double bonds introduced into pentagons per C60 (j) and the number of formed intermolecular bonds (k). Their electronic properties are discussed and compared with those of single C60 . Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Rings constructed of C60 balls have attracted interest due to their special structures. Theoretical studies on two rings composed of 3 and 4 C60 balls have been carried out [1,2]. Recently, based on good-quality experimental data, these two small closed rings were reported in the phototransformation of C60 at 50°C [3]. It is exciting to think that the experiment indicates the possibility of synthesis of a new kind of important C60 derivative, i.e., cyclic oligomers of C60 . C60 -ring structures can also be found in many 2D polymers that have been reported widely. For instance, 3-membered and 4-membered C60 -rings can be found in rhombohedral and tetragonal phases, respectively [4±6]. There are 4-membered singly bonded C60 -rings in the 2D structure of Na4 C60 [7±9]. Moreover, usually 1D C60 polymers can be treated as n-membered C60 -rings with n ! 1. Besides the C60 polymers, some other
*
Corresponding author. Fax: +86-010-62200567. E-mail address: [email protected] (Y. Huang).
fullerene polymers such as s2D or 3D C36 polymers which have been proposed [10±14], can also be regarded as the extension of the ring-based structures. 2D polymerized C74 is predicted to be stable [15], and to consist of 6-membered rings. These rings may become signi®cant in the further study of fullerene solids. Therefore, a systematic study of the C60 -rings is indicated. So far as we know, no such work has been performed though there are theoretical investigations of the two 3- and 4-membered C60 -rings [1,2]. In this Letter, a method for constructing various C60 -rings is given. The geometric structures and electronic properties of C60 -rings with Dn h and Cn v symmetries are investigated using the self-consistent-®eld molecular orbital method (SCF-MO). In particular, we propose a way to express the relationship between the structure and stability of these C60 -rings. 2. Construction of C60 -rings and computation details Consider a C60 -ring C60 n with Dn h symmetry. For convenience, we build a reference frame in the
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 0 6 4 - 1
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
Fig. 1. Reference frame for C60 -rings with Dn h symmetry.
C60 balls constructing the ring as shown in Fig. 1. The center of the C60 ball O is set as the origin point of the reference frame, the x-axis is points to the center of the ring and the z-axis is vertical to the ring plane. Based on the reference frame in Fig. 1, if the C60 balls rotate around the y-axis, the symmetry element rv remains but C2 disappears and the C60 -rings reduce to Cn v symmetry. Likewise, if the C60 balls rotate around z- or x-axes, Cn h or Dn symmetries will appear. Adjacent C60 balls in the ring will form bonds through carbon atoms situated on their adjacent faces and the connected area will move on the surface of the C60 balls with the change of the ring size so that the possibly bonding area on the C60 balls de®nes a `bondingbelt'. One symmetry group may be related to several bonding-belts, as shown in Fig. 2. The C60 -rings composed of n C60 balls here are all have a principal symmetry axis Cn .
525
In the present Letter, we focus on the structures built from the Dn h -A and Cn v -A bonding-belts for a preliminary study. In fact, the C60 -rings constructed through Dn h -B bonding-belt are also calculated, but only one stable D4h ring is obtained. Only bondingbelts resulting in stable C60 -rings are signi®cant. We describe the C60 -rings with the following notation: hsymmetry Dn h or Cn v i-h linking pattern described by indices of bonding atoms in bonding-beltsi. For instance, the ring D6h -cc0 dd0 is a D6h ring that consists of six C60 balls and the connection between neighbor C60 balls in the ring is through the four bonds of Cc ±Cc , Cc0 ±Cc0 , Cd ±Cd and Cd0 ±Cd0 . The superscript primes can be omitted, since bonds with and without primes are equivalent and the ring now is expressed by D6h -ccdd (see Fig. 5). Lots of linking patterns are possible. The geometrical and electronic structures of C60 -rings studied are calculated by using the semiempirical SCF-MO method at modi®ed neglect of diatomic overlap (MNDO) [16] level with the GA M E S S program package [17]. The rings are very large, the optimization work is hard and time consuming. Therefore, the semiempirical method is suitable. Here, the size of the C60 -rings is considered from 3 to 8. Symmetry constraints are applied in the calculations. 3. Results and discussion 3.1. Stability 3.1.1. C 60 -rings with Dn h symmetry The Dn h C60 -rings are constructed by forming covalent bonds on the Dn h -A bonding-belt
Fig. 2. Some bonding-belts as shown by bold line area on the C60 balls. The Greek alphabets denote atom indices.
526
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
Table 1 linking details, the number of double bonds introduced into pentagons per C60 ball j Energy E (kcal/mol), standard deviation r (A), of the Dn h C60 -rings and bond lengths (A) C60 -rings C60 Ih
E
r 0.000
Linking details
j
Bond lengths Ca ±Ca
Cb ±Cb
Cc ±Cc
Cd ±Cd
Ce ±Ce
±
±
±
±
1.601 1.605 1.577
1.576
1.581 1.657 1.748
1.563 1.531 1.528
±
±
0
± 1.620
D3h -a D3h -bb D3h -abb D3h -abbcc
11.499 21.163 )9.551 1.477
0.0950 0.1018 0.0904 0.0882
Singly (5±1; 3) (5±1; 2; 3) (5±1; 2; 3; 4; 5)
0 2 2 8
D3h -bbcc D4h -bbcc D5h -bbcc
11.317 58.544 160.066
0.0966 0.1739 0.2775
(5±1; 2; 3; 4)
8
D3h -cc D4h -cc D5h -cc D6h -cc
37.004 )11.099 )8.928 6.612
0.1216 0.0391 0.0435 0.0691
5=62 2
4
1.583 1.562 1.564 1.574
D4h -ccdd D5h -ccdd D6h -ccdd D7h -ccdd
82.965 )8.152 )36.116 )39.629
0.2578 0.1731 0.1294 0.1132
(6±1; 2; 3; 4)
0
1.523 1.527 1.536 1.548
1.728 1.653 1.622 1.604
D5h -ccddee D6h -ccddee D7h -ccddee D8h -ccddee
97.317 )2.368 )44.713 )59.639
0.2845 0.1908 0.1346 0.1087
(6±1; 2; 3; 4; 5; 6)
0
1.537 1.540 1.550 1.563
1.565 1.569 1.571 1.572
90.251 60.621 49.199
0.1924 0.1594 0.1461
(6±1; 4)
2
1.635 1.623 1.620
3.545
0.1284
(6±1; 2; 3; 4)
2
1.586
37.488
0.1586
6=62 2
0
D5h -dd D6h -dd D7h -dd D8h -ddee D8h -ee
between neighbor C60 balls. 24 stable C60 -rings have been obtained by full optimization from these starting structures, with symmetry constrains. The energy per C60 ball related to that of single C60 (EC60 ) with Ih symmetry is given by E E C60 n =n
EC60 ;
1
where E C60 n is the energy of the C60 -rings. The calculated results are listed in Table 1. In order to describe the linking patterns more clearly, the linking details are also given in Table 1. They represent the relative position of the bonding atoms in the 6- or 5-membered carbon ring on the C60 ball. For example, link (6±1; 4) indicates that two para-carbon atoms on the 6-membered carbon ring form the intermolecular bonds. Some
1.556 1.582
1.732 1.643 1.604 1.581
1.575 1.613
traditional notations are also used, such as 5=6 2 2. Table 1 shows that the bond lengths between neighbor C60 balls are in the range of 1.52±1.75 A. Since there is at least one bond with a bond length among the intermolecular bonds of 1.52±1.64 A for whichever of the rings, the rings can be considered as covalent C60 oligomers. Formation of covalent bonds between neighbor C60 balls will cause distortion of the C60 balls, which results in change of the strain energy. It was estimated that about 80% heat of formation of C60 is made up of strain energy [18]. We thus may suppose that strain energy in C60 -rings still plays an important role in thermodynamic stability. Generally speaking, the distortion of C60 balls in
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
the C60 -ring will lead to an increase of strain energy. For a certain linking pattern, it is reasonable to consider that the intermolecular bonds will determine the overall magnitude of the bonding energy for dierent ring sizes. Then, the strain energy due to distortion becomes a key factor in determining the stable ring size since dierent ring sizes are associated with dierent degrees of distortion. In the following, we estimate the change of strain energy for the C60 -rings studied in the present Letter. We chose the Ih C60 ball optimized at the MNDO level as a benchmark for the distortion of
Fig. 3. E r and E
8:5j
527
C60 balls in the C60 -rings. See Fig. 1, where an is the tangent plane of the C60 ball perpendicular to line OO0 ; Pn is the tangent point and T is the position of a possible bonding atom on the bonding-belt. TQ is the vertical distance from the possible bonding atom to an and concerned with the relative distance of possible bonding atom pair, such as T ; H . If there is an atom at Pn , it is more ready to bond than others. This situation causes the smallest distortion after bonding. At this point, TQ 0. It is reasonable to think that the longer TQ is, the bigger is the distortion introduced on
kEb r distribution for the C60 -rings, numbers (1)±(8) denote the ring size (n). (a) Dn h , (b) Cn v .
528
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
bonding, because the possible bonding atom pairs must become close enough to form the intermolecular bonds. We propose a parameter r, the standard deviation of TQ, to estimate the degree of distortion s P 2 TQ TQ r ; 2 m where m is the number of TQ including reference point Pn TQ 0 at Pn for a certain linking pattern, such as m 5 for `bbcc'. The calculation of r is dependent on the structure of a single undistorted C60 ball, the linking pattern and the ring are also listed in Table 1. size. The values of r (A) According to the data in Table 1, it is found that the energies of C60 -rings with a given linking pattern are nearly linear functions of their r values as shown in Fig. 3a-(1). The average slope is about 785 kcal/ but the intercepts are dierent for dierent mol/A, linking patterns. The smaller the r is, the lower the energy. This indicates that the distortion has a great in¯uence upon the stability. Therefore, the C60 -ring with the smallest r is predicted to be the most stable for a certain linking pattern. This is useful as r can be obtained through simple calculation.
For the comparison of relative stability for all rings with dierent sizes and linking patterns, in addition to the distortion of C60 balls, two factors must be considered: (a) the number of covalent bonds formed; (b) the number of double bonds introduced into pentagons on each C60 ball. We can see the trend in Fig. 3a-(1) that lines for the rings with more intermolecular bonds have small intercepts, indicating that more bonds cause the C60 -ring to be more stable. On the other hand, several previous calculations indicate that the heat of formation increases with the introduction of double bonds into pentagons on C60 balls for C60 H2 , C60 2 O and C60 2 [19,20]. The increment is about 8.5 kcal/mol per double bond. In the C60 rings, this rule still applies. As examples, Fig. 4 shows the changes of the bonds on the C60 balls for D4h -cc and D7h -ccdd rings from a view of valance bonding theory. It is found that the energies de®ned by Eq. (1) for C60 -rings can be expressed approximately as (in kcal/mol) E ar 8:5j
kEb b;
3
where j is the number of double bonds introduced into pentagons per C60 ball, k is the number of
Fig. 4. Changes of bonds on the C60 balls for C60 -rings D4h -cc and D7h -ccdd. The arrows indicate the shift direction of one half bond when the reaction takes place. A double bond forms when two half-bonds shift to the same single bond. The short straight line are also given. The double connected to the C60 ball with one end point is a half intermolecular bond. Calculated bond lengths (A) bonds accompanied by are those introduced into pentagons.
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
covalent bonds formed between two C60 balls, Eb can be considered as the average energy change for forming an intermolecular bond, a corresponds to the change of stain energies per unit of r, and b is from other contributions. Both a and b are constants for the rings constructed from the same bonding-belt. j and k are constants for rings with the same linking pattern. The increase of energies is thus proportional to r for the rings with the same linking pattern (Fig. 3a-(1)). Eq. (3) can also be written as E
8:5j
kEb ar b:
529
vary approximately linearly with r regardless of the linking patterns and ring sizes. The ®tted parameters are Eb 17:0 kcal/mol, a 784:7 kcal and b 52:3 kcal/mol. The most stable =mol/A C60 -rings obtained for dierent sizes are shown in Fig. 5. 3.1.2. C 60 -rings with C n v symmetry For the Cn v rings, the above discussion is also applicable. To take advantage of this method and to test its reliability, only two or three C60 -rings with smallest r are fully optimized by the same MNDO SCF-MO method for each linking pattern. As with Dn h structures, for a certain linking pattern, smaller r is associated with lower energy, as shown in Table 2 and Fig. 3b-(1). The sign `)' in
4
The distribution of E 8:5j kEb r is given in Fig. 3a-(2). All the data concentrate near a straight line, i.e., the values of E 8:5j kEb
Fig. 5. Structures of the most stable Dn h C60 -rings for each size (n 3±8). Table 2 Linking patterns, linking details, the number of double bonds introduced into pentagons per C60 ball j, the size of the ring n, standard and energy E (kcal/mol) for Cn v C60 -rings deviation r (A) Linking patterns
Linking details
j
n
a
Singly
5
r E
0.0493 31.741
0.2536 100.567
0.4522
0.6134
0.7416
0.8444
ab
6=62 2
0
r E
0.0398 )46.881
0.2018 7.301
0.3426
0.4521
0.5386
0.6081
bc
5=62 2
4
r E
0.2860
0.0946 )11.861
0.0542 )13.060
0.1110 4.308
0.1649
0.2098
abc
(6±1,2,3)
3
r E
0.2422 15.652
0.1664 )7.883
0.2935
0.4001
0.4815
0.5449
bcd
(5±1,2,3)
7
r E
0.5802
0.2963
0.1547 37.146
0.1099 21.404
0.1347 30.023
0.1753
3
4
5
6
)
7
8
530
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
Table 3 Energy gaps, HOMO and LUMO levels and symmetries of Dn h and Cn v C60 -rings (in eV) C60 rings
Eg
HOMO
C60
6.569
)9.132
(Hu )
)2.563
(T1u ) (A01 ) (A02 ) (A02 ) E00
LUMO
D3h -a D3h -bb D3h -abb D3h -abbcc
6.251 6.272 6.231 5.293
)8.909 )8.969 )8.925 )8.389
(A02 ) (A002 ) (A002 ) 0 E
)2.659 )2.697 )2.694 )3.097
D3h -bbcc D4h -bbcc D5h -bbcc
5.380 5.306 5.391
)8.444 )8.392 )8.411
E0 (B2g ) (E01 )
)3.064 )3.086 )3.020
E00 (B2u ) (A002 )
D3h -cc D4h -cc D5h -cc D6h -cc
5.668 5.627 5.665 5.646
)8.574 )8.555 )8.563 )8.553
(A001 ) (B1u ) (E002 ) (A1u )
)2.906 )2.928 )2.898 )2.906
E0 (B1g ) (E02 ) (A1g )
D4h -ccdd D5h -ccdd D6h -ccdd D7h -ccdd
6.191 6.324 6.335 6.370
)8.803 )8.901 )8.895 )8.915
(A1u ) (E002 ) (B1g ) (E2 )
)2.612 )2.577 )2.561 )2.544
(B2g ) (A02 ) (A2g ) (A2 )
D5h -ccddee D6h -ccddee D7h -ccddee D8h -ccddee
6.174 6.191 6.240 6.270
)8.795 )8.795 )8.830 )8.849
(E002 ) (B1g ) (E2 ) (A2u )
)2.620 )2.604 )2.591 )2.580
(A02 ) (A2g ) (A2 ) (B1g )
D5h -dd D6h -dd D7h -dd
5.807 5.984 6.098
)8.702 )8.759 )8.814
(A2 ) (A2g ) (A2 )
)2.895 )2.776 )2.716
(A002 ) (A2u ) (A1 )
D8h -ddee
6.286
)8.876
(B1g )
)2.591
(B1u )
D8h -ee
6.087
)8.754
(B2u )
)2.667
(B2g )
C3v -a C4v -a
5.097 4.447
)8.267 )7.900
E (B1 )
)3.170 )3.453
E (B2 )
C3v -ab C4v -ab
6.479 6.261
)9.029 )8.879
E (A2 )
)2.550 )2.618
E (A1 )
C4v -bc C5v -bc C6v -bc
5.937 5.946 5.970
)8.721 )8.746 )8.776
(A1 ) (A1 ) (A1 )
)2.784 )2.800 )2.806
(B1 ) (E2 ) (B1 )
C3v -abc C4v -abc
5.938 5.992
)8.724 )8.697
E (A1 )
)2.786 )2.705
E (A2 )
C5v -bcd C6v -bcd C7v -bcd
4.950 4.896 4.903
)8.218 )8.202 )8.212
(E2 ) (B2 ) (E2 )
)3.268 )3.306 )3.309
(E2 ) (B1 ) (E2 )
Table 2 denotes that no converged result was obtained in the molecular orbital calculations, a situation which occurs when r > 0:29. As with Dn h rings, the values of E 8:5j kEb (Eb 17:0 kcal/mol) for the Cn v
rings increase with r and are distributed about a straight line (see Fig. 3b-(2)). The constants a and and )27.4 kcal/mol, reb are 334.5 kcal/mol/A spectively. C3v -ab has been found by experimental work [3]. It has the smallest r and no double bond
Y. Li et al. / Chemical Physics Letters 335 (2001) 524±532
is introduced into the pentagons on C60 balls, hence it is predicted to be the most stable of the Cn v rings in Table 2. If this C60 -ring is extended to a 2D structure, it will form a rhombohedral phase [4±6]. Since a Cn v < a Dn h , the in¯uence of the distortion of the C60 balls is greater in Dn h than in Cn v structures for the C60 -rings based on the de®nition of Eq. (2). We would like to point out that MNDO calculation overestimated the stabilities of fullerene dimers compared with more realistic ab initio DFT method [21,22], hence it may be worth doing a similar comparison in the future for the rings. 3.2. Electronic structures The calculated energy levels and symmetries of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), as well as the energy gaps (Eg ) between them, are listed in Table 3 for all the C60 -rings studied. All the HOMO levels and most of the LUMO levels of the C60 -rings are higher and lower respectively than that of the single C60 ball. Compared with single C60 , therefore, these C60 rings will both lose and accept electrons more easily. All the C60 -rings discussed here have smaller energy gaps than that of single C60 ball. Smaller energy gaps may lead to lower chemical stability, and thus they may more reactive than the single C60 ball. Moreover, large j results in small energy gap for the C60 -rings. Thus, the double bonds introduced into the pentagons possibly decrease both the thermodynamic and kinetic stability of the C60 -rings. From Table 3, it can also be seen that doubly degenerate HOMOs or LUMOs occur only for the C60 -rings with odd size (n 3; 5; 7) and Dn h or Cn v geometries. It should be pointed out that both singly and doubly bonded 1D C60 polymers have only C2 rotational symmetry and hence nondegenerate frontier bands [23,24]. 4. Conclusions The concept of the bonding-belt is proposed for building C60 -rings with various linking patterns
531
and symmetries. Two kinds of C60 -rings constructed through Dn h and Cn v bonding-belts are investigated using the semiempirical MNDO molecular orbital method. It is found that the stability of the C60 -rings is mainly dependent on the three factors: the change of strain energies due to the distortion of C60 balls, the number of double bonds introduced into the pentagons on C60 balls and the number of covalent bonds formed between neighbor C60 molecules. Distortion of the C60 balls always has a considerable eect on the stability of the C60 -rings. A strain-associated factor r can be used to predict the energy changes of the C60 -rings due to distortion. Since the LUMOs and HOMOs of all but two rings are bracketed by those of the single C60 molecule, the C60 -rings should have correspondingly greater tendencies to lose or accept electrons. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 29873007). References [1] D. Porezag, G. Jungnickel, Th. Frauenheim, G. Seifert, A. Ayuela, M.R. Pederson, Appl. Phys. A 64 (1997) 321. [2] D. Porezag, Th. Frauenheim, Carbon 37 (1999) 463. [3] T. Pusztai, G. Oszlanyi, G. Faigel, K. Kamaras, L. Granasy, S. Pekker, Solid State Commun. 111 (1999) 595. [4] F. Rachdi, C. Goze, L. Hajji, K.F. Thier, G. Zimmer, M. ~ ez-Regueiro, Appl. Phys. A 64 (1997) 295. Mehring, M. N un [5] A.M. Rao, P.C. Eklund, U.D. Venkateswaran, J. Tucker, M.A. Duncan, G.M. Bendele, P.W. Stephens, J.L. Hodeau, ~ ez-Regueiro, I.O. Bashkin, E.G. L. Marques, M. N un Ponyatovsky, A.P. Morovsky, Appl. Phys. A 64 (1997) 231. [6] Y. Iwasa, T. Furudate, T. Fukawa, T. Ozaki, T. Mitani, T. Yagi, T. Arima, Appl. Phys. A 64 (1997) 251. [7] G. Oszlanyi, G. Faigel, G. Baumgartner, L. Forr o, in: Molecular Nanostructures, World Scienti®c, Singapore, 1998, pp. 343. [8] S. Pekker, G. Oszlanyi, G. Faigel, in: Molecular Nanostructures, World Scienti®c, Singapore, 1998, p. 306. [9] M. Hjort, S. Stafstr om, Europhys. Lett. 46 (1999) 382. [10] P.G. Collins, J.C. Grossman, M. C^ ote, M. Ishigami, C. Piskoti, S.G. Louie, M.L. Cohen, A. Zettl, Phys. Rev. Lett. 82 (1999) 165.
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