Molecular orbital studies on the rings composed of D2d C36 cages

Computational Materials Science 36 (2006) 474–479 www.elsevier.com/locate/commatsci

Molecular orbital studies on the rings composed of D2d C36 cages Ya-juan Jin, Bao-hua Yang, Yuan-he Huang

*

Department of Chemistry, Beijing Normal University, Beijing 100875, China Received 1 February 2005; received in revised form 24 May 2005; accepted 2 June 2005

Abstract The structures and electronic properties of the rings composed of D2d C36 cages are investigated using the semi-empirical AM1 molecular orbital method with full geometric optimization. It is found that most of the converged structures are more stable after the rings are formed. Strain plays an important role in the stability of the fullerene rings. Other factors influencing the stability are also discussed, such as the type of the bonded carbon atoms and the size of retained aromatic domains. Comparison of the structures and electronic properties are made between the C36-rings and those composed of D6h C36 isomers. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Fullerenes; Electronic properties

1. Introduction The C36 molecule has attracted much scientific interest [1–6] because of its special structures and properties. The strong tendency of C36 to form intermolecular bonds has led to a number of theoretical investigations of 1D, 2D and 3D polymers of C36 [7–10]. The connection through intermolecular bonds of C36 can also result in a ring structure just as C60 does. Therefore, the investigation on C36-ring is of significance for the basic research and study of possible C36 solid. We have reported the structures and electronic properties of the rings constructed with D6h C36 (36:15) [12] (simply called C36(D6h)-ring). It is well known that there are many different isomers for the C36 fullerene [13]. Further study on the C36-rings formed with different isomers is helpful for the understanding of the structure-property relationship of the rings. However, there have been few reports on C36-ring constructed with other isomers except for D6h C36. Among all the isomers D6h C36 (36:15) and *

Corresponding author. Tel.: +86 10 5880 5425; fax: +86 10 5880 0567. E-mail address: [email protected] (Y.-h. Huang). 0927-0256/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2005.06.001

D2d C36 (36:14) were predicted to be the most energetically favorable structures [14] with almost the same energy at the ground state. Following the study on 1D polymer of D2d C36 [15], we wonder what properties the rings composed of D2d C36 (denoted as C36-ring in the following) have. In the paper we build two series of D2d C36-rings, the structures and electronic properties of which are investigated using self-consistent field molecular orbital (SCF-MO) method and are compared with those of C36(D6h)-rings. 2. Construction of C36-rings and computation detail Like the rings composed of D6h C36 cages in Ref. [12], we consider only the rings that have the highest symmetry. All the rings studied here are formed through two ‘‘bonding-belts’’ which are denoted as belt-a and belt-b as shown in Fig. 1. It should be noted that there are six belts with Dnh symmetry in D6h C36-rings [12,16] while we find only two Dnh belts for the rings made of D2d C36 cages here. We construct and name the rings following Ref. [11] and [12]. The C36-rings are described in the notation: hBonding-belti-hsymmetry (Dnh)i-hlinking pattern described by indices of bonding atom in

Y.-j. Jin et al. / Computational Materials Science 36 (2006) 474–479

475

Fig. 1. (a) Five kinds of non-equivalent atoms and nine kinds of bondlength. The alphabets a and b denote two aromatic hexagon belts. (b) Two Dnh bonding-belts as shown by bold line area on the C36 cages.

bonding-beltsi. For instance, the ring b-D3h-2233 is a D3h ring constructed with three C36 cages, and there are four bonds between the neighboring C36 cages on the belt-b, including two C2–C2 bonds and two C3–C3 bonds. All the ring structures are fully optimized by semiempirical AM1 SCF-MO method [17] with GAMESS program package [18]. Here most of the rings are quite large; hence the semi-empirical method is still suitable for full geometric optimization. Besides, a comparison with previous work can be done based on the same calculation level. Symmetry constraint is applied in the geometrical optimization and the size of the C36-rings is considered for n = 3–8. To examine the reliability of the semi-empirical calculations, the three-member rings (n = 3) are also optimized using ab initio density-function-theory (DFT) method of B3LYP [19,20] with 3–21G basis set. The B3LYP optimized results show that the order of stability, the change of structures and electronic properties are consistent with those of AM1 calculations for the C36-rings with the smallest size, in spite of the numerical difference obtained from the two methods. All the following discussions are based on the AM1 calculations.

3. Results and discussion 3.1. Stability and geometric configuration

where (EðC36 Þn ) is the energy of the C36-rings. The formation of the C36-rings is favorable to the stability of the system when E < 0. The energy values of 20 rings in Table 1 are less than zero, i.e., most of the C36 systems among the converged structures become more stable after forming the rings. This is similar to the case in D6h C36-rings [12], which confirms again that C36 molecules have a strong tendency to form intermolecular bonds, even for those isomers with lower symmetry than D6h. The most stable Table 1 ˚ ) of the C36-rings Energy (kcal/mol) and intermolecular bond length (A C36-rings

E

Bond lengths C1–C1

C2–C2

1.562 1.559 1.626 1.607

1.572 1.593 1.788

C36 a-D3h-1 a-D3h-122 a-D4h-122 a-D5h-122

0.0 24.876 85.239 4.976 104.701

a-D3h-22 a-D4h-22 a-D5h-22

72.541 13.065 66.509

1.556 1.615 1.692

a-D3h-2255 a-D4h-2255 a-D5h-2255

14.781 19.473 32.173

1.523 1.617 1.842

a-D3h-55 a-D4h-55 a-D5h-55 a-D6h-55 a-D7h-55 a-D8h-55

81.027 17.913 37.963 42.812 42.802 40.971

b-D3h-22

66.292

1.565 1.539 1.631

C5–C5

1.610 1.523 1.491 1.626 1.542 1.526 1.522 1.522 1.524

Among the 54 possible rings constructed from the two bonding belts, 25 converged structures of the C36-rings with Dnh symmetry have been obtained by full geometrical optimization. In other words, only 25 Dnh rings can be formed using D2d C36 cages by AM1 calculations. Among the 25 converged structures, 16 rings are constructed from belt-a and 9 from belt-b respectively. The energy per C36 cage related to that of single D2dC36 (EC36 ) is given by

b-D3h-2233 b-D4h-2233

85.723 30.729

b-D4h-33 b-D5h-33 b-D6h-33 b-D7h-33 b-D8h-33

6.192 14.618 12.680 7.329 0.673

1.539 1.533 1.535 1.541 1.548

E ¼ EðC36 Þn =n  EC36

b-D8h-3344

15.132

1.508

1.569 1.512

1.626

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Y.-j. Jin et al. / Computational Materials Science 36 (2006) 474–479

cases in Refs. [11] and [12], the energies of most of the C36-rings with a given linking pattern are nearly linear function of their r values as shown in Fig. 3. The smaller the r is, the lower the energy is. This indicates that the distortion strain has great influence upon the stability. Obviously there exists an upper limit of r for a given linking pattern; those rings having r values exceeding the upper limit cannot lead to a convergent result. But

100 80

Fig. 2. Structures of the most stable rings for the two bonding-belts.

E (Kcal/mol)

40 20

Belt-a Linking patterns: 1 1122 122 22 55

0 -20 -40 -60 -80 0.00

0.05

0.10

0.15

0.20

0.25

σ (Å)

a 0 -10 -20 -30

E (Kcal/mol)

C36-rings constructed from the two belts here have the same D3h symmetry as shown in Fig. 2. However, the sizes of the most stable C36(D6h)-rings change with the bonding belts and usually with larger sizes [12]. But the stability of the C36-rings is not size-dependant. Table 1 shows that the sizes of most stable structures vary with linking patterns. Moreover, the intermolecular ˚ for bond lengths are within the range of 1.491–1.842 A the rings and most of them have at least one intermolec˚ . Therefore, most of these ular bond shorter than 1.60 A C36-rings can be considered as covalent oligomers just like the fullerene-rings in Refs. [11] and [12]. Formation of covalent bond between neighboring C36 molecules will cause the distortion of the cage, which increases the strain energy and reduces the stability of the rings. The parameter r [11,12] suggested by us can be used to estimate the distortion strain of C36 cage in the rings: larger r reflects greater strain. The r is a parameter that is easy to obtain, because it depends only on the structure of the undistorted single C36 cage, the linking pattern and the ring size, not the optimized ring structures. The calculated r values for the rings with different linking patterns and sizes are listed in Table 2. The C36-rings with r values expressed by boldface type are those having converged structures with full geometrical optimization. Obviously the smaller value of r is favorable to the formation of C36-ring. Similar to the

60

Belt-b Linking patterns: 22 2233 33 3344

-40 -50 -60 -70 -80 -90 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

σ (Å)

b

Fig. 3. E  r distribution for the C36-rings.

Table 2 Calculated values of the parameter r for all the possible linking patterns of the C36-rings Belt

Linking patterns

n 3

4

5

6

7

8

a

1 122 22 2255 55

0.05655 0.07839 0.09023 0.14434 0.18168

0.20900 0.15182 0.13129 0.11078 0.04525

0.34255 0.24519 0.19733 0.20040 0.00683

0.44720 0.32011 0.25772 0.26783 0.00000

0.52885 0.37918 0.30844 0.31749 0.00443

0.59360 0.42636 0.35039 0.35543 0.01278

b

22 2233 33 3344

0.09452 0.09002 0.11342 0.53068

0.15061 0.14571 0.01640 0.36657

0.22256 0.23118 0.00014 0.27424

0.28544 0.29172 0.00716 0.21522

0.33717 0.33632 0.02088 0.17528

0.37962 0.37070 0.03587 0.14768

Y.-j. Jin et al. / Computational Materials Science 36 (2006) 474–479

the values of upper limits for different linking pattern are not the same. In fact such a conclusion can also be drawn from our previous reports about C60-rings and C36-rings. In this way we can acquire the relative stability for the C36-rings with same linking pattern but different size by simply calculating r. D2d C36 cage consists of 12 pentagons and 8 hexagons and has five non-equivalent atoms as shown as in Fig. 1(a). The type of the bonding atom also affects the stability of the C36 rings, since bonding through the atoms at different positions will cause different degrees of distortion and reduction of aromatic domains in the carbon cage. For example, among the rings with the same type of intermolecular bonds and the same order of magnitude of r values, those formed through C2–C2 bonds are most stable, followed by those through C5–C5 and C3–C3. The p-orbital axis vector (POAV) angle [21] for the five non-equivalent atoms is C4 (107.3°) > C2 (106.0°) > C5 (105.8°) > C3 (103.3°) > C1 (102.6°), which is in accordance with the order of stability mentioned above. Accordingly, the order of stability seems related to the POAV angle of the non-equivalent atoms. There is no ring formed through pure C4–C4 bond. A large POAV angle causes large strain on the sp2 carbon atom and high reactivity [21], thus bonding through the atoms having larger POAV angle should be favorable to releasing the strain. However, the ring a-D5h-22 has high energy because of the large distortion of C36 molecules. On D2d C36 cage the 8 hexagons are separated into two aromatic belts by the pentagons and each one includes four paratactic hexagons (see Fig. 1(a)). Based on the data of AM1 calculation the bond length of hexa˚ (B3LYP/3-21G, 1.385–1.469 A ˚) gons is 1.384–1.476 A ˚ while the bond length of pentagons is 1.509–1.514 A ˚ ). Hence D2d C36 cage (B3LYP/3-21G, 1.497–1.501 A can be considered to have two aromatic belts named a and b. It is different from the D6h C36 cage which has three aromatic domains [12]. Nevertheless the total aromatic area is the same for the two C36 structures, which to some extent explains why the two C36 molecules with different symmetries have almost the same stability.

477

Kroto [22] has proposed that the derivative stability is increased for small fullerene by maximizing conventional aromatic domains concomitant with alleviating cage strain. When forming the intermolecular bonds the bonding atoms with sp2 hybridization turned into sp3 and the conjugation of bonding atoms with the neighboring sp2 carbon atoms on the same cage is broken. The original aromatic domains are then destroyed, or reduced to smaller aromatic domains. We find from the calculation here that less aromatic domains are destroyed in those stable rings. Fig. 4 shows the ichnographic structure for the most stable and unstable rings with the same D3h symmetry constructed from belt-a respectively. For the most stable ring a-D3h-122, the bonded atoms locate in the middle of a aromatic belt, thus the conjugation of middle area on the a belt is destroyed, but the aromatic area of the two benzenes aside still remains. In addition, few changes present in the b belt due to the absence of bonded atoms. As to a-D3h-55, the intrinsic aromatic domain of b belt is all destroyed though the aromaticity of a belt is retained. Anyhow, the whole conjugated area of the ring a-D3h55 is smaller than that of the ring a-D3h-122. Furthermore, the shared sides of two abutting pentagons in a-D3h-122 are still single bonds, while in a-D3h-55 four of pentagon-pentagon shared bonds shortened from ˚ . The double bonds introduced into 1.514 to 1.465 A pentagons will increase the heat of the formation. ˚ is not a conventional double bond Although 1.465 A length, the shortening of the shared sides of adjoining pentagons may lead to a more unstable structure. For the C60-rings [11], we can deduce the appearance and position of the shared-pentagon double bonds. However it is difficult to do prediction for the C36-rings, since the ground state of C36 molecule is not a closed-shell Kekule structure. Moreover, similar to the rings composed of D6h C36 cages, it is hard to get a quantitative or semiquantitative estimation of the intermolecular bonding energy that is another stable factor for the rings. Nevertheless the formation of intermolecular bonds will also be concomitant with the distortion of C36 cages, it may lead to larger strain energy and reduce the stability.

Fig. 4. Geometric structures of C36 cage for rings a-D3h-122 and a-D3h-55. The dashed lines denote the retained benzenoid aromatic domains, and the spots denote the bonded atoms.

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Y.-j. Jin et al. / Computational Materials Science 36 (2006) 474–479

3.2. Electronic structures The calculated energy levels and symmetry of the higher occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), as well as the energy gaps (Eg) between them, are listed in Table 3. It can be seen that the energy gaps of more than half of the C36 rings become smaller than that of D2d C36. But most of C36(D6h)-rings have larger energy gaps than the single D6hC36molecule [12]. The change of Eg has something to do with the linking patterns. Using Eg of the D2d C36 as benchmark, the rings with pure C3–C3 or containing C5–C5 bonds all have smaller energy gaps, meanwhile those with pure C2–C2 or containing C2–C2 except for a-Dnh-2255 have larger energy gaps. Hence, the type of bonded atoms has influence on not only the stability but the electronic structures for the C36-rings as well. The most stable structure a-D3h-122 has the largest gap, and it may be favorable to both thermodynamic and kinetic stability. From the levels of HOMO and LUMO listed in Table 3, most of the LUMOs are lower after the formation of the C36-rings, indicating that the ability of accepting electrons is increase for the C36 rings, which is contrary to the case in the C36(D6h)-rings. For most of the rings composed of D6h Table 3 Energy gaps, HOMO and LUMO levels (in eV) and symmetry of the C36-rings C36-rings

Eg

HOMO

C36 a-D3h-1 a-D3h-122 a-D4h-122 a-D5h-122

5.306 5.219 5.851 5.638 5.312

8.754 9.094 9.293 9.143 8.996

A2 E0 A001 A1u A001

3.448 3.875 3.442 3.505 3.684

B2 E00 E0 B2g E02

a-D3h-22 a-D4h-22 a-D5h-22

5.850 5.589 5.453

9.295 9.132 8.991

A100 A1u A001

3.445 3.543 3.538

E0 B2g E02

a-D3h-2255 a-D4h-2255 a-D5h-2255

3.244 1.192 3.684

8.158 7.143 8.441

E0 A1u E02

4.914 5.951 4.757

A001 B2g A001

a-D3h-55 a-D4h-55 a-D5h-55 a-D6h-55 a-D7h-55 a-D8h-55

3.559 4.011 4.221 4.302 4.376 4.436

8.517 8.895 8.969 9.004 9.0560 9.116

A02 B2g E02 B2u E00 B1g

4.958 4.884 4.748 4.702 4.6804 4.680

E00 B1u E002 B1g E0 A1u

b-D3h-22

5.796

9.336

A02

3.540

A002

00

LUMO

b-D3h-2233 b-D4h-2233

5.836 5.263

9.178 8.787

E A1g

3.342 3.524

E00 B2u

b-D4h-33 b-D5h-33 b-D6h-33 b-D7h-33 b-D8h-33

3.121 3.627 3.723 3.658 2.017

7.671 7.954 8.006 7.987 6.376

A2g A2 B2U E00 B2g

4.550 4.327 4.283 4.329 4.359

B1u E002 B1g E0 A1g

b-D8h-3344

3.518

7.973

A2U

4.455

B1g

C36 cages, the LUMOs have higher energy than that of the single D6hC36 cage [12]. It can also be found that the doubly degenerate frontier orbitals (HOMO and LUMO) exist only in the rings with odd number of C36 cages. Therefore, the electronic structures of the fullerene rings also depend on the odd or even symmetry of fullerene cage numbers. This is similar to the situation of the C60 rings [11], but different from that of the C36(D6h)-rings, among which a few of even member rings can have doubly degenerated HOMOs. Hence, the C36-rings constructed from different C36 isomers display different electronic properties.

4. Conclusion Two series of Dnh C36-rings constructed with D2d C36 cages are investigated using semi-empirical AM1 molecular orbital method. From a view of energy, the formation of the rings enhances the stability of most of the systems. The strain-associated factor r can be regarded as an indicator of stability order for the C36-rings with the same linking pattern. This manifests again that the strain energy plays an important role in the formation of the fullerene rings. Other interaction factors involved in the stability of the C60-rings and the C36(D6h)-rings previously studied also have impacts on the C36-rings studied in this paper. Such factors include the size of conventional aromatic domains and the position of the bonding atoms. Furthermore, the electronic structures are discussed and compared with C36(D6h)-rings. It is found that the changes of electronic properties for the C36-rings constructed from D2d C36 cages are different from those for the C36(D6h)-rings. The calculations show that most of the rings have smaller Eg and larger electron affinity compared with D2d C36 molecule, i.e., most of the rings may be excited more easily and have stronger ability of accepting electron than the 36:14 isomer. In addition, the doubly degenerate frontier orbitals occur only in the rings with an odd number of C36 cages.

Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 20373008) and Major State Basic Research Development Programs (Grant No. 2002CB613406).

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