Structures and Stabilities of Rings Composed of C20 Cages

Ying Liu*1, Guo Wang*1 and Yuanhe Huang*2

Department of Chemistry, Beijing Normal University, Beijing 100875, P. R. China

The ring structures with Dnh, Dn, Cnv and Cnh symmetries composed of C20 cages have been investigated using the self-consistent fieldmolecular orbital method. It is found that the formation of the C20-rings is favorable to stability of the C20 systems. The influence of structuresymmetries and distortion on the stabilities of the C20-rings is explored. The strain change due to the structure distortion greatly affects thestabilities of the C20-rings. Furthermore, the bonding and electronic properties of the C20-ring are also discussed and compared with those ofother fullerene-rings such as C60-rings and C36-rings. [doi:10.2320/matertrans.48.670]

(Received November 20, 2006; Accepted February 6, 2007; Published March 25, 2007)

Keywords: fullerene, structure, electronic property

1. Introduction

The C20 cage is considered to be the smallest fullerene. Itonly consists of twelve pentagons which lead to largecurvature and contravention of the ‘‘isolated pentagonrule’’.1–3) Possibly, its special structure makes it one of theunstable fullerenes. In fact, the products of C20 cage obtainedfrom perhydrogenation and gas-phase debromination proc-esses have a lifetime of only about 0.4ms.1) Moreover, it isknown that a larger �-orbital axis vector (POAV) angle cancause a larger strain on the sp2 carbon atom and high reactiveactivity.4) The POAV angle on the atoms of Ih C20 cage isequal to 110.90� which is larger than those of C60

5) and C366,7)

fullerene balls. Therefore, the C20 cages should have astronger tendency to form the intermolecular bonds forreleasing the cage strain. Theoretical studies on variousstructures consisting of C20 cages such as dimer, trimer andtetramer structures have been carried out, as well as one-dimensional (1D), two-dimensional (2D) and three-dimen-sional (3D) polymers.8,9) Moreover, connection throughintermolecular bond should also result in ring structures,similar to the case in C60 and C36 cages.5–7) So far as weknow, there is no report on the ring structures composed ofC20 cages, but study of the C20-ring structures would help usto understand more about the structures and electronicproperties of the C20 cage and the possible ring structures. Inthis paper, we investigate the structures, stabilities andelectronic properties of C20-rings using the self-consistentfield molecular orbital (SCF-MO) method and compare theresults with those of C60 rings and C36 rings. Four symmetriesDnh, Dn, Cnv and Cnh are considered for the C20-rings here.

2. Construction of C20-Rings and Computation Details

We have constructed a series of C20-rings with Dnh, Dn,Cnv, and Cnh symmetries using the same method used forthose C60 and C36 rings.5–7) Similarly, we can build areference frame for a ring composed of n C20 cages with Dnh

symmetry as shown in Fig. 1. The center of the Ih C20 cage Ois set as the origin point, the Z-axis is vertical to the ring (XY)

plane. If the C20 cages rotate 30� around X-axis, a Dn ringwill appear. If the cages rotate 30� around Y-axis, Cnv

structure is obtained. If the cages rotate 30� around Z-axis, aCnh ring will be formed. Adjacent C20 cages in the ring willform intercage bonds through carbon atoms situated on theiradjacent faces and the connected area will move on thesurface of the C20 cages with the change of the ring size.Likewise the possibly bonding area on the C20 cages defines a‘‘bonding-belt’’.5) The five bonding-belts for constructing theC20-rings are shown in Fig. 2. The size of the C20-rings isconsidered from three to eight (n ¼ 3{8).

We denote the C20-rings as followings: <Bonding-belt> -<linking pattern described by indices of bonding atoms inbonding-belts>. For example, the ring C6v-B-����� ex-presses a C6v ring that consists of six C20 cages and theconnection between adjacent C20 cages is through two C�-C�

bonds, one C�-C� bond, and two C�-C� bonds on Cnv-Bbonding-belt. Since Cnh rings contain the intercage bonds ofC�-C� and C�-C�0 , we use two Greek letters to describe theseintercage bonds. For example, there are two C�-C� bonds andone C�-C�0 bond in the ring C5h-��-��-��

0.The structures and electronic properties of C20-rings are

calculated using the ab initio Hartree-Fock (HF) method at3-21G level10) with fully geometric optimization and theGAMESS package.11) From the viewpoint of the time

O

y

x

n

nPβ

T

O'

Qα

Fig. 1 Reference frame for C20-rings of Dnh symmetry.

*1Graduate Student, Beijing Normal University*2Corresponding Author, Beijing Normal University

Materials Transactions, Vol. 48, No. 4 (2007) pp. 670 to 674Special Issue on ACCMS Working Group Meeting on Clusters and Nanomaterials#2007 Society of Nano Science and Technology

consuming and computational efficiency, this calculationmethod is suitable for the calculations of the C20-ringsystems. The symmetry constraints are applied in all thecalculations. In order to further confirm the HF results, ahybrid density-function-theory (DFT) method B3LYP12,13)

with 6-31G (d) basis set is also used to optimize the three-member rings structures. The B3LYP optimized resultsdemonstrate that the sequence of stabilities, the change ofgeometrical structures and electronic properties are nearlyconsistent with those of HF calculations for the smallest sizeC20 rings, in spite of some numerical differences. Thus thefollowing discussions are all based on the HF/3-21G results.

3. Results and Discussion

3.1 Stability and geometric configurationThe C20-rings are constructed by forming intercage bonds

on the bonding-belts shown in Fig. 2 between adjacent C20

cages. The full optimization gives twenty-nine stable ringstructures. Among these optimized ring structures, there aresixDnh rings, elevenDn rings, six Cnv rings, and six Cnh rings,respectively. The binding energy per C20 cage related to asingle C20 with Ih symmetry is defined as

Eb ¼ EðC20Þn=n� EC20ð1Þ

Here EðC20Þn is the energy of the C20-rings, and EC20is the

energy of the single C20 cage calculated with the samemethod. The formation of the C20-rings is favorable to thestability of the systems when Eb is less than zero. Thecalculated results are listed in Table 1. From Table 1, we cansee that the intercage bond lengths between adjacent C20

cages are in the range of 1.450–1.644 A. Accordingly, therings can be considered as covalent C20 oligomers, similar to

C60-rings and C36-rings.5–7) The binding energies of all the

rings except for the C6h-��-��-��0 and C7h-��

0 rings are lessthan zero, namely most of the systems become more stableafter the formation of the ring structures. This indicates thatthe C20 cages, indeed, have a tendency to form intercagebonds from a view of thermodynamics. Since the C20 cage issmaller, the number of rings with the highest symmetryDnh ismuch less than those of C60-rings and C36-rings. Figure 3shows the most stable C20-ring structures for differentsymmetries. It can be seen that the size of the most stablerings is different for different symmetries. The most stablerings with lower symmetries (Cnv and Cnh) are three-memberrings, but the corresponding Dnh and Dn rings are seven-member and five-member structures respectively. It seemsthat the most stable C20-rings with higher symmetries arefavorable to the structures with larger size.

From the binding energies of the rings, we can find that therings with Cnh symmetry have the lowest stability whosebinding energy averages 2.831 eV higher than the others, andthe energy differences of rings among the other threesymmetries (Dnh, Dn and Cnv) are small. In fact, for Cnh

rings, the intercage bonds are not parallel to the ring planebefore optimization but become parallel after the formationof stable rings, which is different from those in the Dnh, Dn

and Cnv rings and thus leads to larger distortion of the C20

cage and higher energy structures. Thus the symmetry of thering structures has something to do with the stability of C20-rings, since different symmetries cause different degrees ofstructure distortion.

The change of the strain energy is an important factor forthe stabilities of C20-rings. To estimate the distortion strain, aparameter � has been suggested and successfully applied tothe fullerene rings.5–7) Likewise, as shown in Fig. 1, we builda ball which touches every vertex on the C20 cage. Thetangent plane of the ball, �n, is perpendicular to line OO0

where O and O0 is the center of the two adjacent balls,respectively. Pn is the tangent point and T is the position of apossible bonding atom on the bonding-belt. TQ is the verticaldistance from the possible bonding atom to �n. Then theparameter � can be expressed by5)

� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðTQ� TQÞ2

m

sð2Þ

where m is the number of TQ. More detailed description ofeq. (2) is in Ref. 5). Larger � reflects greater strain, higherenergy and lower stability, and vice versa. Hence, the smallervalue of � is favorable to the construction of the C20-ring. Wealso like to point out that the calculation of parameter � isonly dependant on the structure of a single undistorted C20

cage, the linking pattern and the ring size, and need not tohave the optimized ring structures. Figure 4 shows therelation of the binding energies of C20 rings and the standarddeviation parameter �. With the same linking pattern, thebinding energies of most C20-rings increase with theincreasing of the � values, even some are nearly linearfunctions of their � values. This demonstrates that theparameter � can also be used to estimate the relative stabilityof the C20-rings with the same linking pattern. Thus thechange of strain always has an important impact on thestability of the ring structures consisting of fullerene cages.

δ β

γδ

Cnv-A

Cnh

α'

β

β

αγ

γ

β

β

α'

Cnv-B δ

β

γ

β

δ

Dn

δ βγ

β δγ

Dnh

α

β

Fig. 2 Five bonding-belts shown by bold line area on the C20 cages. The

Greek letters denote atom indices.

Structures and Stabilities of Rings Composed of C20 Cages 671

The optimized bond lengths of the single Ih C20 cage are allequal to 1.447 A. After formation of the rings, the intracagebonds related to the bonding atoms (C-Cbonding) are elon-gated, which leads to the distortion of the cage. It seems that

D7h-α D5-β

C3v-A-βγ C3h-βγ-βγ

Fig. 3 Structures of the most stable C20-rings for each symmetry (Dnh, Dn,

Cnv, Cnh).

0.0 0.1 0.2 0.3 0.4

-5

-4

-3

-2

-1

0

1

2

43

8

65

454

678

3

6

54

3

7

8

7

Eb/e

V

/Åσ

Dnh

-αC

nh-βγ -βγ

Cnh

-βγ -βγ -αα'C

nh-αα'

Cnv

-βγD

n-β

Fig. 4 Eb � � distribution. The numbers are the size of rings.

Table 1 Binding energy Eb (eV), distortion parameter � (A) and intercage bond lengths (A) between adjacent C20 cages.

C20-rings Eb �Intercage bond lengths

�-� �-� �-� �-� �-� �-�0

D3h-��� �1:315 0.1894 1.604 1.629

D4h-� �3:745 0.0883 1.513

D5h-� �3:922 0.0350 1.551

D6h-� �4:163 0.0128 1.535

D7h-� �4:235 0.0036 1.531

D8h-� �4:234 0.0000 1.530

D3-� �4:079 0.0160 1.529

D4-� �3:122 0.0000 1.595

D3-� �2:969 0.0191 1.569

D4-� �4:084 0.0219 1.516

D5-� �4:206 0.0574 1.508

D6-� �4:113 0.0943 1.513

D7-� �0:197 0.1267 1.587

D8-� �3:786 0.1537 1.529

D5-�� �2:507 0.3677 1.619 1.651

D6-� �2:385 0.0242 1.548

D8-�� �1:493 0.1421 1.611 1.644

C3v-A-�� �5:420 0.0901 1.549 1.545

C4v-A-�� �4:623 0.1407 1.463 1.604

C5v-A-��� �4:450 0.1624 1.577 1.562 1.625

C6v-B-����� �4:319 0.2615 1.631 1.614 1.556

C7v-A-�� �3:659 0.0953 1.621 1.496

C8v-A-� �3:686 0.0460 1.479

C3h-��-�� �2:656 0.2088 1.464

C4h-��-�� �1:340 0.2798 1.495

C5h-��-��-��0 �1:091 0.3248 1.530 1.518

C6h-��-��-��0 2.216 0.3975 1.609 1.450

C7h-��0 0.227 0.0423 1.570

C8h-��0 �2:283 0.0277 1.576

672 Y. Liu, G. Wang and Y. Huang

the more unstable rings have longer C-Cbonding lengths. Forexample, the C-Cbonding lengths of the most unstable C6h-��-��-��0 ring are 1.613–1.637 A, but those of the most stableC3v-A-�� ring are 1.543 A and 1.526 A. In fact, thecalculations show that the lengths of C-Cbonding bonds forrelatively more stable rings (Eb < �4:084 eV) are about1.508–1.577 A, and at least there is one C-Cbonding bondlonger than 1.583 A for relatively more unstable rings. Theseindicate that the C-Cbonding bonds in more unstable C20-ringsare easier to be broken than those in more stable ones. Inother words, the C20 cages seem to be disrupted more easilyfor rings with higher energy.

Previous studies8,14) show that the open C20 cage polymerswith broken intracage bonds are most stable among the C20

linear oligomers, 1D polymers and 3D solid-state structures.And Miyamoto et al.9) also showed that the simple-cubic-likecondensed phase with opened C20 cages is stable. Howeverthe situation is different in the C20-rings. The rings withclosed intracage bonds are relatively more stable, but the twoenergetically unfavorable rings have broken intracage bonds.The intracage �-� bonds (see Fig. 2, Cnh bonding-belt) in themost unstable C6h-��-��-��

0 are broken and elongated to2.791 A after formation of the rings. This is a distinctdifference between the ring structures and the 1D or 3Dpolymers composed of C20 cages. In addition, energeticallyfavorable 3D solid-state structure8) forms intercage bondssimilar to the C�-C� intercage bonds on Dnh bonding-belt inFig. 2, but only C�-C� intercage bonds can be obtained byour optimization for theDnh C20-rings, except for the smallestD3h ring. The bonding characters are quite different betweenthe C20-rings and the 1D, 3D C20 polymers.

3.2 Electronic propertiesThe calculated energy levels and irreducible representation

of the highest occupied molecular orbital (HOMO) andlowest unoccupied molecular orbital (LUMO), as well as theenergy gaps (Eg) between them, are listed in Table 2 for theC20-rings studied. It can be seen that the HOMOs of mostC20-rings are lower than that of the single C20 cage from0.180 eV to 1.336 eV with only four exceptions, leading to adecreasing of the ability to lose electrons in these systems.But the changes of LUMO levels are not as consistent, thoughmore than half are higher than that of the single C20 cage.Moreover, the energy gaps for most of the rings obviouslybecome larger compared with that of the single C20 cage,which is similar to the case in D6h C36-rings

6) but contrary tothe C60-rings

5) and the D2d C36-rings.7) A larger energy gap

may be favorable to the kinetic stability for the systems.Besides, the two most unstable C6h-��-��-��

0 and C7h-��0

rings have the smallest energy gaps, hence the two rings arenot favorable to both thermodynamic and kinetic stabilities.We notice that the energy gap has some relation to thethermodynamic stability for all the C20-rings. The energy ofthe C20-rings decreases with the increase of energy gapfor the same linking patterns. Thus it seems that higherthermodynamic stability is accompanied with higher kineticstability for the C20-rings. In addition, it can also be foundthat the doubly degenerate frontier orbitals exist in the ringswith not only odd numbers of C20 cages but also evennumbers. This is similar to the situation of the D6h C36-rings,

but different from that of the C60-rings and D2d C36-ringsamong which only odd number rings have doubly degeneratefrontier orbitals. Moreover, the energy gaps of energeticallyunfavorable C60-rings and the C36-rings are not the smallestones. Therefore, different fullerene-rings have differentcharacteristics for their electronic properties.

4. Conclusion

Four series of C20-rings with Dnh, Dn, Cnv and Cnh

symmetries are investigated by using the HF SCF-MOmethod. The calculations show that the formation of the C20

rings is energetically favorable for most of the systems. It isfound that the change of the strain energy plays an importantrole in the stability of the C20-rings. The relative energychange due to the C20 cages for the C20-rings can beestimated by the strain-associated parameter � which hasbeen successfully used for the C60-rings and the C36-rings.The most stable ring sizes are different for different

Table 2 HOMO and LUMO levels (in eV), corresponding irreducible

representations, and energy gaps (eV) of C20-rings.

C20-rings HOMO LUMO Eg

C20 �6:971 Au �1:036 FU 5.935

D3h-��� �4:309 A02 �1:912 E0 2.396

D4h-� �7:964 A2U �2:040 B1G 5.924

D5h-� �8:337 A02 �0:767 E00

2 7.570

D6h-� �8:266 E2G �0:732 B1G 6.174

D7h-� �8:119 E03 �0:675 E00

3 7.445

D8h-� �7:956 B1G �0:661 B1U 7.295

D3-� �8:133 A1 �0:974 E 7.159

D4-� �7:951 A2 �1:346 B2 6.604

D3-� �8:040 A2 �0:558 A2 7.483

D4-� �7:945 A2 �0:588 A2 7.358

D5-� �7:921 A2 �0:653 A2 7.268

D6-� �7:893 A2 �0:718 A2 7.175

D7-� �7:502 A1 �0:944 A2 6.558

D8-� �7:801 A2 �0:808 A2 6.993

D5-�� �8:220 A1 �0:046 E2 8.174

D6-� �8:193 A1 �0:103 B1 8.089

D8-�� �7:738 E2 �0:114 B1 7.624

C3v-A-�� �8:152 E �0:868 A1 7.284

C4v-A-�� �7:923 A1 �0:481 B2 7.442

C5v-A-��� �8:146 E1 �1:436 A1 6.710

C6v-B-����� �6:207 A1 �1:855 A2 4.352

C7v-A-�� �7:167 A1 �1:273 E2 5.894

C8v-A-� �7:298 B1 �0:968 B2 6.329

C3h-��-�� �8:021 A00 �1:107 E00 6.914

C4h-��-�� �7:719 AG �2:214 BU 5.505

C5h-��-��-��0 �7:426 A0 �1:757 E00

2 5.669

C6h-��-��-��0 �5:704 E2U �2:954 E1U 2.750

C7h-��0 �4:512 E0

1 �2:938 E02 1.574

C8h-��0 �7:151 BU �1:053 BG 6.098

Structures and Stabilities of Rings Composed of C20 Cages 673

symmetric structures. The electronic properties show thatmost of the energy gaps of C20 rings are larger than that of thesingle Ih C20 cage, and the change tendency of kineticstability is in accordance with that of thermodynamicstability for the C20 rings with the same linking patterns.Moreover, the degeneracy of the frontier orbital is differentfrom those of C60 rings and D2d C36 rings.

Acknowledgments

This work is supported by the Research Fund for theDoctoral Program of Higher Education (20060027001) andMajor State Basic Research Development Programs(2002CB613406).

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