Energetic analysis of 24 C20 isomers

Cody Allison, Kyle A. Beran*

Department of Science and Mathematics, The University of Texas of the Permian Basin, 4901 E. University Blvd., Odessa, TX 79762, USA

Received 24 February 2004; revised 28 April 2004; accepted 28 April 2004

Abstract

Twenty-four individual geometric structures of the 20-carbon system are investigated using hybrid Hartree–Fock/density functional

theory (DFT-B3LYP) in combination with the 6-31G and 6-311G* basis sets. These computations are carried out in order to evaluate the

relative energies of the various C20 isomers utilizing both geometry optimization and single-point energy calculations. The relative energies

of the three most widely studied C20 isomers (ring, bowl, and cage) are compared to the energies of other cyclic, bowl-like, and miscellaneous

isomers. Although the ring and the bowl isomers are predicted to be the two most stable species, we have identified three to four additional

cyclic structures that are predicted to be energetically competitive with the fullerene structure.

q 2004 Elsevier B.V. All rights reserved.

Keywords: C20; Isomers; Fullerenes; DFT calculations; Geometry optimizations; Potential energy; Relative energy

1. Introduction

It has only been in recent years that the C20 caged-

fullerene, the smallest and least stable of this class of

molecules, has been identified experimentally. Prinzbach

et al. [1] reported in 2000 the preparation of the fullerene

cage from brominated dodecahedrane in the gas phase using

mass-selective anion photoelectron spectroscopy. The

vibrational spectrum and electron affinity of the caged

structure was compared to the spectra of the bowl, which

was obtained from brominated corannulene, and ring [2]

isomers. In the following year, Wang et al. [3] reported the

synthesis of solid crystals of the cage isomer by irradiating

samples of ultra-high molecular weight polyethylene

(UHMWPE), which are indirectly cooled in liquid nitrogen,

with an Arþ beam. In spite of the historical difficulties

associated with the experimental synthesis of the cage

structure of the 20-carbon fullerene, theoretical studies on

this system have been extensive ever since Kroto [4] first

proposed its existence in 1987.

In addition to theoretical studies on the electronic and

vibrational properties [5,6] of C20 and its ions, theoretical

studies on narrow nanotubes (5 A diameters) have proposed

that C20 fragments may be responsible for capping these

slender tubes [7]. The theoretical studies on the tubes were

based on the experimental production of very narrow carbon

nanotubes that possess diameters on the order of 4–5 A

[8,9]. There has also been a considerable amount of

computational time devoted to analyzing the relative energy

of isomers in the 20-carbon system [10–18]. Primarily, the

focus of these studies has been directed toward the ring,

bowl, and cage structures. According to the literature, it has

been difficult to identify the most stable isomer due to the

variation of relative stabilities of the structures as a function

of the theoretical model applied. This uncertainty has been

attributed to the structural diversity and chemical unsatura-

tion, which yields contributions from both the static and

dynamic electron correlation effects. However, Grimme

et al. [10] have recently employed the most accurate

theoretical methods to determine that, within experimental

uncertainty, the bowl and the cage isomers are isoenergetic

with the ring approximately 2.0 eV higher in energy.

Regardless of these findings, there is very little mention in

the literature [2,14,16] of other possible stable isomers that

may compete with the energetic stability of the ring, bowl,

and cage structures.

Our previous studies on the C20 system [19–21]

focused on the construction of the potential energy surface

that exists around the ring, bowl, and cage isomers.

During the course of analyzing the output generated from

0166-1280/$ - see front matter q 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2004.04.042

Journal of Molecular Structure (Theochem) 680 (2004) 59–63

www.elsevier.com/locate/theochem

* Corresponding author. Tel.: þ1-432-552-2238; fax: þ1-432-552-2236.

E-mail address: beran_k@utpb.edu (K.A. Beran).

saddle-point calculations between the three primary

isomers, we have located and identified over 40 additional

20-carbon isomers, each of which is predicted to occupy

an energy minimum. Consequently, the research presented

in this paper goes beyond the theoretical investigation of

simply the ring, bowl, and cage isomers by considering 21

additional structures. These isomers were chosen amongst

the 40 þ isomers based upon their respective heats of

formation ðDHfÞ that we had determined using the semi-

empirical PM3 model. Five of the isomers were con-

structed via the pencil-on-paper method and incorporated

into this research based on their near bowl-like, or near

sheet-like structure, each of which has a central six-

membered ring.

2. Experimental method

Geometry optimizations and single-point energy calcu-

lations were conducted using the Spartan ’02 Windows [22]

software package running on a Dell Precision 450 Work-

station. Using the Cartesian coordinates obtained in our

previous work, the 24 isomers, see Fig. 1, were geome-

trically optimized using the split-valence 6-31G [23] basis

set with the B3LYP functional [24] for the purpose of

analyzing the relative energies of these isomers in terms of

density functional theory. These optimizations were carried

out without any symmetry constraints in order to locate the

local minimum for each isomeric structure. Vibrational

analysis of the 24 isomers was also conducted.

Fig. 1. The 24 isomers of C20 that are theoretically investigated. To save space, the molecules pictured do not accurately depict theoretically calculated

molecular volumes.

C. Allison, K.A. Beran / Journal of Molecular Structure (Theochem) 680 (2004) 59–6360

The eight isomers with the lowest energy were

subsequently subjected to full geometry optimization

using the B3LYP functional with the 6-311G* basis set.

As with the 6-31G optimization process, analysis of the

vibrational frequencies (B3LYP/6-311G*) were performed.

This theoretical method was chosen for comparison with

Ref. [15]. Based on these optimized geometries, single-

point energies were determined for the eight chosen isomers

using DFT (B3LYP/6-311G*//B3LYP/6-311G*).

3. Results and discussion

The 21 isomers that were incorporated into this study, in

addition to the ring, bowl, and cage isomers, have been

categorized into five groups. Besides the group consisting of

the three primary isomers, which have been studied

extensively, the cycloadducts all exhibit ring-like charac-

teristics either in two- or three-dimensional space, of which

only the bowtie isomer has been reported previously [2,14].

The five additional bowls provided an internal comparative

study to the more well-known bowl structure, with its five-

member central ring structure and C5v symmetry. Of the five

new bowl structures, only c20-6om is optimized to a planar

geometry and, therefore, is representative of a carbon sheet.

The c20-5m isomer is a near-planar bowl possessing an

approximate curvature of 10–158, whereas the other three

bowls possess a much larger curvature (30–408). The cage

isomer with C2v symmetry (cage-c2v) isomer was also

included for an energetic comparison to the D2h symmetri-

cal version of the cage isomer. The bowl-like derivatives

(Group IV) are all representative of the atomic rearrange-

ment of the bowl isomer due to the breaking of bonds

between the central pentagon and the external hexagons.

The bowl-loop8 isomer was previously generated by a

genetic algorithm developed by Deaven and Ho [16]. The

other four derivatives were observed during our previous

studies. The three miscellaneous isomers were included

simply due to their unique structural appearance. These 24

isomers represent approximately one-half of all the C20

isomers that we have located and identified through our

semi-empirical studies.

Total geometry optimizations of the 24 isomers were

carried out by implementing the B3LYP/6-31G model and

are tabulated in descending order in Table 1, along with

their respective point groups and multiplicity. A consider-

able amount of time and energy has been directed toward

determining the relative energies of the three primary

isomers, with the assumption being that these three

structures occupy the three lowest-energy potential minima.

While our work predicts that the ring and bowl isomers

occupy the two lowest energy minima, the cage (D2h)

isomer is not the next in line.

We propose that there may be four additional isomers

that possess a relative energy that is comparable to that of

the cage (D2h) isomer. Consequently, there are five cyclic

isomers, including the ring, that lie lower in energy than the

cage (D2h) isomer. This is entirely consistent with

arguments that support the ring structure being lower in

energy due to the larger total entropy that is associated with

the ‘floppier’ structures. Excluding the ring isomer, the

three-dimensional cyclic structure of 3-6loop isomer is

lower in energy by ,0.4 eV in relation to the planar and

isoenergetic bowtie and halfbow isomers. Even though

2-6loop-tri has a unique carbon triangle, suggesting a large

ring strain, it is still predicted to be lower in energy than the

fullerene structure.

Vibrational analysis of the 24 isomers indicates that all

occupy local minima (all frequencies .0), except for the

cage (D2h) isomer that has one imaginary frequency

(2199i cm21). The fullerene isomer was re-built and

optimized without any symmetry constraints. The resulting

structure possesses a C1 symmetry and is not predicted to

possess an imaginary frequency. However, these two

symmetries of the cage isomer, along with a structure

with Cs symmetry, are predicted to be isoenergetic.

Consequently, we have chosen to identify the higher

symmetry structure (D2h) as the local minimum for the

cage isomer, which is consistent with that reported by

Grimme et al. [10]. In contrast, the cage-c2v isomer is

expected to be 1.03 eV higher in energy than the other

isoenergetic symmetries of the cage structure.

Table 1

Relative energiesa of the 24 C20 isomersb based on B3LYP/6-31G

calculations

Isomeric name Point group Multiplicity Relative energy (eV)

Ringc C10h 1 0.00

Bowl C5v 1 1.30

3-6loop D3h 1 2.27

Bowtie D2h 1 2.67

Halfbow C2v 1 2.68

2-6loop-tri C1 1 3.21

Cage D2h 1 4.05

c20-6om C2v 1 4.07

naphth þ 10 C1 1 4.35

C20-5m C2v 3 4.88

Cage-c2v C2v 1 5.08

Bowl-loop8 C1 1 5.22

c16 þ 4 C1 1 5.52

c20-6m C1 3 5.73

11 þ 2 þ 7 C2 3 5.78

c20-6p C2v 3 5.91

c12 þ 8 C1 1 5.92

4 þ 4cross C1 1 6.29

c13 þ 7 C1 1 6.77

Tank C1 1 7.02

c20-6o C1 3 7.13

c15 þ 4 þ 1 C1 1 7.56

Shuttle C1 3 7.80

c19 þ 1 C1 3 8.53

a Uncorrected total energies.b Cartesian coordinates or z-matrix for each isomer is available upon

request.c Total electronic energy ¼ 2761.3428 hartree.

C. Allison, K.A. Beran / Journal of Molecular Structure (Theochem) 680 (2004) 59–63 61

Of the additional bowl structures, only c20-6om is

predicted to be a ground-state singlet and is only 0.02 eV

higher in energy than the cage (D2h) isomer. The stability of

this isomer in comparison to the other bowl-like structures

can be attributed to the location of the five-membered rings.

Having the five-membered rings nested between six-

membered rings, as occurs in the other four bowl-like

structures, not only induces a curvature of the geometry but

also increases the instability of the isomer. Since the five-

membered rings in c20-6om are isolated, the energy of this

isomer is 0.9 eV lower in energy in relation to its nearest

bowl-like structure with a central six-membered ring.

To bring the study to a higher level of theory, we reduced

the number of isomers in question to the eight isomers

lowest in energy, ring ! c20-6om. These eight isomers

were geometrically optimized using the B3LYP/6-311G*

method and incorporated symmetry constraints. This

method was chosen so that we could compare the relative

energies of the ring, bowl, and cage (D2h) with the results

reported by Grossman et al. [15]. As was the case in the

previous calculations, a vibrational analysis of each isomer

was conducted. Based on these optimized geometries,

single-point energy calculations were performed using

B3LYP/6-311G*. Vibrational analysis reveals that the

bowtie isomer, with its D2h symmetry, has two imaginary

frequencies (142i and -89i cm21). Re-optimization of this

isomer without symmetry constraints results in isoenergetic

C2h and Cs symmetries that occupy local minima. However,

they are predicted to lay 0.2 eV higher in energy than the

D2h symmetry. Consequently, we do not interpret the

imaginary frequencies to be indicative of a second-order

saddle-point for the D2h structure of the bowtie isomer.

Vibrational analysis also showed that the cage (D2h) and the

halfbow (C2v) isomers have imaginary frequencies of 233i

and 2179i cm21, respectively. In the case of the cage (D2h)

isomer, the small imaginary frequency did not lead us to

conclude that this structure occupies a first-order saddle-

point. For the halfbow isomer, re-optimization without

symmetry constraints produced a local minimum structure

with Cs symmetry. An energy comparison between the two

symmetries of the halfbow isomer revealed that they are

isoenergetic. Based on vibrational considerations, we have

included the Cs structure in Table 2. Table 2 lists the relative

energies for the eight isomers as determined by geometry

optimization and single-point energy calculations. The

relative energies are also represented graphically in Fig. 2.

The B3LYP/6-311G* calculations performed by Gross-

man et al. resulted in the ring being identified as the most

stable isomer, followed by the bowl (þ0.40 eV) and the

cage structures (þ2.33 eV). In comparison, our results do

show that the ring is indeed predicted to be the most stable

isomeric structure, followed by the bowl and the cage.

Although the relative single-point energy of the bowl

reproduces that reported by Grossman et al., we are 0.4 eV

higher in relative energy based on the total optimization.

For the cage isomer, we are 0.2 eV higher in relative

energy with respect to total optimization and 0.4 eV lower

in energy in terms of the single-point energy. Since

Grossman et al. [15] do not report symmetries for their

isomers, we are uncertain as to the cause of this slight

discrepancy, whether it is in fact due to symmetry

considerations or differences in methodology.

Table 2

Relative energiesa (eV) based on B3LYP/6-311G* geometries and

B3LYP/6-311G* single-point calculations

Isomers Point group B3LYP/6-311G* B3LYP/6-311G*//

B3LYP-6.311G*

Ring C10h 0.0b 0.0

Bowl C5v 0.8 0.4

3-6loop D3d 2.0 1.5

Bowtie D2h 2.3 1.8

Cage D2h 2.5 1.9

Halfbow Cs 2.5 1.8

2-6loop-tri C1 2.6 2.1

c20-6om C2v 3.3 2.7

a Uncorrected total energies.b Total electronic energy ¼ 2761.6833 hartree.

Fig. 2. Graphical representation of the relative energies compiled in Tables 1 and 2 for the eight isomers of C20 based on the levels of theory considered in this

work.

C. Allison, K.A. Beran / Journal of Molecular Structure (Theochem) 680 (2004) 59–6362

Even at this higher level of theory, the relative stability of

the cage isomer is still receiving competition from four

other cyclic isomers using the hybrid exchange-correlation

functional. The single-point energy calculations at this level

of theory suggest that the cage, halfbow, and bowtie isomers

are nearly isoenergetic, with the 3-loop6 isomer lying only

0.3 eV lower in energy. The graphical depiction of the

relative energies suggest that by increasing the rigor of the

basis set, while utilizing the same DFT method, decreases

the difference in the total electronic energies of these

isomers.

The structural parameters for the ring, bowl, and cage

isomers obtained at this level of theory are in very good

agreement with that reported by Grimme et al. [10]. They

optimized the geometries of the three isomers at the MP2

level using a large polarized valence triple-z ([5s3p2d1f]

TZV2df) [25] atomic orbital basis set. For the ring isomer,

the geometry comparison between B3LYP/6-311G* and

Grimme et al., where their bond lengths are contained

within parenthese, yields alternating bond lengths of 1.226

(1.251) and 1.344 (1.337) A. The DFT correlation function,

therefore, views the electron distribution as being slightly

less uniform, accounting for the shorter ‘triple’ bond and a

slightly longer ‘single’ bond. Our studies show that the bond

lengths in the five-membered ring of the bowl isomer are all

1.428 (1.423) A. The length of the bonds extending from the

central pentagon into the six-membered rings is also

predicted to be 1.428 (1.434) A. The lengths of the external

bonds alternate between 1.241 (1.269) and 1.417 (1.411) A.

As with the ring, the DFT method exhibits a more localized

description of the electronic distribution, particularly along

the external bonds. Of the 15 different bond lengths reported

by Grimme et al. for the cage isomer, eight of the bond

lengths predicted using B3LYP/6-311G* are within

0.003 A, five bond lengths are with 0.015 A, and the final

two are within 0.028 A. As an aside, the same structural

parameters for these three isomers determined using the

B3LYP/6-31G method are only slightly less precise.

Consequently, there is not a dramatic gain in geometric

precision utilizing a more rigorous basis set when compared

to the geometric dimensions obtained with the B3LYP

method.

4. Conclusions

The present study focuses on evaluating the relative

stability of 24 unique isomers of the 20-carbon system.

Although this is only a small selection of the total possible

isomers, we have shown that based on this level of theory,

the ring structure with the C10h symmetry is the most

energetically stable. We have also shown that even though

there are several different potential symmetrical structures

for the cage isomer, the D2h symmetry is at least iso-

energetic with structures of lesser symmetry (C1 and Cs).

Of greater interest, however, we have determined that there

are several C20 isomers that are energetically competitive

with the cage isomer at this level of theory. The bowtie

(D2h) and the halfbow (Cs) isomers are nearly isoenergetic

to the cage, and the 3-6loop (D3d) isomer is predicted to be

0.4–0.5 eV lower in energy than the caged structure.

Additional investigations at higher levels of theory are

warranted to further evaluate relative energies of these other

cyclic structures.

Acknowledgements

This work was entirely supported by the Sponsored

Project Development Fund (SPDF) grant from The

University of Texas of the Permian Basin, Odessa, TX.

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