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The Symmetry of the Boron Buckyball and a Related Boron Nanotube

N. Gonzalez Szwacki1 and C. J. Tymczak2*

1Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Hoża 69, 00-681 Warsaw, Poland

2Department of Physics, Texas Southern University, Houston, Texas 77004, USA

Abstract

We investigate the symmetry of the boron buckyball and a related boron nanotube. Using large-scale ab-initio calculations

up to second-order Møller–Plesset perturbation theory, we have determined unambiguously the equilibrium

geometry/symmetry of two structurally related boron clusters: the B80 fullerene and the finite-length (5,0) boron nanotube.

The B80 cluster was found to have the same symmetry, Ih, as the C60 molecule since its 20 additional boron atoms are located

exactly at the centers of the 20 hexagons. Additionally, we also show that the (5,0) boron nanotube does not suffer from

atomic buckling and its symmetry is D5d instead of C5v as has been described by previous calculations. Therefore, we predict

that all the boron nanotubes rolled from the α-sheet will be free from structural distortions, which has a significant impact on

their electronic properties.

*E-mail: tymczakcj@tsu.edu

1. Introduction

Only recently the most energetically stable boron sheet,

the so called α-sheet [1], has been theoretically described.

This sheet is closely related to the very stable boron

fullerene, B80, which is predicted to be the boron analog of

the famous C60 fullerene [2, 3]. The α-sheet is also a

precursor of boron nanotubes [4] whose theoretical study is

very important in the light of the recent experimental

verification [5]. The boron nanotubes had been investigated

theoretically (see Refs. [6-8] and references therein)

previous to the first boron tubular forms being synthesized.

Additionally, small planar and quasi-planar boron clusters

have also been extensively studied both experimentally and

theoretically [9]. Together, these efforts have made

possible the deeper understanding of the most likely stable

structure of all-boron nanotubes, fullerenes and sheets, but

more work still needs to be done.

The structural analogy between the B80 and the boron

nanotubes has been demonstrated in Refs. [10, 11].

Furthermore, Zope et al. have shown the link between B80

and the α-sheet [12]. Despite all the success of the

theoretical description of the B80 cluster, whose structure

“inspired” many other investigations, the description of its

symmetry is still controversial. The B80 cluster was

originally predicted to have the full icosahedral symmetry

[2]. These calculations were done using the DFT-GGA

approach. In a later publication, Gopakumar et al. have

shown that the symmetry of the boron structure is not Ih,

but instead, the cluster slightly distorts into the Th

symmetry [13] where two such structural distortions, called

A and B, have been identified at both Hartree-Fock (HF)

and hybrid B3LYP levels of theory. The symmetry of B80

was also addressed in several other later papers [14-16].

For instance, Sadrzadeh et al. demonstrated that in fact

there is not one but three isomers, of C1, Th, and Ih

symmetries, which are close in energy and have almost

identical structures [15].

The ambiguity in the description of the symmetry of the

B80 fullerene using pure DFT or hybrid approaches

motivated us to investigate the structure of this cluster

using

perturbation method (MP2). This method, although

computationally expensive, is characterized by a much

more accurate description of electron correlation effects

than the DFT or hybrid HF/DFT methods can achieve.

Additionally, we have extended our investigation to the

finite-length boron (5,0) nanotube using the MP2 approach.

the ab-initio second-order Møller–Plesset

2. Computational details

The calculations have been carried out using both

symmetry restricted and unrestricted methodologies. The

computations with restricted symmetry have been done

using the NWChem code suite [17]. We have done pure

DFT (BLYP), hybrid DFT (B3LYP), HF, and MP2

calculations. The vibrational analysis and IR spectrum

where obtained using tight convergence criteria. The

symmetry unrestricted calculations have been done using

the FreeON code suite [18]. The PBE, PBE0, and X3LYP

functionals [19] have been used for these calculations.

Additionally, the FreeON code suite has been used for

nudged elastic band calculation (using the B3LYP

functional) of the minimum energy paths (MEP) between

two B80 isomers of Th symmetry.

3. Results and discussion

3.1 Boron buckyball

Several tests for the structure and symmetry of B80 have

been performed at the MP2/STO-3G level of theory. First,

we did several computations at the B3LYP/STO-3G level

of theory. At this level, the total energy difference between

clusters confined to Ih and Th (isomer A from Ref. [13])

symmetries is ΔE= 36.41 kcal/mol (see Table I), with the

structure with Th symmetry being energetically more

favorable. The 20 atoms that are located above or below the

hexagonal rings of the B60 frame are divided in two groups

of 8 and 12 atoms. The 8 atoms are inside the frame with a

dihedral angle 16.9° and the 12 atoms are outside the frame

with a dihedral angle 7.2°. The optimized Th-B80 cluster is

shown in Fig. 1(a) (left). In this figure both groups of

hexagonal pyramid units are highlighted. The vibrational

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frequency analysis of the Ih structure gives rise to 7

imaginary frequencies whose values range from -287i to

-264i cm-1. This picture is qualitatively, and in some cases

quantitatively, similar to

calculations that had been done using the B3LYP/6-31G(d)

level of theory [13].

The next step was to optimize the two clusters obtained

at the B3LYP/STO-3G level of theory using the same bases

set and the ab-initio MP2 approach. The optimization of

the Th structure gave rise to a cluster of exactly the same

inter-atomic bonds, symmetry, and total energy as the

optimized Ih structure (this is not surprising since Th is a

subgroup of Ih). The optimized Ih-B80 cage is shown in Fig.

1(a) (right). To validate this result, we have randomized the

positions of the atoms of the Ih cluster making small atomic

displacements and after re-optimization the inter-atomic

distances and angles where within 0.001 Å and 0.1°,

respectively, of the Ih cluster.

In Fig. 2 we have shown the MEP between two B80

isomers of Th symmetry described in Ref. [13]. The cluster

images (structures) where first calculated at the

B3LYP/STO-3G level of theory and then for these

structures single energy calculations using HF, BLYP, and

MP2 methods where done. Interestingly enough, although

the B3LYP images are not the true images for the other

methods, still we obtained correct energy barriers for the

transition from isomer A, through Ih symmetry, to isomer

B. We found the energy barriers to be ~106 and ~25

kcal/mol for the HF and BLYP methods, respectively (see

the previously reported

Table I). As is also shown in Fig. 2 the HF, BLYP, and

B3LYP theories predict that the ground state structure is

isomer A, whereas, in contrast, the MP2 theory clearly

predicts that the ground state structure is of Ih symmetry

with an energy ~26 kcal/mol bellow the energy for isomer

B.

It is noteworthy to mention that the transition states

energy barriers shown in Fig. 2 decrease as basis set

completeness is approached for the HF and DFT theories.

Considering Table I, we see that the ΔE values (energy

barriers) for the BLYP or B3LYP functionals decrease with

increasing basis set completeness, e.g. the aug-cc-pVDZ

basis set energy barriers are very small which implies very

flat energy surfaces along the MEPs. This is more likely the

reason for the coexistence of several low-lying isomers that

are very close in energy as reported in the literature [20].

We believe that the ambiguity in the description of the

structure of B80 is a consequence of correlation effects that

cannot be fully captured by DFT. It is well known that DFT

does not always accurately describe correlations, especially

Van der Waals interactions [19]. If we assume that part of

the attractive interaction between the central boron atom

and the six-member boron ring is of dipole-dipole

character, then DFT will fail to capture these interactions

effectively, and higher-level correlation theories have to be

used in order to correctly predict the structure of B80.

To ensure that our predictions are independent on the

choice of the basis set, we have repeated the MP2

calculations but now with the 4-31G basis set. The starting

point for those calculations were the structures obtained at

the B3LYP/4-31G level of theory. Again the optimized Th

and Ih clusters have exactly the same structure and energy.

Our last step, in the MP2 calculations, was to optimize the

Ih structure at the MP2/6-31G(d) and MP2/cc-pVDZ levels

of theory. At those levels, the B80 cluster was found to have

the same topology as the C60 molecule since the 20

additional boron atoms are located almost exactly at the

centers of the 20 hexagons of the B60 frame.

We summarize our results in Table I where we show the

equilibrium inter-atomic distances of Ih-B80 for HF, BLYP,

B3LYP and MP2 levels of theory at increasing

FIG. 1 (a): The Th and Ih symmetries of the boron fullerenes, and (b) the

C5v and D5d symmetries of the (5,0) boron nanotubes. The structures at the

right (left) are optimized using the MP2/STO-3G (B3LYP/STO-3G) level

of theory. The central atoms of the highlighted in red and blue units are

closer or father away, respectively, from the center (or axis) of the cluster.

FIG. 2: The transition states for the A to B isomers for HF, BLYP,

B3LYP, and MP2 theory levels using a STO-3G basis set. Δr is the

difference between the radial distances of the boron atoms belonging to

the two groups of 8 and 12 atoms located near the centers of the hexagonal

rings of the B60 frame.

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completeness of the basis set. We also list the energy

difference, ΔE, between B80 clusters of Ih to Th symmetries.

From columns 3-5 of Table I, we can see that HF, BLYP,

and B3LYP methods clearly underestimate the hexagon-

hexagon (dhh) and overestimate the hexagon-pentagon (dhp)

B-B distances of the B60 frame respect to the results

obtained using the MP2 method. However, all the

approaches tend to give similar results for the distance, dc,

from an atom of the hexagonal ring to the central boron

atom. It is worth mentioning that the MP2/STO-3G and

MP2/6-31G(d) theories predict that all the B-B inter-atomic

distances are close to 1.71 Å, which is the bond length of

the flat boron triangle sheet [1, 10]. We can also see from

Table I that even for a very large basis set, aug-cc-pVDZ,

the theories, BLYP and B3LYP, predict that the Th

symmetry corresponds to the equilibrium geometry of the

cages (the HOMO-LUMO gaps are 0.99 and 1.93 eV at the

BLYP/aug-cc-pVDZ and B3LYP/aug-cc-pVDZ levels of

theory, respectively). However, it should be pointed out

that such small energy difference, between the Ih and Th

structures, is at the limit of the accuracy of the DFT and

hybrid approaches [21]. And finally we again mention that

the MP2 theory, for all the basis sets we have used, predict

that the Ih symmetry is the ground state symmetry of the

boron buckyball. In Fig. 3, we show the IR spectrum of the

B80 cage calculated at the MP2/4-31G level of theory. All

the obtained vibrational frequencies are, contrary to the

B3LYP/4-31G case, positive which is an additional

indication of the stability of the Ih-B80 cluster. The infrared

peaks at frequencies 372, 425, and 874 cm-1 correspond to

weak IR modes that are localized on the B60 frame and are

equivalents of the T1u(1), T1u(2) and T1u(3) modes,

respectively, present in the infrared spectrum of the C60

fullerene [22]. The strong modes at frequencies 329, 741

and 968 cm-1 involve all the 80 atoms of the boron cage,

although the first frequency (329 cm-1) is mainly localized

on 20 atoms that are located in the centers of the hexagons

of the B60 frame.

For completeness of our investigation, we have also done

computations unrestricting the symmetry of the structures.

For those calculations, we have used several functionals,

6-31G(d) and 6-311G(d) basis sets, and tight convergence

criteria. The results are essentially the same as reported in

Ref. [15]. Depending on the symmetry of the input

structures, after structural optimization we have obtained

clusters with symmetries close to Th or Ih. Although the

energy differences between those structures were very

small (within 2.7 kcal/mol), the Th cages were always the

lowest in energy.

3.2 Boron nanotubes

First principle calculations predict that boron nanotubes

are metallic or semiconducting depending on their radii [4,

20]. The band gap depends also on the chirality of the tube

[4]. The presence of the band gap in boron nanotubes with

diameters smaller than 17 Å is associated with the buckling

of certain number of boron atoms which occurs when the

α-sheet is rolled to obtain the nanotubes [4]. Structures

with smaller radii tend to have larger structural distortions

(buckling of atoms) and as a consequence larger band gaps

[4, 20]. It was also shown that without this buckling the

nanotubes would be metallic [11, 20]. The close similarity

between the structural distortions that suffer boron

fullerenes and nanotubes motivated us to investigate the

structure of a finite-length (5,0) boron nanotube using the

MP2 approach. This nanotube was reported to have one of

the largest band gaps (~0.6 eV) among all studied

nanotubes [4, 20]. It is also closely related to the B80 cluster

[10, 11]. It should be noted that in the literature we can find

three different indexing schemes for boron nanotubes and,

as a result, there are three different notations for the same

nanotubes: (5,5) in Ref. [20], (5,0) in Refs. [4, 11], and

(15,0) in Ref. [10]. In this work we use the (n,0) notation

for zigzag nanotubes which is consistent with the indexing

scheme adopted in Ref. [6].

For our investigation, we chose a fragment of the (5,0)

boron nanotube long enough, ~10 Å, to reproduce the

structural properties of an infinite nanotube, but still

computationally feasible. This finite-length nanotube, of

FIG. 3: The IR spectrum of the Ih-B80 fullerene. The vibrational

frequencies have been obtained at the MP2/4-31G level of theory.

TABLE I: Equilibrium inter-atomic distances of Ih-B80 at various levels

of theory. dhh and dhp refer to hexagon-hexagon and hexagon-pentagon

bond length, respectively. dc refers to the distance from an atom of the

hexagonal ring to the central boron atom. The six-member ring and the

central atom define a dihedral angle that is also provided. A negative

(positive) angle means that the central atom is shifted towards (away

from) the center of the cage. ΔE is the total energy difference between

clusters confined to Ih and Th (isomer A) symmetries.

NN interatomic

distances (Å)

dhh dhp

HF

1.607 1.752

1.652 1.762

1.656 1.774

BLYP

1.675 1.748

1.684 1.739

1.691 1.751

1.688 1.749

B3LYP

1.654 1.741

1.669 1.732

1.677 1.746

1.672 1.745

MP2

1.709 1.713

1.735 1.713

1.711 1.706

1.745 1.704

ΔE

(kcal/mol)

dc

dihedral

angle

STO-3G

4-31G

cc-pVDZ

STO-3G

4-31G

cc-pVDZ

aug-cc-pVDZ

STO-3G

4-31G

cc-pVDZ

aug-cc-pVDZ

STO-3G

4-31G

6-31G(d)

cc-pVDZ

(*) only the Ih-B80 cluster has been optimized

106.42

63.12

14.87

1.710

1.716

1.731

-11.3°

-6.5°

-8.7°

24.68

7.89

0.13

0.20

1.719

1.712

1.724

1.721

-5.9°

-1.1°

-3.6°

-3.4°

36.41

13.46

0.15

0.11

1.707

1.701

1.715

1.712

-6.8°

-1.4°

-4.3°

-4.3°

0.0

0.0

*

*

1.712

1.725

1.708

1.725

-1.7°

+2.1°

0.0°

+1.5°

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110 atoms, is shown in Fig. 1(b). To minimize the effects

of the edges on the calculations the tubular fragment has

boron double rings at the edges. The cluster has been first

optimized at the B3LYP/STO-3G level of theory. The

resulting structure, of C5v symmetry, is shown in the left

part of Fig. 1(b). In this figure, we have highlighted a

central fragment of the cluster which has a diameter of 8.47

Å (B3LYP/431G: 8.44 Å; B3LYP/cc-pVDZ: 8.42 Å), a

close value to that reported for an infinite nanotube [20].

We can also see from that figure that there are two groups

of nonequivalent hexagonal pyramid like units, which are

building blocks of the nanotube (this was also reported for

an infinite (5,0) nanotube [4, 11, 20]). Next, we have

farther optimized this structure at the MP2/STO-3G level

of theory and found that the equilibrium structure is of D5d

symmetry (see Fig. 1(b) (right)). We have done several

tests to ensure that the predicted equilibrium structure is a

true local minimum of energy. The increase in cluster

symmetry is accompanied by a flattening of the hexagonal

pyramid units that are now all equivalent. This result may

have important consequences on the electronic properties

of boron nanotubes. We predict that not only the (5,0)

nanotube is metallic, but it is very likely that the rest of the

boron nanotubes that were previously classified as

semiconducting do not suffer from structural distortions

(buckling of atoms) and as a consequence are, in fact,

metallic. This is important for nanotechnology since it

indicates that boron nanotubes would be superior

candidates, as opposed to carbon nanotubes, for electrical

interconnects in nano-electronics.

At the limit of a very rich basis set the B3LYP results

tend to be qualitatively the same as those obtained using

the MP2 method and the minimal basis set. Indeed, the

energy difference between the structures of D5d and C5v

symmetries is only 2.34 kcal/mol at the B3LYP/cc-pVDZ

level of theory, whereas for the STO-3G and 4-31G basis

sets the values are 40.02 and 10.33 kcal/mol, respectively.

This result is expected since more complete basis sets favor

better description of correlation effects. Finally, it should

be mentioned that the HOMO-LUMO gap slightly

decreases (B3LYP/cc-pVDZ: from 0.81 to 0.79 eV for

clusters of C5v and D5d symmetry, respectively) with

increasing symmetry of the tubular cluster.

4. Conclusions

In conclusion, we have done extensive calculations at

various levels of theory to determine the equilibrium

geometry of the B80 cage and a related boron nanotube. We

have determined that the equilibrium geometry of B80 is Ih,

the same as for C60, and of the boron nanotube is D5d. From

these results, we have asserted that a high level description

of the correlation effects is essential for the correct

description of the structure of these and other hollow boron

nanostructures.

Acknowledgment

We would like to thank the Robert A. Welch Foundation

(Grant J-1675) for their support of this project. The authors

would also like to acknowledge the High Performance

Computing Center (URL: http://hpcc.tsu.edu/) at Texas

Southern University for providing resources that have

contributed to the research results reported within this

paper.

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