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
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.
Only recently the most energetically stable boron sheet,
the so called α-sheet , 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  whose theoretical study is
very important in the light of the recent experimental
verification . 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 . 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 . 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
. 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  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 .
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
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 . 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 . The PBE, PBE0, and X3LYP
functionals  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. )
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
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 .
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. . 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
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 .
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 . 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.
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 . 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 . 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. . 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
. 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 . 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. , (5,0) in Refs. [4, 11], and
(15,0) in Ref. . In this work we use the (n,0) notation
for zigzag nanotubes which is consistent with the indexing
scheme adopted in Ref. .
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.
(*) only the Ih-B80 cluster has been optimized
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 .
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.
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
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
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