Theor Chem Acc (2017) 136:20
DOI 10.1007/s00214-016-2042-2
REGULAR ARTICLE
Studying lowest energy structures of carbon clusters
by bond‑order empirical potentials
S. K. Lai1 · Icuk Setiyawati2 · T. W. Yen1 · Y. H. Tang2 Received: 2 August 2016 / Accepted: 16 December 2016
© Springer-Verlag Berlin Heidelberg 2016
Abstract A very recently developed optimization algorithm for carbon clusters (Cns) (Yen and Lai J Chem Phys
142:084313, 2015) is combined separately with different
empirical bond-order potentials which were proposed also for
carbon materials, and they are applied to calculate the lowest energy structures of Cns studying their structural changes
at different size n. Based on predicted structures, we evaluate
the practicality of four analytic bond-order empirical potentials, namely the Tersoff, Tersoff–Erhart–Albe, first-generation
Brenner and second-generation Brenner (SGB) potentials.
Generally, we found that the cluster Cn (n = 3–60) obtained
by the SGB potential undergoes a series of dramatic structural
transitions, i.e., from a linear → a single ring → a multi-ring/
quasi-two-dimensional bowl-like → three-dimensional fullerene-like shape; such variability of structural forms was not seen
in the other three potentials. On closer examination of the Cns
calculated using this potential and further comparing them with
those obtained by the semiempirical density functional tightbinding theory calculations, we found that these Cn are more
realistic than similar works reported in the literature. In this
respect, due to its potential applications in the study of chemically complex systems of different atoms especially chemical
reactions (Che et al. Theor Chem Acc 102:346, 1999), the SGB
potential can, moreover, be used to investigate larger size Cn,
and calculated structural results by this potential are naturally
input configurations for higher-level density functional theory
* S. K. Lai
[email protected]
1
Complex Liquids Laboratory, Department of Physics,
National Central University, Chungli 320, Taiwan
2
Computational Materials and Nanoscale Transport Groups,
Department of Physics, National Central University,
Chungli 320, Taiwan
calculations. Another most remarkable finding in the present
work is the Cn results calculated by Tersoff–Erhart–Albe
empirical potential. It predicts a two-dimensional development
of graphene structure, exhibiting always a zigzag edge in the
optimized clusters. This empirical potential can thus be applied
to study graphene-related materials such as that shown in a
recent paper (Yoon et al. J Chem Phys 139:204702, 2013).
Keywords Carbon cluster · Optimization algorithm ·
Topological transition · Fullerene
1 Introduction
The structural and electronic distributions of a cluster consisting of n atoms depend on how the valence electrons are
coupled to their oppositely charged ions. These Coulombic interactions among electrons and ions are inherently
quite intricate. To account for these inherent many-body
interactions, several practical analytic models such as the
Gupta potential [1], embedded atom method model [2, 3]
or Finnis–Sinclair potential [4, 5], Sutton–Chen potential
[6], glue potential [7] and quantal distance-dependent tightbinding model [8] were proposed in the literature, mainly
devoted to metallic clusters. All of these analytical empirical potentials fit parameters in their energy functions to
bulk properties and have been applied successfully to study
bulk solid-state properties and to understand as well various equilibrium properties of metallic clusters. From the
characteristics of the electronic bondings, these empirical
models should be less well appropriate for studying covalent clusters such as silicon, boron or carbon clusters. In
fact even in metallic clusters, the above empirical potentials fail disastrously also to describe the structural traits of
those clusters whose valence electrons are playing delicate
13
20 Page 2 of 13
and subtle roles in driving them to undergo varied forms
of shape transformation. Well-known example is the smallsized metallic gold cluster which was reported [9] to show
a two- to three-dimensional transition. We have examined
this extraordinary structural transition using one of the
above-mentioned empirical potentials; our calculations
failed to predict this dimensionality turnover. For covalent
clusters, extra efforts are certainly required, for now we
have to include more quantitatively the complicated bondorder interactions.
Abell [10] reported one very early quantum-based theory for studying molecular clusters. He was inspired by
an illuminating observation of Ferrante et al. [11, 12] that
a universal relation exists between the binding energy
and interatomic distance, and showed in his work [10]
that such a bonding relation prevails over disparate systems as molecules and simple metals. Applying the Austin-type pseudopotential theory [13] which he followed
closely the works of Anderson [14] and Weeks et al. [15],
Abell [10] derived a binding-energy expression that provides the theoretical framework for a very general study
of structural and atomistic properties of covalent systems. In the following years, Tersoff [16–20] independently fleshed up Abell’s method [10] by introducing a
general analytic expression suitable for numerical computations. The explicit parametrized form of Tersoff’s
formula found justification in his subsequent efforts to
describe in group IV elements their bonding-related
properties in a number of surface and solid-state calculations of defect energies and in ambient and high-pressure
phases of these elements and their alloys [16–20]. This
solid-state or bulk-based empirical scheme of Tersoff is
impressive and apparently quite successful because it
motivated Brenner [21, 22] to develop his first-generation
Abell–Tersoff or Brenner (FGB) empirical potential in
the following year. Brenner tested this potential by applying it to molecular mono-elemental systems of carbon,
hydrogen and oxygen and also to reactive molecular solids [23, 24]. His calculated results are encouragingly successful; these studies and diverse applications by many
others [25] in nearly a decade have paved his way to formulate the SGB potential [26]. A few years later, Erhart
and Albe [27] (to be referred to as TEA) proposed a simpler but more practical bond-order analytic expression
for studying covalently bonded materials as demonstrated
for the carbon system by Jakse et al. [28] and Yoon et al.
[29] in their applications of this potential to grow by
molecular dynamics (MD) simulation the epitaxial graphene layers on the silicon carbide substrate. Since then,
different empirical bond-order expressions have been
published (see Ref. [26] for some earlier bond-order
forms before 2002, Ref. [30] for subsequent bond-order
forms in years 2002–2008 including perhaps Ref. [31,
13
Theor Chem Acc (2017) 136:20
32], Ref. [33]1 for the analytical quantum-based bondorder potential, and the very recent review of bond-order
potential in the context of density functional theory
(DFT) by Drautz et al. [34]). Many of these empirical
bond-order potentials were developed for the understanding of various properties of carbon-/silicon-/hydrogenrelated materials. We should perhaps mention also that a
more refined SGB-type bond-order empirical potential
has appeared four years ago [35]. However, this latter
bond-order empirical potential has yet to be critically
evaluated for carbon clusters.
Despite a more accurate study of Cn has very recently
been done [36], we notice that the empirical bond-order
potentials continue to receive much attention in the literature [33, 34, 37] due mainly to their robustness in investigating chemically complex entities of different atoms,
and in calculating the properties of large statistical systems
especially their applications to chemical reactions and barriers [38] by MD simulations. Such studies are not possible
or impractical by using the density functional tight-binding
(DFTB) theory method [36], although the latter approach is
elegant in producing more accurate structures of Cn. Motivated by these observations, we shall, in this paper, examine four specific bond-order empirical potentials reported
independently by Tersoff [20], Erhart and Albe [27], and
Brenner and coworkers [21, 22, 26], perform a systematic
study of Cns at different size n, scrutinize their respective
shape change and compare the varied forms of structural
changes of Cn among themselves and with other theoretical calculations. To ensure the consistency and quality of
energy optimization, each of these empirical potentials
was combined with the modified basin hopping (MBH)
method which is developed by us [36] very recently and
used to calculate the optimized structures of Cn within the
DFTB framework. The interested readers are referred to
Ref. [36] for further technical details and on how the BH
was skilledly modified for covalent clusters. In this work,
we perform a thorough analysis of the structural evolution
of Cn obtained for each selected bond-order potential given
that the MBH algorithm has been critically evaluated [36].
The calculated Cn will be compared among themselves and
with other theoretical works applying the same bond-order
potential. Through these careful diagnoses, we shall evaluate the appropriateness and suitability of the bond-order
empirical potential in finding the optimized lowest energy
minima of Cn. These putative global energy minimum
structures which we obtained by an unbiased optimization
method can be considered also as initial configurations in
1
Zhou et al. [33] developed their quantum-based bond-order
potential from a series of works by Pettifor DG and coworkers (see
this reference for references cited therein).
Page 3 of 13 Theor Chem Acc (2017) 136:20 more quantitative DFT calculations of the electronic and
magnetic properties.
This paper is organized as follows. In the next section,
we shall present a general expression of the bond-order
potential from which we summarize, in Appendix 1, a few
specific empirical potentials indicating their key features
and differences. Then, in this same Sect. 2, we briefly
describe the BH technique and the modifications that we
have made. In Appendix 2, we give more details of the
MBH method by a flowchart. Section 3 presents our calculated Cn results for the Tersoff and TEA potentials in the
size range n = 3–20, and for the FGB and SGB potentials
in the size range n = 3–60, 72. Main comparisons of our
calculated Cns were made to only works done by Hobday
and Smith [39, 40]2 and Cai et al. [41] because these
authors performed a systematic studies of Cn structures
employing also the same potential as us [42–46].3 In this
same Sect. 3, we discuss furthermore the consequences of
different bonding characteristics and conduct more thorough analysis on the findings of Kosimov et al. [47, 48]
who used the SGB potential to calculate the Cn structures.
To our knowledge, these are the only calculations that have
performed more extensive studies of Cn employing [39–41,
47, 48] the same Brenner empirical potentials as us in the
same size range. Any similarity/difference of results therefore tells apart also the accuracy of the MBH method used,
and hence, the possible conclusions reach. The quality of
the calculated Cn will thus shed light on the appropriateness of these bond-order empirical potentials in structural
studies given an optimization algorithm. Finally, in Sect. 4,
we give a recapitulation of this work.
2 Empirical potential for a covalent cluster
In this section, we give a general expression of an empirical potential for studying covalent systems. This empirical
2
The lowest energy values in Hobday and Smith [40] have been
improved by the Hobday and Smith [39].
3
There are several calculations in the literature also employing the
FGB potential to study Cn clusters either for a particular size n or in
a specific range of n. Among them, Halicioglu [42] considered Cn < 6
using exactly the same Brenner potential expressions as us. The binding energies of the few Cn obtained by him are generally higher than
ours due to his crude means of energy minimization. Others such as
Wang et al. [43] applying a time-going-backward quasi-dynamics
method were less systematic in structural studies of Cn or Zhang
et al. (Ref. [44] below) who used genetic algorithm associated with
simulated annealing method, and Yamaguchi and Maruyama [45,
46] performing molecular dynamics simulations; all these authors
employed a modified FGB potential in which πijRC (see Eq. (10)) was
omitted and their calculated Cns were therefore inappropriate for
direct comparison with our results.
20
potential is cast in a form that readily leads to specific
interaction potentials (see Appendix 1) to be used in calculations of the lowest energies of Cn. Each of these seemingly disparate interaction potentials will be combined with
the same MBH optimization algorithm.
2.1 General form of an empirical potential for covalent
systems
As mentioned in Introduction, the bond-order empirical potentials put forth independently by Abell [10] and
Tersoff [16–20] were quite general interaction potentials because they have been formulated on a fundamental basis. These potentials have been widely used to study
many properties of covalent systems, even applying very
recently to understand the melting of nanoparticles varying in size [49]. In the following, we present its most
general form which we show in Appendix 1 that it leads
quite naturally to four different forms of bond-order potentials depending on how the parameters in the latter were
determined.
A general expression of the Abell–Tersoff empirical
potential is given by
Ei
E = (1/2)
(1)
i
where Ei =
j� =i
V (rij ) which is written
V rij = f C rij VR rij + bij VA rij
(2)
is the total potential energy of atom i, in which rij is the
ij bond-length distance. The repulsive VR(rij) and attractive VA(rij) parts are usually expressed in the form of Morse
potential, i.e.,
(0)
VR (rij ) = A exp −(rij − rij )
(3)
(0)
VA (rij ) = −B exp −µ(rij − rij )
(0)
where A, B, λ, μ and rij are constant parameters. A brief
description of Eq. (2) is in order. In the first place, bij measures the bond order describing the coordination of atoms i
and j. It has been cast in the general form [16, 17, 19]
bij = (1 + αiℓi ςijℓi )−1/(2ℓi )
(4)
where αi and ℓi are constant parameters and are set equal to
one in all of the empirical potentials studied in this work.
Next,
ςij =
fC (rik )g(θjik ) exp ν m (rij − rik )m ,
(5)
k� =i,j
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20 Page 4 of 13
Theor Chem Acc (2017) 136:20
Fig. 1 Schematic diagram
showing a: the original PES
and b: its transformed staircase
PES. The V(X) is the original
potential energy, and Ṽ (X) is the
transformed one
in which m and ν are also constant parameters and g is the
three-body interaction function which reads [20, 27]
2
2
cik
cik
g(θjik ) = γik 1 + 2 − 2
,
(6)
dik
dik + (hik − cos θjik )2
where θjik is the bond angle between bonds ij and ik for any
atom at k (≠ i, j) that is bonded to atom i, and constants γik,
cik, dik and hik are accordingly determined by three-body
interactions. The ςij therefore gives contributions of all
other bonds which are indexed by k to atom i (besides the j
atom in the ij bond); both the bond angle (between bonds ij
and ik) and bond length ik must be explicitly included. We
should mention that the bij defined by Eq. (4) is not equal
to bji since the bond angle ijk at atom j is in general not
equal to jik at atom i for k ≠ i, j; they are generally different because of their respective difference in coordination neighbors. Finally, the fC in Eq. (2) is a cutoff function
varying continuously from 1 at the first-neighbor shell R
decreasing to 0 at S. The R and S are thus the positions of
the inner and outer ranges, respectively. Following Tersoff
[16, 17, 19], we choose

 1,
�
� rij < R
f C (rij ) = 21 + 21 cos π(rij − R)/(S − R) , R < rij < S

0,
rij > S
(7)
where both i and j stand for carbon atoms. The fC therefore
describes the cutoff distance of an atom k located at the
first-neighbor shell of atom i (or atom j for bji) and it spans
the width (S–R). Physically, it limits the range of the covalent interactions and ensures that the interactions between
C atoms included only the nearest neighbors [50, 51].4 In
4
Extension to include long-range interactions has been reported by
Stuart et al. [50] (see also references cited therein), and in a more
recent work by Monteverde et al. [51].
13
Appendix 1, we describe briefly the four bond-order potentials, i.e., Tersoff [20], Tersoff–Erhart–Albe [27], FGB [21,
22] and SGB [26] and how they separately come from the
same Eqs. (1)–(3).
2.2 Optimization methodology: modified basin hopping
technique
The BH method [52, 53] is a technique developed for finding optimized structures of metallic and nonmetallic clusters. The basic idea of the method is to fully exploit the
potential energy surface (PES) V(r1, r2, …, rn) = V(X) of
say n atoms (Fig. 1a) whose potential barriers are strategically truncated (Fig. 1b) to become a transformed PES,
Ṽ (X). The resulting Ṽ (X) contains therefore only the local
energy minima among which should, in principle, include
the lowest energy minimum. In other words, in this Monte
Carlo-based method, the original PES V(X) of position
coordinates X = {r1, r2, …, rn} of n atoms is first transformed into a multi-dimensional staircase topography Ṽ (X)
given by
Ṽ (X) = min[V (X)]
(8)
where the min denotes local energy minima, which are
here obtained by effecting L-BFGS method [54]. Then, the
canonical Monte Carlo (MC) simulation is applied to the
transformed staircase Ṽ (X) to search for its lowest energy
value. Technical details of its operation are described in
Refs. [52, 53] or in our previous works [55, 56].
For Cn, the valence-electron-sharing interactions among
carbon atoms are far more complicated; the BH method
described above is no longer useful especially when it is
combined with the sp-type empirical potential which does
not account so well for the directional bondings. In our
very recent work [36], two modifications were imposed on
the BH method. The first modification is the
√introduction of
a confinement radius Rd∗ = ξ [1 + (3n/4π 2)1/3 ]r0 where
ξ is an adjustable variable and r0 is the nearest neighbor
Theor Chem Acc (2017) 136:20 distance. When ξ = 1, the adjustable radius Rd∗ corresponds
to the case of a face-centered cubic packing for a cluster of size n and is the value customarily used for studying sp-type metallic clusters. For an efficient and realistic
determination of the lowest energy structure of a carbon
cluster, we find that it is crucial to restrict it to lie within a
“compressed” region (ξ < 1) as discussed and demonstrated
earlier by Hobday and Smith [40, 39] and very recently by
Yen and Lai [36]. The second modification introduced to
increase the searching performance is to amend the angular
displacement and random move procedure used to generate
subsequent new configurations of atoms in BH algorithm
with an additional cut-and-splice genetic-operator-(GO-)
like operation. We refer the interested readers to Appendix
2 for the flowchart of this MBH operation. More technical
description is given in Ref. [36].
3 Numerical results and discussions
We have applied equations given in Sect. 2 above to calculate the lowest energy structures of Cn employing Tersoff
[20], TEA [27], Brenner first-generation [21, 22] and second-generation [26] bond-order empirical potentials (see
Appendix 1). In all of these calculations, the BH method
with only the adjustable constant ξ, except for a few clusters, is sufficient to locate the lowest energy minima of
Cns. For Cn obtained by Tersoff and TEA potentials, we
consider Cn for size n = 3 up to 20 since these two potentials are somewhat approximate in their fitting of parameters to their respective version of Eq. (1) (see Appendix
“Tersoff empirical potential”). As the physical content of
valence-electron-shared bondings becomes quantitatively
more realistic, as in the constructed first and second generations of the Brenner potentials, we begin to see varied
changes of Cn structures some of which are in line with
available experiments and with other more quantitative theoretical results. For Cn calculated by the two generations
of the Brenner potentials, we began at n = 3 and went up
to n = 60. We have checked that the MBH optimization
algorithm works efficiently for all of Cn clusters yielding
equal or even lower energy results than those published
previously [39–41, 47, 48] using the same FGB and SGB
potentials.
Before presenting our results of calculations, it is
instructive to present two case studies in order to show the
robustness of our MBH algorithm. To this end, we have
applied the MBH method together with the SGB potential to find the lowest energy values for C60 and C72. The
computing time used for successfully obtaining optimized
C60 with ξ = 0.92 is within 5000 Monte Carlo steps (one
cyclic computation), whereas for C72 with the same input
ξ two cyclic computations are required. Both the isolated
Page 5 of 13 20
Fig. 2 The carbon clusters C60 (left) and C72 (right) calculated
as described in Ref. [36] using the SGB potential applying MBH
method without the cut-and-splice genetic operator included. The isolated pentagon rule for C60 goes all around the cluster, whereas for
C72 the violation of the rule (with two adjacent pentagons) is positioned to show it clearly. The pentagonal carbons are colored purple
for easy visualization
[57] and violation of isolated [58] pentagonal rules are confirmed in C60 and C72 (Fig. 2), respectively. We now present
our Cn results calculated with the four empirical bond-order
potentials.
3.1 Tersoff and TEA potential: clusters Cn=3–20
Figure 3 displays the lowest energy structures of Cn calculated using the Tersoff [20] and TEA [27] empirical potentials (see Appendix 1) which are separately combined with
the MBH optimization algorithm without including the cutand-splice GO-like operation for most of the clusters. The
values of their lowest energies are displayed in Fig. 4 [Tersoff [20] (green solid circles), TEA [27] (red full line)]. A
quick glance at Fig. 3 shows that none of these potentials
predicts a linear structure although they do yield planar
structures for some of Cn (C5, C6, C10, C13, C16 for both
potentials but the TEA has the C19 as well). The Tersoff and
TEA potentials predict a single ring structure first showing up at n = 5, and both potentials continue this planar
topology only up to 6 because their respective next cluster changes to a wavy planar geometry at n = 7. These latter wavy planar structures of both potentials also only last
till n = 8 and their respective cluster then turnovers to a
shape resembling two seven-atom planar single rings with
both rings clinging side-by-side and bending into an angle
to become a “λ-shape” structure at n = 9. At n = 10, the
Tersoff and TEA potentials yield two planar hexagons
sharing one side, and their clusters C11 return to the same
“λ-shape” geometry as n = 9 except that each of the two
planar single rings now has eight carbon atoms. One sees
a first sign of cage-like signature predicted by TEA potential at n = 12, whereas for the Tersoff potential it predicts
a triple-ring bowl-like structure. At n = 13, both potentials
predict a structure showing a planar triple hexagonal rings.
13
20 Page 6 of 13
Theor Chem Acc (2017) 136:20
Fig. 3 Carbon clusters Cn
calculated using Tersoff (left
column) and TEA (right
column) bond-order potentials
for size n = 3–20. These Cn
were obtained using the MBH
without the cut-and-splice
genetic operator following the
procedure described in Ref.
[36]. Notice that the optimized
Cn by TEA displays 2D graphene structures for n = 6, 10,
13, 19, …, each with a zigzag
peripheral edge
From this n onward, the Tersoff potential continues with
its bowl-like (n = 14, 15, 17) or graphitic planar (n = 16)
geometries and it captures its first cage-like structure at
n = 18 and remains to be cage-like for C19 and C20. The
TEA potential, on the other hand, predicts similar multiring geometries in the range n = 14–16, with a slightly
twisted plane at n = 14, bowl-like shape at n = 15, and the
same graphitic planar geometry as the Tersoff potential at
n = 16. We see that for the TEA potential the real cage-like
structure occurs at n = 17, at a cluster size earlier than the
Tersoff potential. This cluster C17 and the next cluster C18
13
display cage-like topologies following that at n = 12, and
at n = 19 the TEA potential shows again a planar hexagonal rings. The cluster evolution sequence of the TEA potential manifests a multi-ring bowl-like structure at n = 20
quite different from that of the Tersoff potential.
There are two aspects of the TEA potential that deserve
emphasis. Firstly, the multi-ring bowl-like structure at
n = 20 obtained by this potential has much relevance to
C15 if one scrutinizes Fig. 3. The C20 there is in fact an
extension of the C15 (in the orientation displayed) with
five atoms zigzagging along the peripheral edge on top.
Theor Chem Acc (2017) 136:20 Page 7 of 13 20
the Tersoff and TEA empirical bond-order potentials has
indicated unambiguously the role played by sp1 bonding
since no linear structure is predicted by both potentials.
Proper account of bji in addition to bij is necessary for
locating structure whose origin is associated with sp2 such
as the 2D graphene layer predicted in calculations using the
TEA potential. Nevertheless, these two potentials by and
large yield similar Cn structures except for C4, C8, C12 and
C17–C20.
3.2 Brenner first‑ and second‑generation potentials:
clusters Cn=3–60
Fig. 4 Lowest energy values determined by combining separately the
MBH with Tersoff [20] (green solid circles), TEA [27] (red full line),
first-[21, 22] (solid triangles) and SGB [26] (open circles) empirical
potentials. The data for C20, C22–C25 were obtained with cut-andsplice genetic-operator-like operation included in the MBH. Note
that the energy values of C15, C16, C17 and C39 are lower than those
obtained by Cai et al. [41]; in particular, the C15 and C16 structures
display, respectively, a linear and a spiral-like linear shapes instead of
same single rings in both clusters by Cai et al. Units of energy are in
10−19 J. The energy of C72 is −837.8284323 × 10−19 J lying outside
the energy range of this figure and hence not included. Numerical
data of all these energies are available on request
These clusters C15 and C20 constitute a portion of the buckminsterfullerene shown in Fig. 2. The bowl-like structure
of C20 is very similar to our very recent calculation performed within the DFTB theory calculation except that
here it assumes a larger curvature contrasting to the smaller
one obtained by Yen and Lai [36]. Secondly, we notice a
regularity in the predicted geometries for n that runs as
C6 → C10 → C13 → C16 → C19 displaying the sequence of
2D graphene growth and these changes in Cn give one concrete piece of evidence supporting our previous successful
growth of epitaxial multilayer graphene on silicon carbide
by the simulated annealing technique [29]. This sequence
of 2D graphene structure continues in fact with C22, C24,
C27, …, always taking on a zigzag edge in the optimized
clusters. We have not, however, pursued it further in this
work. In summary, suffice it to say that the thorough comparison of the optimized Cn structural results obtained by
We come to study the structural development of Cn
obtained in the context of the FGB and SGB potentials.
For clusters C15–17 and C39, our present calculations using
the FGB potential in combination with the MBH algorithm
yield energy values (solid triangles in Fig. 4) lower than
those obtained by Cai et al. (Table 3 of Ref. [41]) applying exactly the same FGB potential and, for all other clusters, their calculated energy values and ours are in excellent
agreement. We do not therefore depict the geometries of Cn
(readers interested may refer to Fig. 5 of Cai et al. [41]).
In Fig. 5, we show all of Cn calculated using the SGB
potential and display in the same Fig. 4 (open circles) their
lowest energy values. Compared with the Tersoff and TEA
potentials, the SGB potential shows more varied forms
of shape transitions. For C3–C5, the cluster is linear and it
changes into single ring structure for C6–C13. At n = 14, we
see the first planar triple rings geometry and then the planar
quadruple topology for n = 15–17, and the Cn transforms
into planar higher multiple rings (penta, hexa rings, etc.)
up to n = 24 except at n = 20 where it assumes a bowl-like
geometry. The first fullerene is captured at n = 25 which differs from n = 24 predicted by Yen and Lai [36], and the Cns
for n > 25 continue as fullerene-like cages until at n = 60 at
which size it becomes a highly symmetrical buckminsterfullerene. At this point, one can see the differences between
TEA and SGB potentials. The planar double-rings structure
predicted in the former at n = 10 is not seen in the latter.
As mentioned above, the cage-like geometry first appears at
n = 25 in the latter, whereas it is at n = 17 in former.
Generally, the optimized Cn predicted by the SGB potential is even more varied in the structural transition and these
topological variations are summarized in Table 1. As can
be read there, the SGB potential predicts planar multi-ring
structures at a much earlier size at n = 14 compared with
the n = 19 in Yen and Lai [36]. Notice in particular that
the geometries of Cn in the size range n = 20–23 between
the SGB potential and those obtained by the DFTB theory
(Fig. 2a in Ref. [36]) are remarkably similar. This size
range n has been pointed out also by Hobday and Smith
[39, 40] that they are clusters requiring proper attention
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20 Page 8 of 13
Theor Chem Acc (2017) 136:20
Fig. 5 Carbon clusters Cn calculated using SGB bond-order potential
for the range of sizes n = 3–60. Most of these Cn were obtained using
the MBH without cut-and-splice genetic operator, i.e., for n = 3–19,
21, 22, 26–58. For clusters C20, C23–C25, and C60 our calculations
show that the MBH supplemented by cut-and-splice genetic operator
yield lower energy values
Table 1 Topologies of the lowest energy structures of carbon clusters
Cn obtained by the SGB potential with MBH (without cut-and-splice
genetic operator) for C3–C19, C21–C22, C26–C58 and MBH with cutand-splice genetic operator for C20, C23–C25 and C60 according to the
flowchart given in Appendix 1
n
Geometry
n
Geometry
n
Geometry
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Linear
Linear
Linear
PSR
PSR
PSR
PSR
PSR
PSR
PSR
PSR
PTR
PQR
PQR
PQR
PPR
PPR
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
PH6R
PH6R
PH7R
PH7R
3DC
3DC
3DC
3DC
3DC
3DC
3DC
3DC
3DC
3DC
3DC
3DC
3DC
39
40
42
44
46
48
50
52
54
60
3DC
3DC
3DC
3DC
3DC
3DC
3DC
3DC
3DC
Buckminsterfullerene
20
BHR
38
3DC
PSR planar single ring, PTR planar triple rings, PQR planar quadruple rings, PPR planar penta rings, BHR bowl-like hexa rings, PH6R
planar hexa rings, PH7R planar hepta rings 3DC three-dimensional
cage-like
13
Fig. 6 The binding energy per atom for the FGB (open squares) and
SGB (solid squares) potentials optimized by the MBH algorithm Ref.
[36] compared with Kosimov et al.’s [47, 48] (solid circles) results
obtained by the SGB potential. The crosses are experimental data [59]
to be paid, i.e., these clusters Cn cluster must be confined
to lie within “compressed” region (see Sect. 2) if reliable
energy minima were to be located.
The SGB potential has been applied more recently by
Kosimov et al. [47, 48] who studied the lowest energy
minima of Cn by ad hoc restricting the minimization region
to be either planar or non-planar, depending on the size n.
Theor Chem Acc (2017) 136:20 For typically 500 initial random configurations, they then
sought for the energy minimum of each Cn by performing
the conjugated gradient minimization [54]. Based on this
procedure, they calculated Cn for n = 2–55. In Fig. 6, we
present our calculated results (solid squares) together with
theirs (solid circles) [47, 48]. As far as the energy minimum
values are concerned, those reported by Kosimov et al. are
systematically and substantially higher than ours, starting from n = 14 up to 55. These disparities in the lowest
energy values of Cn are by no means small, and the magnitudes explain the blatant differences in cluster geometries
between Cn obtained by them and ours (cf. Fig. 5 or Table 1
and Fig. 1 in Refs. [47, 48]). This comparison is instructive and a relevant comment appears in order. In their
works, Kosimov et al. [47, 48] have compared their calculated binding energies with the theoretical calculations of
Zhang et al. [44] who used the FGB potential instead of
the SGB potential. Strictly speaking, this energy comparison by Kosimov et al. is unsound even though the authors
skilledly ignored the πijRC term in Eq. (10) as Zhang et al.
[44] did. The reason is that the contents of the total energy
expressions in these two generations of Brenner potentials
differ somewhat not only in the functional forms used for
fitting but also in details of data chosen to determine the
parameters. Furthermore, the SGB potential includes πijDH
which is none in the FGB potential. Attributing their calculated lowest energy minima higher than those obtained
by Zhang et al. (see Fig. 2a of Refs. [47, 48]) just based on
the omission of πijRC term in the SGB potential is an inappropriate explanation and can not justify the reasonableness
of their calculated Cn. In our opinion, one possible cause
that led Kosimov et al. obtaining comparatively higher
binding energies of Cn and hence different geometries,
apart from the uncertainties mentioned above, is due also
to their crudeness in the energy optimization. To demonstrate this point, we display also in the same Fig. 6 our calculated lowest energy values for the FGB potential. We find
that the SGB potential (at the same quantitative level as
that used by Kosimov et al. [47, 48]) relative to that of the
FGB in which both generations include πijRC [see Eq. (10)]
still yields energy values of comparable order and it is
even lower especially for n > 40. At this point, we should
emphasize that we make no attempt to compare thoroughly
with the Cn results obtained by Kosimov et al., apart from
their approximate means in the determination of lowest
energy structures, but also because these authors did not
include the πijRC term in the SGB potential.
In the present work, we do not comment either on the
differences in magnitudes of energy values between the
FGB and SGB potentials because, for a given empirical
potential, the lowest energy minimum should be unique,
independent of any optimization algorithm used. Strictly
speaking, there is no physical basis for comparing (the
Page 9 of 13 20
absolute values of) the FGB and SGB potentials by simply dropping a given term in any of these potentials even
though subsequently these energy values are determined by
a same minimization algorithm, because the expressions
and the number of parameters that are fitted to experimental
or other known data are generally not the same. Nevertheless, in the present cases, browsing the differences between
them does shed light on the importance of contributions
from the dihedral angle in carbon–carbon double bonds
[Eq. (12)] for Cn in the size range n = 5–17. It is perhaps
worthwhile to point out that the calculated lowest energy
values using the FGB potential appear to agree slightly better with an older experimental data of Drowart et al. [59].
More experimentally measured data are needed to confirm
the quality of the two Brenner potentials.
4 Concluding remarks and perspectives
Considering the practicability of the classical bond-order
empirical potential in many successful applications to
solid-state and molecular systems, we have revisited this
kind of the bond-order potential with due emphasis put on
the theoretical calculations of the lowest energy structures
of carbon clusters. We carried out thorough investigations
of four Abell–Tersoff-type potentials which were combined separately with a same accurate, reliable and efficient optimization algorithm developed very recently by
us [36], because a highly regarded interparticle potential
alone is necessary but not a sufficient requirement for predicting a stable cluster. To this end, we have employed the
recently developed MBH optimization algorithm. In this
modified algorithm, modifications were done on the BH
method which consists in introducing a scaling parameter whose value monitors the region within which the C
cluster is to be encapsulated. This idea of constraining the
carbon cluster to certain region for energy-minimization
searching is physically in line with the valence-electronsharing characteristics of carbon atoms as well as the
original idea that Abell [10] formulated his quantumbased bond-order potential. In view of the covalent bonding nature of the carbon cluster that leads valence electrons the propensity to disperse around carbon atoms in a
directional way, we have proposed another modification
to the BH method, i.e., the incorporation of the cut-andsplice GO-like operation to generate new cluster configurations in addition to the built-in random move and atomic
displacement in the BH method. The incorporation of this
GO-like procedure will certainly add much credence to
the searching of minimized energy as indeed it was in
the calculations of Cn. Among the four empirical potentials tested, we found that the Tersoff [20] and TEA [27]
potentials display less varied in cluster shapes. However,
13
20 Page 10 of 13
some of the Cns predicted by the TEA potential show an
inspiring sequence of Cn at sizes n = 6, 10, 13, 16, 19,…,
which is none but the stable 2D graphene with a zigzag
peripheral edge. It would be an interesting endeavour
if one were to continue increasing the size n and check
further whether the optimized cluster using TEA/MBH
method could evolve into a larger sheet of graphene such
as in an earlier generalized tight-binding MD technique
calculation of 126 carbon atoms by Menon et al. [60]. For
the FGB potential obtained by us, we found that it captures the essential features of the geometries of Cn, and
generally, they compare very well with other theoretical
calculations. The more refined SGB potential has furthermore predicted topologies showing more regularity and these conformational structures agree quite well
with most of the Cn obtained by the semiempirical DFTB
method that goes beyond the empirical potential approach
[36]. The SGB potential is therefore more promising in its
applications to carbon-related materials as concluded by
Mylvaganam and Zhang [61] in their MD simulation of
the mechanical properties of carbon nanotubes, and commented also by Zhou et al. [33] in their very recent evaluation of bond-order potentials.
Acknowledgements This work is supported by the Ministry of Science and Technology (MOST103-2112-M-008-015-MY3), Taiwan.
We thank Prof. R. Smith for sending us the code of the first-generation Brenner potential.
Appendix 1: Empirical bond‑order potentials
In this appendix, we briefly summarize some of the essential features of the four selected empirical potentials to be
used in this work. They are presented all with reference to
Sect. 2 above.
Tersoff empirical potential
Starting from the general formula of Eq. (1) and the equations that follow, the form proposed by Tersoff [20] is
(0)
that he sets rij = 0 in Eq. (3), γik = 1 in Eq. (6) and also
assumes bij = bji which implies that the parameters γik, cik,
dik and hik in Eq. (6) are independent of the atom at site
k, i.e., one may write (γik, cik, dik, hik) simply as (γi, ci, di,
hi). With these approximate replacements, the author determined the parameters in the remaining formulas by fitting them to bulk properties. Two further comments are in
order. Firstly, the bond order considered in this potential
only works well in the graphite and diamond environments
(through fitting parameters to them), and it is a lack of the
non-local environment (conjugation and radical effect) [21,
22]. Secondly, the bond angle θjik is fitted only to optimal
13
Theor Chem Acc (2017) 136:20
bulk bond angle that satisfies θjik < 180o which means that
only bent structures of sp2 and sp3 bonds are allowed.
Tersoff–Erhart–Albe empirical potential
The empirical potential of Erhart and Albe [27] is basically of Abell–Tersoff-type described by the same Eq. (1).
(0)
Referring to Eq. (3), here the parameter rij is not zero in
the TEA potential and it is fitted to the dimer bond length,
a term omitted in the Tersoff potential. The authors also
assume γik ≠ 1, bij ≠ bji and quantities A, B, λ and μ in
Eq. (3) are now expressed as
√
A = D0 /(Z − 1), = ω 2Z
(9)
B = ZD0 /(Z − 1),
µ=ω
2/Z
where D0 is the dimer energy, ω is associated with the
ground-state oscillation frequency of the dimer, and Z is
a parameter adjusted to the slope of the Pauling plot [62].
The TEA potential is thus constructed through fitting D0 to
(0)
the measured dimer energy and rij to dimer bond length
as well as the cohesive energy of cubic (simple cubic,
body-centered and face-centered lattices) and diamond
structures. By this numerical procedure, the TEA potential
[27] improved on the approximations that Tersoff potential
made on parameters γik, cik, dik and hik mentioned above
since more realistic angular bonding factor such as bij ≠ bji
has now been explicitly taken into account in these parameters by fitting to cohesive energies and bond lengths of
several high-symmetry structures as well as to the elastic
constants of ground-state structures. Finally, as in Tersoff
potential, the optimal bond angle hik used in Eq. (6) corresponds to optimal angle of the bulk system implying that
only sp2 and sp3 bonds with θjik < 180o are included. Further detailed comparison of the parameters in the Tersoff
and TEA empirical potentials are compiled in Table I of
Ref. [29].
Brenner potential: first generation
The FGB empirical potential [21, 22] has essentially the
same mathematical structure as the Tersoff [20] and TEA
[27] potentials (see Appendix “Tersoff empirical potential” and Tersoff–Erhart–Albe empirical potential). The
major modification is the author’s observation that the
bond-order interaction in either Tersoff or TEA potential
is mainly fitted to bulk graphite and diamond solids where
only single- and double-bond characters appear, notably
in graphite. These latter two empirical potentials do not
therefore work well when the radical effects have to be
taken into proper account and in circumstances for conjugated systems. To include these features, he wrote
Theor Chem Acc (2017) 136:20 Page 11 of 13 20
Fig. 7 Flowchart showing
the modified BH method used
in locating the lowest energy
structure
13
20 Page 12 of 13
b̄ij = bij + bji /2 + πijRC
Theor Chem Acc (2017) 136:20
(10)
where the function πijRC is introduced to rectify the unphysical situation of overbinding of radicals and the apparently
contradictory feature that could happen between conjugated and nonconjugated double bonds in the absence of
non-local effects. In other words, πijRC depends on whether
the bond between atoms i and j that each with its own total
number of C neighbors has a radical character and is part
of a conjugated system. This term therefore represents the
influence of radical energetics and π-bond conjugation on
the bond energies, and it describes radical structures correctly as well as including also for non-local conjugation
effects such as those governing different properties of the
carbon–carbon bonds. In this FGB potential, the first term
[cf. Eq. (4)] in Eq. (10) reads

−δi
��
�
�


�
� � αijk rij −rije −(rik −rike )
c
bij = 1 +
fik (rik )gi θjik e
+�
(11)


k�=i,j
where fC(rik) is a cutoff function similar to Eq. (7) and bji
is defined by writing i → j. The bij or bji depends on the
bond angle through gi (gj) between bond ij (ji) of atom i
(j) and ik (jk), and the local coordination through the Δ
in {…} of atom i (j). Both Δ and αijk are identically zero
in this FGB potential, and values of δi and δj are both set
equal to 0.80469 [21, 22]. Note that the function gi(θjik)
has the same form as Eq. (6), but the value of hik has
been now chosen to be cos180o. For other parameters that
appear in Eq. (11), the author determined them from the
binding energies, lattice constants of graphite, diamond,
simple cubic and face-centered cubic lattices of pure carbon, and also from the vacancy and formation energy of
diamond and graphite. All of these numerical values are
given in Ref. [21, 22].
Before proceeding further, a relevant comment is perhaps relevant. It is that this FGB potential does not take into
account properly the angular dependence of the optimal
angle for the planar ring structure. Neither does it include
the rotations about carbon–carbon double bonds and nonbonded interaction.
Brenner potential: second generation
One major modification in the SGB empirical potential
[26] is to add to Eq. (10) on the right-hand side a term


�
� ��
�
�
�
�
πijDH = Tij 
1 − cos2 Θkijℓ fikC (rik )fjℓc rjℓ 
k�=i,j ℓ�=i,j
(12)
13
where the function Tij is a tricubic spline, cosΘkijℓ = ejik·eijℓ
with ejik a unit vector in the direction of the product
Rji × Rik, Rji being the vector connecting atoms j and i.
The f C(rij) has the same meaning as that defined by Eq. (7).
Equation (12) describes the dihedral angle for carbon–carbon double bonds. We should point out that in this SGB
potential values of δi and δj in Eq. (11) are both 0.5, and
gi(θjik) or gj(θijk) does not have the same functional form
as Eq. (6) but is calculated instead as sixth-order polynomial splines [26] in cosΘkijℓ. The latter change in functional
form was carried out to rectify the above-mentioned deficiency in the FGB potential and it considered more realistic
dependences on the angular interaction. In this work, we
shall examine also this SGB potential which is well coded
and readily for use in the LAMMPS software (See http://
lammps.sandia.gov for LAMMPS code). A similar potential which was generalized to include oxygen has been
reported earlier by Kutana and Giapis [62], and Harrison
et al. [37] revisited this potential very recently and gave a
brief overview.
Appendix 2: modified basin hopping method
The numerical procedure of the BH method and its modifications is summarized as follows. For a cluster of n atoms,
we randomly generate an atomic configuration and confine
the n atoms inside a sphere of radius Rd∗ (see Sect. 2) whose
origin is positioned at the center of mass of the cluster. The
searching of the lowest energy value follows the BH procedure originally proposed for metallic and nonmetallic
clusters. We depict schematically in Fig. 7 a flowchart that
describes in greater detail how the BH method is modified.
The definition of all variables in the chart is the same as in
Lai et al. [55], and readers interested in technical details
are referred to this reference.
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