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The2D-to-3Dgeometryhoppinginsmall
boronclusters:Thechargeeffect
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Chemical Physics Letters 577 (2013) 32–37
Contents lists available at SciVerse ScienceDirect
Chemical Physics Letters
journal homepage: www.elsevier.com/locate/cplett
The 2D-to-3D geometry hopping in small boron clusters:
The charge effect
Hung Tan Pham a, Long Van Duong a, Buu Quoc Pham a, Minh Tho Nguyen a,b,⇑
a
b
Institute for Computational Science and Technology at Ho Chi Minh City (ICST), Quang Trung Software City, Ho Chi Minh City, Viet Nam
Department of Chemistry, University of Leuven, B-3001 Leuven, Belgium
a r t i c l e
i n f o
Article history:
Received 4 March 2013
In final form 20 May 2013
Available online 26 May 2013
a b s t r a c t
DFT TPSSh/6-311+G(d) calculations are carried out on a series of 2D and 3D forms of Bn, n = 20, 22 and 24
in different charge states. For a certain size, the relative energy within a pair of two-dimensional quasiplanar (2D) and three-dimensional staggered double-ring (3D) boron cluster isomers may shift the sign
as they reach a certain charge state. Specifically, electron addition tends to enhance the stability of the 2D
over the corresponding 3D isomer irrespective of the available electrons. Linear correlations between
2D–3D relative energy and net charge are established. Along with 2D-to-3D geometry hopping at critical
size, our results suggest a local 2D–3D geometry hopping via critical charge.
Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
Boron is well-known to exist in solid state in the forms of amorphous, a- and b-rhombohedral allotropies, and in other boron-rich
compounds such as boranes. A number of these materials have
important applications in, among others, high temperature semiconductors at up to 530 °C [1], super-conductors at abnormal conditions of 6 K and 175–250 GPa [2], and/or high density energy
fuels [3]. Recent discoveries of boron clusters [4], boron sheets
and boron nanotubes [5,6] have opened up new horizons of boron-based materials with many interesting characteristics. A boron
paradox is that it has one electron less than carbon to form cluster
bonds but prefers highly coordinated centers. A peculiar feature of
finitesize boron clusters is their aromaticity which has intensively
been studied during the last decade [7]. Such aromaticity apparently reveals a nano-confinement effect, i.e., properties that exist
at a nanoscale but are not found in solid bulk. For instance, while
the dominant cells in bulky boron materials are three-dimensional
(3D) cages, low-lying energetic isomers of small boron clusters
mostly adapt two-dimensional (2D) quasi-planar (QP) or perfectly
planar structures [8–14]. This characteristic was initially introduced for the global minimum of the Bþ
13 cluster which was presented with a perfect 2D planar structure instead of a 3D cage as
previously assumed [14,15]. The p-electron delocalization was
identified as a factor facilitating the planarity and aromaticity.
Electron delocalization of the planar Bþ
13 isomer was subsequently
confirmed by its molecular orbital framework [14,15].
⇑ Corresponding author at: Department of Chemistry, University of Leuven, B3001 Leuven, Belgium. Fax: +32 16 32 79 92.
E-mail address: [email protected] (M.T. Nguyen).
0009-2614/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.cplett.2013.05.041
Small boron clusters were found to have not only p- but also
r-aromaticity, called a double aromaticity. For instance, the global
minima of B2
8 and B9 clusters exhibit both p and r electrons obeying the classical Hückel (4N + 2) electron rule [7,16]. They are together responsible for the interesting planar wheel structures
with highly coordinated central boron atom. While small boron
clusters Bn, with n being up to 23, have planar or quasi-planar
structures resembling unsaturated hydrocarbons [4,7,17–21], the
well-known boron hydrides BnHn prefer cage-formed 3D structures
[22]. For circular and elongated structures, a trend of boron clusters is to adapt p-electron patterns of unsaturated hydrocarbons.
Accordingly, the B16–18 neutral and negatively charged clusters
were known to resemble naphthalene [18,19], that of B
19 with
10-annulene [17], and B
and
B
clusters
with
anthracene
and
22
23
phenanthrene [4], respectively. Recently, the concept of disk aromaticity [21] has been introduced to rationalize the electronic
structure of the B2
20 dianion. It appears that for circular species,
the classical Hückel electron count is a particular case of the more
general disk aromaticity.
Furthermore, the aromatic character has also been anticipated
to be involved in the 2D-to-3D geometry hopping of boron clusters
as they reach a transition size, i.e., size at which the global minimum shifts from a 2D to a 3D structure. For instance, at the size
of 20-atoms, the global minimum of the neutral boron cluster
has a 3D staggered double-ring (SDR) structure [10,23]. The geometry hopping was implicitly assigned for a marvelous double aromaticity of 3D SDR structure contributed from two separate
tangential and radial MO stacks, whose occupied electrons obeying
the Hückel count [11]. This geometrical transition was confirmed
by the fact that 3D SDR structures remain the global minima for
the immediately larger sizes [12,13,24].
33
H.T. Pham et al. / Chemical Physics Letters 577 (2013) 32–37
However, the critical size for 2D-to-3D geometry hopping,
quantitatively represented by the 2D–3D relative energy, turns
out to vigorously vary with respect to the implemented net charge.
For instance, the critical size of the B20 system is solely applicable
for the neutral state. In the cationic state, a geometrical transition
was found to occur earlier, i.e., the 3D SDR isomer is already the
global minimum of Bþ
16 clusters [15,25]. In the anionic state, the
2D QP isomers, however, remain iso-energetic with 3D SDR struc
tures for B
19 and B20 [21,26]. By adding one extra electron giving a
dianionic state, the 2D QP structure becomes a distinguishable global minimum with large energy gap with respect to the 3D SDR
isomer [21,26]. Boron clusters of larger sizes, containing 22–24
atoms could have a global minimum having either a 2D QP [4] or
a 3D SDR structure [13], apparently depending on the implemented charge of the system. Thus, the 2D–3D relative energy,
which intuitively depends on their charged state, is likely to directly be affected by the number of itinerant electrons. However,
to the best of our knowledge, no previous study has analyzed a
charge effect on 2D–3D relative energies of boron clusters. Most
of them mainly considered the geometrical transition at a specific
charged state for a critical size. In this context, we set out to elucidate such an effect by investigating a series of 2D and 3D boron
clusters Bn in a range of n = 20–24 atoms with an effective net
charge varying from +2 (dication) to 2 (dianion).
2. Computational methods
The present Letter mainly concentrates on the relative stability
between both 2D QP and 3D SDR isomers of a size with respect to
the implemented net charges. All electronic structure calculations
are performed at the same level for the purpose of comparison. For
each pair, relevant structures are obtained, when available, from
the literature [10]. When a structure at a certain size and charge
is missing, we carry out a search for its global minimum. Both
low-lying 2D QP and 3D SDR structures can be built up either manually starting from the highest symmetry and then relaxed to their
correct symmetry, or generated using a stochastic search algorithm
[27].
The genetic algorithm used [27], which is implemented with
some of our own modifications, involves a generator of random
number that is now supplemented by an additional module physically checking the structures generated. The additional module
aims to eliminate fragmented and/or superpositioned structures
by assigning each atom with avoiding and valence spheres. Both
spheres have radii that are equal to the half of the possibly shortest
and longest valence bonds. By counting the number of superpositioned (Figure 1a), normal (Figure 1b) and fragmented (Figure 1c)
overlaps, unphysical structures can thus be eliminated. Using the
kick algorithm [27,28], another simple bash shell script is also
implemented to automatically build inputs from guessed
(A)
(B)
geometries, and compute their energies at a desired theoretical level [28]. In eliminating the failed and redundant points, a new set
of possible structures can be considered for higher level
computations.
For 2D isomers, many degrees of freedom can be reduced by
keeping the structures strictly planar (Cs point group). All optimized structures are subjected to normal mode vibrational analyses at the same level. Imaginary frequencies are eliminated by
geometry reoptimization slipping along the corresponding normal
modes.
Due the large size of the clusters considered, high accuracy MO
methods such as the second-order perturbation theory (MP2) and/
or coupled-cluster theory (CCSD(T)) could not be applied within
our limited computational resource. In order to obtain relatively
reliable energy but with more reasonable computational costs,
density functional theory (DFT) is used. Instead of the popular
hybrid B3LYP functional, the TPSSh functional [29] is used in conjunction with the 6-31G(d) basis set for initial computations, and
the 6-311+G(d) basis set for refined structural and energetic determinations. Extensive recent benchmark computations demonstrated that the B3LYP functional tends to underestimate the
energy differences between 2D and 3D boron clusters [25,26,30–
32]. The TPSSh functional has also been calibrated and shown to
give better relative energies for boron clusters than other available
functionals, as compared to results obtained from high level MO
methods (CCSD(T)). All calculations are carried out using the GAUSSIAN 09 suite of program [33].
3. Results and discussion
3.1. The 20-atoms boron clusters
The B20 system was previously found to be a critical point of the
2D-to-3D geometrical transition of pure boron clusters. For the
neutral, the 3D SDR isomer was identified by DFT computations
as the global minimum, which is also the possible thinnest embryo
of boron nanotube [10]. This result was subsequently confirmed by
coupled-cluster (CCSD(T)) calculations.
DFT calculations at the TPSSh/6-31+G(d) level show that the
pairs of low-lying 2D QP and 3D SDR isomers for B20 are similar
to the previously reported ones [10,25,26,34] (Figure 2a). Similar
results on 2D–3D relative stabilities are also obtained with calculated relative energies presented in Table 1. This quantity appears
to be noticeably sensitive to the actual net charge. For instance,
neutral and positively charged 3D SDR isomers consistently remain
the energetically lowest-lying isomers with about 10–30 kcal/mol
lower than 2D QP counterparts [25,34]. In the anionic state, it becomes ambiguous to assign the global minimum as the 2D–3D energy difference is faded within the range of expected error margin
of DFT computations of about 4–5 kcal/mol [21,26]. However, such
(C)
Figure 1. Three typical relative position between a pair of atoms (green points) in term of avoiding (red) and valence (blue) spheres. (a) The superposition position with the
superposition overlap between two avoiding spheres; (b) the normal position with normal overlap between two valence spheres; and (c) the fragmented position with no
(fragmented) overlap between two valence spheres. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
34
H.T. Pham et al. / Chemical Physics Letters 577 (2013) 32–37
SDR
QP-1
QP-2
(a) B20
(b) B22
(c) B24
Figure 2. Geometries of the 2D QP and 3D SDR isomers of boron clusters considered.
Table 1
2D–3D relative energy (DE kcal/mol) with respect to implemented charges.a
Charge
B20-QP-1
B20-QP-2
B22-QP-1
B22-QP-2
B24-QP-1
B24-QP-2
2
1
0
1
2
22.6 (10, aar)
18.5
14.1 (12, ar)
3.6
20.9 (14, aar)
30.4 (10, aar)
24.6
9.9 (12, ar)
5.0
18.4 (14, aar)
36.9 (12, ar)
30.6
14.9 (14, aar)
0.9
17.8 (14, ar)
32.4 (12, ar)
20.4
7.2 (14, aar)
0.9
3.2 (14, ar)
41.6 (12, aar)
36.2
26.7 (14, ar)
8.5
11.0 (16, aar)
38.5 (12, aar)
37.8
32.7 (14, ar)
10.2
18.7 (16, aar)
a
A 2D QP structure is more stable than the corresponding 3D SDR isomer with a negative relative energy. Total number p of electrons and corresponding aromaticity (ar)/
anti-aromaticity (aar) of the 2D QP isomer according to Hückel rule are presented in parentheses.
a result appears to be in agreement with the fact that only the anionic 2D QP isomers were detected by photoelectron spectroscopy
under experimental conditions [26]. By adding one extra electron,
the dianionic 2D QP isomers are now found to be 18–20 kcal/mol
clearly more stable than the 3D SDR ones [21].
3.2. The 22-atoms boron clusters
The 2D–3D geometry transition identified for the size of 20atoms is confirmed as the B22 system keeps the 3D SDR isomer
as the global minimum of the neutral state. In varying the net
charge from cationic to anionic states, SDR isomers are always calculated to be 8–50 kcal/mol higher in energy than the QP isomer
[12].
Our results obtained from an extensive structural search using
the genetic algorithm are generally consistent with the literature
reports [12] in which the global minimum of either the neutral
B22 or the positively charged Bþ
22 boron clusters have a 3D SDR
structure (Figure 2b). The corresponding 2D–3D energy difference
varies from 7 to 37 kcal/mol as shown in Table 1. Similarly to B22
system, the 2D QP isomer in the anionic B
22 clusters is nearly
iso-energetic with the 3D SDR counterpart, e.g., energy difference
being equal to about 1.0 kcal/mol (cf. Table 1). As one more electron is added to form the dianionic state, the elongated isomer
B22-QP-1 becomes much more stable than 3D SDR one, the former
being 18.0 kcal/mol lower in energy than the latter. The second 2D
isomer of the dianion B22-QP-2 remains even iso-energetic with
the 3D SDR structure.
3.3. The 24-atom boron clusters
Structures of the B24 clusters have recently been determined
including certain geometries of monocycles, 3D SDR and other
forms [24]. The 3D SDR isomer was found as the most stable neutral form.
The two low-lying 2D QP isomers, namely the B24-QP-1 and
B24-QP-2 shown in Figure 2c, are calculated to be less stable than
their 3D SDR isomers not only for the positively charged and neutral states but also for the anionic state. In contrast to the B20 and
B22 systems, the 3D SDR anions of B24 turn out to be more stable
than the corresponding 2D QP anions. In this charged state,
3D structures are about 9–42 kcal/mol lower in energy than their
2D QP isomers. There is no iso-energetic domain between 2D and
3D structures for B24 clusters as it is the cases of B20 and B22 clusters. When adding one extra electron to form the doubly negatively
charged state, 2D QP isomers are now energetically less favored
than 3D SDR structure by up to 11–19 kcal/mol (Table 1).
3.4. An analysis of the geometrical transition
In observing the evolution of various pairs of 2D and 3D boron
clusters having different size and charge, the corresponding 2D–3D
relative energy and their trend of variation with respect to the
implemented net charge can be established. Geometry and electronic structure of the clusters considered also provide us with
some insights into the driving force for the energetic changes.
Our calculated results point out an intrinsic interdependency
between shape (2D versus 3D) and charge (from dicationic to dianionic states) in a small range of size from 20 to 24 B-atoms. The 20atom boron cluster is now firmly established as the transition point
where neutral 3D SDR structure becomes more stable than the 2D
QP isomers. Such a geometrical transition generally occurs early for
positively charged states, and later for negatively charged states. In
other words, calculated results point out that the 2D–3D relative
stability inherently depends on the implemented charge of the
clusters. Figure 3 shows a plot illustrating the variation of 2D–3D
relative energy with respect to net charge. The correlation between
2D–3D relative stability and implemented charge indicates that
addition of electrons tend to stabilize the 2D QP over the 3D SDR
isomers. Furthermore, such a 2D–3D stability dependency can be
H.T. Pham et al. / Chemical Physics Letters 577 (2013) 32–37
Figure 3. Plots of relative energy (kcal/mol) between 2D QP and 3D DR isomers
with respect to implemented charges of B20 (a); B22 (b); and B24 (c) boron clusters.
fitted into linear equations with relatively high correlation coefficients. These relationships are derived in a series of equations from
B20-1 to B24-2:
DEð0:98Þ ¼ 12:73C þ 8:31
ðB20-1Þ
DE
ð0:91Þ
¼ 10:92C þ 6:17 ðB20-2Þ
DE
ð0:98Þ
¼ 14:09C þ 12:73
ðB22-1Þ
DEð0:95Þ ¼ 9:09C þ 11:546
ðB22-2Þ
DEð0:95Þ ¼ 13:30C þ 20:40
ðB24-1Þ
DE
ð0:83Þ
¼ 14:20C þ 20:09
ðB24-2Þ
While B20-1 denotes the linearly fitted equation for B20-QP-1
isomer, DE (in kcal/mol) is the 2D–3D relative energy formulated
in such a way that a 2D QP isomer is more stable than the
35
corresponding 3D SDR with a negative value and vice versa. The actual net charge is denoted with capital C varying in the range of
[2, 2+]. The superscript given on the DE 2D–3D relative energy
is the correlation coefficient of the fitted equation. In all cases studied, the correlation between relative energy and implemented
charge are relatively high. The lowest correlation coefficient
amounts to 0.83 (Eq. B24-2), whereas the highest one of 0.98
(Eqs. B20-1 and B22-1) is very close to unity.
The magnitude of the slopes and intercepts of the fitted lines
represents the dependency of 2D–3D relative stability on implemented charge, i.e., the larger the slope, the higher the dependency, and reversely for the intercept value. As seen in equations
B20-1 to B24-2, all slopes are positive, and relatively larger than
the intercepts. The positive slope values prove that electron addition consistently decreases the stability of the 3D SDR isomer as
compared to its corresponding 2D QP isomer. This rationalizes
the finding of an early geometrical hopping for positively charged
states (e.g. for the cationic state, the critical size being around 16atom system [15,25]), while it becomes later for negatively
charged states (e.g. a critical size being larger than 20-atom in anionic state). Therefore, the 2D-to-3D geometry hopping of boron
clusters occurs not only at a critical size but also with a critical
charge. With a certain size the relative stability of a pair of 2D–
3D isomers could changes as the system reaches a critical charge.
Accordingly, a full description of geometry hopping of small boron
clusters needs to include a coupling of both critical size and charge
parameters.
The stability of 2D versus 3D boron clusters was reportedly interpreted in term of the delocalization of p electrons heading toward
the classical Hückel rule for aromaticity, and the resemblance with
unsaturated hydrocarbon in their p patterns [4,7,18,19,35]. The first
step of aromatizing and resembling electronic structure of hydrocarbons is to flatten the B-clusters facilitating an overlap of p-atomic
orbitals. To observe geometrical changes of 2D isomers as being
charged, the peripheral bonds and side-projected geometry of 2D
QP of the B24 clusters are considered as for an example. Geometrical
and structural parameters of B24 are depicted in Figure 4. With
change in the charged state, Figure 4 shows that both the bondlength and corresponding occupancy of the peripheral bonds tend
to be conserved. While internal atoms are more vigorously reordered to adapt the molecular planes, most of the peripheral bonds
are saturated with two electrons giving 2c–2e bond systems. The
planarization with conservative boundary obviously increases the
atomic repulsion significantly. Such a pattern is expectedly stabilizing the structure by maximizing the exchange energy obtained from
p-electron delocalization within molecular planes [7]. In a sense, the
planarity is favored as more electrons are added. For instance, while
the cationic B24-2D-1 isomer remains buckled, the corresponding
anion is almost planar. Nevertheless, the dicationic B24-2D-1 isomer becomes also perfectly planar and this might come from a side
effect of Coulomb repulsion [36].
The 2D QP structures are iteratively supported upon planarization and the consequent delocalization of electrons occupying p-,
or even r-MO. Intuitively, the total number of p electrons is expected to obey the Hückel count as a prerequisite for energetically
favorable electron delocalization. Interestingly, the Hückel electron
count seems not always to correlate with the 2D–3D relative stability. As for a typical example, the p-system of the well-known
B20 clusters is plotted in Figure 5. In the neutral state, the B20QP-1 isomer is found to be anti-aromatic with a total number of
12p electrons, while the corresponding neutral 3D SDR isomer features a double aromaticity [11]. This gives rise to a suitable 2D–3D
relative energy, i.e., the neutral B20-QP-1 being about 14 kcal/mol
higher in energy than the 3D SDR isomer (cf. Table 1). With two
additional electrons yielding the dianionic state, the B20-QP-1 isomer is now Hückel aromatic with 14p electrons, while the 3D SDR
36
H.T. Pham et al. / Chemical Physics Letters 577 (2013) 32–37
Figure 4. Bond-length (Å) and bond-order (values in parentheses) of 2D QP B24 boron cluster. From inside out, the values are depicted for dianionic to dicationic isomers.
Planarization of 2D QP B24 clusters is also presented from sid-projected figures of neutral and charged isomers.
Di-anion
(14π
π)
Neutral
(12 π)
Di-cation
(10 π)
π−HDOMO (51)
π (50)
π−HDOMO (50)
π (49)
π (49)
π−HDOMO (49)
π (47)
π (47)
π (47)
π (43)
π (43)
π (43)
π (38)
π (38)
π (39)
π (34)
π (34)
π (35)
Figure 5. Plots of occupied p/r-MOs of B20-QP-1 isomer with respect to
implemented charge. HDOMO, SOMO and LUMO stand for the highest doubly,
singly occupied MO, and lowest unoccupied MO. Position of plotted MOs is given by
number in parentheses. Total number of p electrons at each charged state is
presented in the second row.
contains a mixed aromatic radial (10p electrons) and anti-aromatic
tangential (12p electrons) stack of MOs [11]. This reverses the
energy ordering in favor of the negatively charged state, e.g. the
B20-QP-1 isomer being now 21 kcal/mol lower in energy than
the 3D SDR. On the contrary, upon removal of two electrons from
the neutral, the dication B20-QP-1 becomes aromatic (e.g. with
10p electrons shown in Figure 5). The dicationic 3D SDR isomer
again contains a mixed aromatic radial (10p electrons) and antiaromatic tangential (8p electrons) MO stacks. However, the B20QP-1 dication is calculated to be 23 kcal/mol higher in energy than
the corresponding 3D SDR cation, in contrast to the prediction
based on the classical electron count.
By performing the similar MO analyses on the p-systems of the
other 2D QP isomers, the added electrons are found to occupy pMOs in priority inducing a p-delocalization favoring planarization
[8,9]. Total numbers of counted p electrons of 2D QP isomers and
the assigned aromaticity (ar) as well as the anti-aromaticity (aar)
in term of Hückel rule are presented in Table 1. It can be seen that
assignment of Hückel aromatic character does not always correlate
with energy ordering. This result is consistent with previous analyses [21,37] where the Hückel aromaticity for p-delocalization are
not always applicable to boron clusters.
4. Concluding remarks
In the present theoretical investigation, we carried out DFT calculations on a series of pairs of energetically lower-lying 2D QP and
3D SDR structures of small boron clusters having different sizes
from 20 to 24 atoms and net charges (from dication to dianion).
The 2D–3D relative energy is generally found to linearly vary with
respect to the actual charge of the cluster, i.e., a 2D QP isomer becomes stabilized, or even more stable, as compared to its corresponding 3D SDR following addition of extra electrons. Our
results thus unveil the underlying background for the early 2Dto-3D geometry hopping for positively charged boron clusters,
and reversely for the negatively charged counterparts.
H.T. Pham et al. / Chemical Physics Letters 577 (2013) 32–37
Planarization of the molecular skeleton appears obviously as a
driving force facilitating electron delocalization in 2D structures.
Within the range of implemented charge of [2, 2+], the added
electrons to 2D structures tend to reinforce the p-MOs and thereby
stabilize planar structures irrespective of the classical Hückel electron count.
Acknowledgments
The work carried out at ICST has been supported by the Office of
Science & Technology of Ho Chi Minh City. M.T.N. thanks the ICST
for supporting his stays in Vietnam.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, in
the online version, at http://dx.doi.org/10.1016/j.cplett.2013.05.
041.
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