Chemical Physics Letters 608 (2014) 295–302

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Chemical Physics Letters

journal homepage: www.elsevier .com/ locate /cplet t

The boron conundrum: Bonding in the bowl B30 and B36, fullerene B40

and triple ring B42 clusters

http://dx.doi.org/10.1016/j.cplett.2014.05.0690009-2614/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Fax: +32 16 32 79 92.E-mail address: minh.nguyen@chem.kuleuven.be (M.T. Nguyen).

Hung Tan Pham a, L.V. Duong a, Nguyen Minh Tam a,b, M.P. Pham-Ho a, Minh Tho Nguyen b,⇑a Institute for Computational Science and Technology (ICST), Quang Trung Software City, Ho Chi Minh City, Viet Namb Department of Chemistry, University of Leuven, B-3001 Leuven, Belgium

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 April 2014In final form 23 May 2014Available online 7 June 2014

Geometries and bonding of B30, B36, B40 and B42 clusters were studied using quantum chemical compu-tations. The bowl B30 and B36 and planar B42 clusters exhibit disk aromaticity. Diatropic ring current isstrong in B30 and weaker in B42. A fullerene-like B40 (D2d) having two hexagons and four heptagonswas found as the lowest-lying isomer. Such a fullerene whose MOs closely mimic those of the buckyballB80, represents novel structural feature of boron clusters. The most stable B42 (C2h) isomer is a triple ringtube with consistent r + p diatropic magnetic responses making it a tubular aromatic species.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Boron compounds continue to provide us with full of surpriseabout their structural features. Regarding boron clusters, a pleth-ora of two-dimensional (planar, quasi-planar) and three-dimen-sional (tubular, cage, spherical, fullerene. . .) shapes apparentlysuggests several distinct growth patterns. Extensive effort has beendevoted to the search for new forms of boron allotropes andnanocages [1,2], and most boron clusters appear to exhibitpolymorphism and multi-center bonding [3,4]. In relation to C60,particular attention has been paid on the boron buckyball B80 [5],and larger fullerene derivatives such as B100 and B112 [6] have beenpredicted by computations. Some smaller all-boron fullerenes havealso been identified by computations including B14 [7], B32, B38, B44,B50 and B56 [8,9], B40, B48,. . . (B32+8k, with 0 6 k 6 7) [10]. In manycases, the fullerene-like form reported is however not the most sta-ble isomer of the size considered. In addition, the buckyball B80 hasso far not been detected by experiment yet [11].

Boron has been proved to be a rare element whose atomic Bn

clusters retain either planar or quasi-planar (QP) geometries evenwhen the cluster size n goes up beyond >20, depending on thecharge state. The cations Bn

+ prefer to adopt 3D structures at smal-ler sizes (at n � 18) [12,13], whereas the anions Bn

� and dianionsBn

2� species tend to have planar or QP structures with n beingup to >30. There is an effective charge effect in which addition ofextra electrons consistently favors QP boron cluster anions, andremoval of electrons leads to 3D cations [14].

Let us briefly summarize the main geometrical features of thebare clusters Bn, with n being an even number, in their neutralstate: (i) most of the sizes up to B18 have planar or QP geometries[15–18], except for the surprising B14 fullerene [7]; (ii) the moststable B20 isomer is a tubular double ring (DR) in which the two10-membered rings are connected together in an antiprismbonding motif [19]; (iii) Similar to B20, each of the neutrals B22,B24, B26 and B28 also exists in a DR ground state [20,14]. Theseobservations resulted in a popular belief that intermediate-sizedBn with n P 30 still prefer tubular geometries; (iv) we recentlyfound that B30 has a bowl structure [21] which is composed of apentagon base and successively built up by two strings of 10 and15 boron atoms yielding a fivefold symmetry (C5v). This5@10@15 strings bowl turns out to be �5 kcal/mol more stablethan the tubular 3�10 triple ring (TR). The slightly higher thermo-dynamical stability of the bowl B30 has been shown to arise fromits disk aromatic character [22,23]. In spite of its non-planarity,the 20 P-type electrons of bowl B30 are distributed in a way similarto those of the planar disk aromatic B20

2� dianion [25,26]. Althoughthe bowl B30 is not established yet as the lowest-lying isomer, itconstitutes an appropriate building-block whose recombinationeventually leads to B80 and larger buckyballs [24], (v) previouscomputations [19,24] also predicted that B32 has a DR shape com-posed of two 16-membered ring. There is no previous report on theB34 isomers yet, but it seems that the latter seemingly has the low-est-lying DR structure, and (vi) more recently, B36 has been foundby computations to have a bowl structure which is based on a hex-agonal background and successively caped on it by two strings of12 and 16 B atoms. The 6@12@18 strings thereby lead to a sixfoldsymmetry (C6v) bowl form which is calculated to be the lowest-lying structure being �20 and �32 kcal/mol more stable than the

296 H.T. Pham et al. / Chemical Physics Letters 608 (2014) 295–302

triple ring (TR) 3�12 and QP (having a hexagon and a heptagon)isomers, respectively [25]. Existence of the neutral B36 was indi-rectly detected from a photoelectron spectrum of B36

� anion [25].As far as we are aware, no careful search on the neutral B38 isreported yet. Only a B38 cage [8] and a fullerene structure [26] werementioned but without a clear status (whether it is a lower-lyingisomer).

In this context, a legitimate question is as to whether the bowlB36 also has a disk aromatic character, and the bowl shape persistsas a general structural feature of B-clusters. The following bowlcandidate is an isomer of the B42 size consisting of a superpositionof the 7@14@28 strings with a heptagon base.

Although a fullerene B40 was previously reported as a member ofthe B32+8k series of clusters [10] which is based on four-memberedrings, no thorough search for structures of this size has beenexplored. If confirmed, the point of interest is about the stabilityof such B40 fullerene. Motivated by such key issues, we set out toperform in the present Letter an investigation on some Bn clusterswith n = 30, 36, 40 and 42 in their neutral state. Using DFT methods,we find a fullerene-type form as the lowest-lying isomer of B40, butthe one located in this Letter differs much from that reported in Ref.[13]. For its part, B42 turns out to have a tubular TR form as theground state. A B42 isomer formed by 7@14@21 strings also exists,but is less favored than the TR counterpart, and more importantly,such a B42 isomer becomes a planar disk (C7v).

In the present Letter we report for the first time some newgeometrical features of both B40 and B42 clusters, and also an anal-ysis of chemical bonding of B30, B36, B40 and B42. We make use ofthe partition of the electron densities and the magnetic ringcurrents in order to probe their bonding pattern and aromaticcharacter.

2. Computational methods

Standard electronic structure calculations are carried out usingthe GAUSSIAN 09 program [27]. Due to the relatively large clustersizes, only DFT methods are employed in the searches for equilib-rium structures, and subsequent determination of relative ener-gies. The searches for energy minima are conducted using twodifferent approaches. In the first, we use a stochastic genetic algo-rithm to generate all possible structures [14,28]. The equilibriumstructures that are initially optimized using computations withsmall basis sets, are reoptimized using larger basis sets. In the sec-ond approach, initial structures of a Bn cluster are manually con-structed by adding necessary B-atoms at all possible positions onsurfaces of the Bn�2 or smaller clusters. The use of the genetic algo-rithm is effective for producing non classical 3D structureswhereas the manual search is more focused. While the large num-ber of initial structures makes the genetic search more tedious,only a combination of different search approaches allows newstructures to be discovered and a consistent set of lower-energystructures to be obtained.

Previous calculations demonstrated that the B3LYP functionalunderestimates the energy differences between 2D and 3D boronclusters [18,29–31]. In recent studies, we found that the TPSShfunctional gives better results on relative energies of boron clustersthan other available functionals, as compared to the high accuracyMO method (such as CCSD(T)) [13,14]. While computations on ini-tial guess geometries are carried out using the hybrid TPSSh func-tional [27], in conjunction with the 3-21G and 6-31G(d) basis sets,all selected equilibrium geometries with relative energies up to�3 eV are fully optimized using the same functional but with thelarger 6-311 + G(d) basis set [32]. In order to confirm the identityof true local minima obtained, their harmonic vibrational frequen-cies are also calculated at the same level of theory. For the analysisof chemical bonding, in particular the aromaticity, we make use of

canonical MOs, electron localization function (ELF) [33] and elec-tron localizability indicator (ELI-D) [34] maps, as well as the mag-netic ring current approach [35] along with the ipsocentric model[36,37]. Calculations of magnetic responses and ring currents arecarried out using the Gamess-UK program [38] in conjunction withthe SYSMO package [40].

3. Results and discussion

The calculated results are mainly presented in different figures.Figures 1 and 2 are devoted to the bowl-shaped clusters, Figures 3–5 are for B40 and Figures 6–8 for B42.

3.1. Disk aromaticity of the bowl B30 and B36 and disk B42 clusters

Figure 1 displays in two different views the shape of B30 30.b,B36 36.b and B42 42.b, in which b stands for bowl. Cartesiancoordinates of their optimized structures are given in the Supple-mentary Information (ESI) file. The main characteristic is that thebase of the bowl is successively enlarged from 5 in 30.b to 6 in36.b and then 7 in 42.b. In each cluster, the number of B-atomsin the outer string is a multiple of that of the inner base. The B-atoms in each string are thus increased from x to 2x and then 3x,with x = 5 in 30.b, 6 in 36.b and 7 in 42. Both clusters 30.b and36.b were reported to be the lower-lying isomers of B30 (C5v)[21] and B36 (C6v) [25], respectively. Their geometries are abun-dantly discussed and thus do not warrant further comments. Letus however note that incorporation of six extra boron atoms intothe bowl brings in four p electrons in such a way that 30.b, 36.band 42.b contain 20, 24 and 28 valence p electrons, respectively.

The B42 42.b displayed in Figure 1 is thus composed of three cir-cles containing 7, 14 and 28 boron atoms having the same origin,and turns out to be a planar disk (C7v). Starting from a bowl form,geometry optimization invariably leads to a planar form (or nearlyplanar depending on the basis set employed). It appears that a hep-tagon base becomes large enough to overcome the inherent strainin a bowl shape, and finally makes the cluster planar. However, asgiven in a following section, 42.b is not the global minimum of B42,being �13 kcal/mol above the corresponding tubular TR (value atTPSSh/6-311 + G(d) + ZPE).

We now analyze the aromatic character of both 36.b and 42.busing the same approach described before for 30.b [21]. The con-cept of disk aromaticity, which has been described in detail inour recent work [21–23], is based on a simple model of a particlein a circular disk, in which a free particle is moved on a planar diskencircled by infinite walls. In polar coordinates, the Schrödingerequation for this electron is written as follows:

��h2

2l@2

@r2 þ1r@

@rþ 1

r2

@2

@u2

!Wðu; rÞ ¼ EWðu; rÞ ð1Þ

where r = R is the radius of the disk, ⁄ the Plank constant and l themass of particle. Because of the circular symmetry, the Wðu; rÞ canbe written as R(r).U(u), with U(u) = exp(imu)/(2p)1/2. The cyclicboundary condition requires the angular part to be periodic. As aresult, the cylindrical quantum number must be integer, namelym = 0, ±1, ±2,. . . Substitution into the Schrödinger equation givesus the radial part, with ⁄2k2 = 2lE:

@2RðrÞ@r2 þ 1

r@RðrÞ@rþ k2 �m2

r2

� �RðrÞ ¼ 0 ð2Þ

This equation is well known as the Bessel’s differential equa-tion, and its solutions are the integer Bessel functions Jm(kr) [39].The potential wall at r = R requires the radial function to vanishat the boundary of the box, Jm(kR) = 0. The radii that correspondto the zeroes of the Bessel function are denoted as am,n, that are

3B

30.B3

.b 30

CC5vv 36B

6.bB36

b C6

C66v 442B

2.b42

b CC7vv

Figure 1. Molecular structures (TPSSh/6-311 + G(d)) of the bowl B30 30.b, bowl B36 36.b and planar B42 42.b, from two different views.

Figure 2. Calculated p MOs of the bowl-shaped B30 30.b (C5v) and B36 36.b (C6v) and planar B42 42.b (C7v) clusters, and shape of orbitals derived from the model of a particleon a circular disk (upper panel).

H.T. Pham et al. / Chemical Physics Letters 608 (2014) 295–302 297

dimensionless. In this case, n is a radial quantum number thatcounts the zeroes. The Bessel zeroes give rise to the quantifiedenergies as:

E ¼�h2ðam;nÞ2

2lR2 with : n ¼ 1; 2; 3; . . . m ¼ 0; �1; �2; �3; . . .

ð3Þ

Rotational quantum numbers are usually denoted by Greek let-ters as m = r, p, d, /, c. . . States with non-zero values for m will betwofold degenerate. The lowest eigenstates in ascending orderingare 1r, 1p, 2r, 1d, etc. (cf. Figure 2). The orbital ordering dependson the radius of the disk. A molecule containing a number of

electrons which fully occupy degenerate eigenstates of the modelis disk aromatic. These numbers are 2, 6, 10, 12, 16, 20, 24, 28. . .

A species containing a number of electrons which only singlyoccupy one of two highest degenerate eigenstates will be diskantiaromatic. The numbers are 4, 8, 14, 18. . .

Figure 2 displays a comparison of the shapes of selected p MOsof 30.b, 36.b and planar 42.b. The 2pz atomic orbitals recombine togive delocalized p MOs, and their shape is not susceptible to thestructural composition. The orbital plots show a strong resem-blance within the sequences. 30.b contains 20 valence p-electronsthat fully occupy 10 orbitals (cf. Ref. [21]). The 24 valencep-electrons of 36.b and 28 valence p-electrons of 42.b also fullyoccupy degenerate MOs. In this case, the two additional MOs of

aa) To

d

ot

d) H (d

al

HOdeg

l

OMgen

MnerMO

rate 2e π

δπ)

b)) ππ –– MMMOOs

e)) H (dHOdeg

c

OMgen

c)

MOner

σ

O-rat

σ -

1 te π

M

1γπ)

MO

γ

Oss

Figure 3. Characteristics of the planar disk B42 42.b (C7v). Current density maps computed using HF/6-31G(d) wavefunctions: (a) total p + r, (b) p and (c) r electrons. Shapeof (d) HOMO and (e) HOMO-1.

D40

D2d

0

0.1- 1

0.0

1A1

0

4C40Cs -

14

0.21A

4.3

2A'

3

4Cs

2

40.s -

24.

.31A

.6A' D

4D20d

2

0.4d -

8.9

41A

9A1

Figure 4. Some lower-lying isomers of B40. Optimized geometries, point groups, electronic states and relative energies (in kcal/mol obtained from TPSSh/6-311 + G(d) + ZPEcomputations).

298 H.T. Pham et al. / Chemical Physics Letters 608 (2014) 295–302

36.b belong to the 1c orbitals, and those of 42.b to the 2d orbitalsof the disk model. For the sake of clarity, Figure 3 displays again thedegenerate HOMO (2d) and HOMO-1 (1c) of 42.b.

When using the classical Hückel’s rule, these species can beconsidered as antiaromatic (4N electrons, N = 5, 6 and 7). However,as the valence p-electrons of each disk fully occupy the corre-sponding lowest-lying eigenstates (Figure 2), the model of a parti-cle in circular box confers to each isomer a disk aromatic character:

B30 30.b with n = 2, l = ±1. . . (1r)2 (1p)4 (1d)4 (2r)2 (2p)4 (1/)4,B36 36.b with n = 1, l = ±4. . . (1r)2 (1p)4 (1d)4 (2r)2 (2p)4 (1/)4

(1c)4, andB42 42.b with n = 2, l = ±2. . . (1r)2 (1p)4 (1d)4 (2r)2 (2p)4 (1/)4

(1c)4 (2d)4.

In order to probe further their aromaticity, we plot the ring cur-rent maps that are the response of a molecule with respect to anexternal magnetic field. The ring current of 30.b has been dis-cussed in our previous Letter [21]. The ring current maps of 30.band 36.b are presented in Figure S1 of the ESI file. Here we displayin Figure 3 only the ring current of 42.b. The ipsocentric model is

an effective model which was used to evaluate aromaticity of pla-nar compounds [40,41]. In the framework of this model, an excita-tion from an occupied to an unoccupied molecular orbital canresult in a contribution to the ring current which can be either dia-tropic, paratropic or null. Accordingly, a diatropic current arises ifthe product of symmetries of occupied and unoccupied orbitalscontains the in-plane translational symmetry. A paratropicresponse to an external magnetic field results when the productof symmetries of occupied and unoccupied orbitals contains thein-plane rotational symmetry. This rule is relatively straightfor-ward for planar species.

The electron density of each species can be partitioned in termsof r and p electrons. The main difference between the clusters con-sidered in that 30.b has only diatropic current for all components.36.b presents a conflicting response, with a paratropic current fromp electrons but a diatropic current from r electrons. The resultingtotal current shows a paratropic response along the inner 6-ringand the outermost 18-ring, and a diatropic along the central 12-ring(Figure S1, ESI). It is clear that more appropriate plots for visualisingthe current of bowl structure [42] need to be carried out before aclearer picture for both 30.b and 36.b can be obtained.

cc) EL

H

LI

Hex

I -

xa

D

ago

Dto

on

ota

nal

l

l faacce

a

b

d

a)

b)

d) E

L

H

EL

UM

HO

LI

M

M

-

MO

MO

Dσ

O

O

σ

HHeepptaagoon

E

nal

EL

l h

LI

hol

I -

le

DDπ

Figure 5. Characteristics of the fullerene B40 40.1 (D2d). Frontier orbitals (a) HOMO a1 and (b) LUMO b2 viewed in two different faces. Electron localizability indicator maps (c)ELI-Dtotal, (d) ELI-Dr and (e) ELI-Dp viewed from a heptagonal face.

H.T. Pham et al. / Chemical Physics Letters 608 (2014) 295–302 299

For its part, the planarity of 42.b provides us with clear-cut ringcurrent maps (Figure 3). Its 28 delocalized p electrons are effectivefor the magnetic response of their density, and the total currentincludes all core, p and r MOs. They induce a relatively weak con-tribution to the total ring current. The r electrons induce a stron-ger diatropic current around each boron center, and the total ringcurrent mainly arises from contributions of r electrons. In partic-ular the inner 7-circle is locally characterized by a paramagneticcurrent, irrespective of electron type, but with a predominant rcontribution. Such conflicting ring currents are comparable to thesituation of the dianion B20

2� [23]. Analysis of the excitationsinvolved shows the main contributions arises from the transitionsHOMO ? LUMO + 1 and HOMO-1 ? LUMO (the virtual 2c/3dpairs). It can thus be concluded that B42 42.b has a weaker disk aro-matic character than B30, and that is presumably a reason for itslower thermodynamic stability.

3.2. A fullerene-like B40: a thermodynamically stable isomer

As stated above, although the isomers of B40 size have not thor-oughly been searched, a fullerene B40 form has been identifiedusing a leapfrog transformation [10]. A relationship between car-bon and boron fullerenes can be established in that a three-con-nected carbon in C-fullerene can be transformed into a hexagonalin B-fullerene. The C24+6k fullerenes can thus be transformed intothe B32+8k ones. Starting from a C10, these authors obtained by

capping a C30 which can further be modified using leapfrog givingB40 [10]. Nevertheless, the identity of such B-fullerenes (global orlocal minimum) could not be determined. For example, a B32 (Oh)fullerene was found, but for this size a DR 2�16 isomer was foundto be by far the lowest-lying isomer [24].

We carry out a search for B40 structures and the geometries ofsome lowest-lying isomers of this size are displayed in Figure 4(their optimized coordinates are given in the ESI file). It turns outthat the fullerene 40.1 is calculated to be the lowest-lying isomer(of the structures located so far), being 14 and 25 kcal/mol belowboth planar isomers 40.2 (Cs) and 40.3 (Cs), respectively. The DR40.4 (D20d) is about 29 kcal/mol higher in energy than 40.1. In spiteof intensive search, we could not relocate the fullerene reported inRef. [10] within a relative energy of <3 eV. Accordingly, after thefullerene-type B14 [7], the B40 40.1 represents the second boron ful-lerene which is the thermodynamically most stable isomer of thecorresponding size. 40.1 is mainly characterized by a high pointgroup symmetry (D2d, 1A1) and contains two staggered hexagonalfaces and four heptagonal faces forming a kind of distorted cylinderconnecting the two hexagons. The C2 axis is going through the twohexagons. As in B80, each boron is a multi-coordinated centre, thuscreating about 48 triangles fulfilling the cylinder surface.

Figure 5 displays the frontier orbitals and ELI-D maps of 40.1. Tofacilitate the comprehension, the orbitals are present in two differ-ent views through a hexagonal and a heptagonal face. The densitiesof states of 40.1 that consist of the atomic contributions to the MOs

a)

c)

) 4si

) E

42.ide

EL

.t e v

I –

(Cview

– D

C2hw

Dto

h)

tal d) ELI

b

I –

b)

– D

42to

Dπ

2.top

t (vie(CewC2hw

e

)

e) ELLI – Dσσ

Figure 6. Optimized geometry (TPSSh/6-311 + G(d)) of the B42 triple ring (TR) in two different views: (a) side view, and (b) top view, and its ELI-D isosurfaces: (c) total p + relectrons, (d) p electrons, and (e) r electrons.

c)) ttottall σσ ++ ππ bb) π ellecctrronns a)) σσ eeleecttroons

Figure 7. Maps of magnetic responses of the triple ring B42 42.t (C2h). Contributions are from (a) r electrons, (b) p electrons, and c) total r + p electrons (HF/6-31G(d)).

= 2

HOMO – 1

= 3

LUMO

= 5

LUMO + 1

= 4

HOMO

Figure 8. Main excitations responsible for the magnetic responses of the triple ring B42 42.t (C2h).

300 H.T. Pham et al. / Chemical Physics Letters 608 (2014) 295–302

H.T. Pham et al. / Chemical Physics Letters 608 (2014) 295–302 301

(given in Figure S2 of the ESI file) clearly point out that severalHOMOs are built up mainly from px/py and in part from pz atomicorbitals. The contributions of s atomic orbitals appear to be negli-gible in the ten highest energy MOs. Although the cage is not hav-ing an icosahedral form, both degenerate HOMO (a1) and LUMO(b2) shown in Figure 5 contain some features closely mimickingthose of the boron buckyball B80 or even those of C60 [43]. Theorbital lobes are mainly distributed over the triagonal faceswithout components around the 6- and 7-faces, and located out-side the cage surface. The energy gap between the HOMO andLUMO of 40.1 amounts to �3 eV (Figure S2 of ESI), which is muchlarger than that of �2 eV in the boron buckyball B80 [43,44], thusmaking the species kinetically stable or the excitation is expectedto lie in high energy spectrum.

The electron localizability indicator (ELI-D, Figure 5c–e) is asimple measure of the electron localization in a molecular systemand thus gives information about the molecular spaces, calledbasins, where electrons are likely to occupy. It is useful to addresslocalization domains which correspond to cores, bonds or lonepairs. The ELI-D description can also be partitioned into differentr and p components, and this bonding description is comparableto the picture given by the electron localization function (ELF)approach. It can be seen that both type of electrons, either p orr, equally contribute to the bond formation around the faces ofthe fullerene. Although integration of electron densities over thebasins leads to the corresponding populations, we are not able inthis case to assign the electron numbers to the basins, due to thecomplex and ambiguous electron distribution. Note that from theELI-D(total) map, there is in 40.1 a bond between two adjacentboron atoms.

In summary, the existence of both hexagonal and heptagonalfaces in 40.1 represents a novel geometrical feature of boron fuller-ene, and this aspect deserves more detailed examination in subse-quent studies.

3.3. B42 triple ring: a tubular aromaticity

Let now examine the B42 cluster. As stated in Section 3.1, theplanar disk 42.b is calculated to be less stable than the triple ring(TR) 42.t whose optimized geometry is presented in Figure 6, againin two different side and top views. 42.t is composed of threestrings of 14 boron atoms connected each other in an antiprismmotif (C2h). Such an arrangement creates several triangular, rhom-bic and hexagonal faces. Each hexagon is apparently doped at thecenter by an atom giving rise to a block of B7. The distance betweentwo strings amounts to �1.50 Å, which is slightly larger than thedistance of �1.35 Å between the two strings of the correspondingDR 42.d formed by two strings of 21 (2�21) B-atoms. The TR 42.t iscalculated at �13 kcal/mol more stable than the disk 42.b and�53 kcal/mol more stable than the DR 42.d (values at TPSSh/6-311 + G(d) + ZPE).

Figure 6 also displays the ELI-D maps of 42.t. The map of densi-ties of states are given in Figure S4 of the ESI file. The r electronsare mainly located in tangential orbitals, connecting the B-atom ofeach string (Figure 6c). The p MOs, arising from radial atomic orbi-tals, are mostly delocalized also along each atom string but point-ing toward outside the tubular surface (Figure 6d). The sum ofcontributions of both types of electrons, as seen in Figure 6e,clearly result in the basins located between each boron-boronbond within either each string or between two strings. The energygap between both frontier orbitals of the TR 42.t amounts to�1.6 eV, which is close to that of �1.4 eV of the planar 42.b.

Figure 7 displays the maps of the magnetic ring currents of 42.tdue to the contributions of r electrons (Figure 7a) and p electrons(Figure 7b) and of the total density (Figure 7c). It is obvious thatconsistent and unambiguous responses occur inducing strong

diatropic current, irrespective of the electron type. Excitationswithin the frontier orbitals, as shown in Figure 8, namelyHOMO ? LUMO + 1 and HOMO-1 ? LUMO, bring in essential con-tributions to magnetic responses. The change of the K numberaccompanying these electronic transitions obeys the selection rule,DK = 1, thus making the TR tube aromatic. A similar situation waspreviously observed in the boron toroids, that are in fact the doublering structures of B20, B24, B28 characterized by a double aromatic-ity [19,29,45]. We would conclude that the B42 TR exhibits a tubu-lar aromaticity, which is seemingly a general feature of boronhollow cylinders. Such a tubular aromatic character tends to stabi-lize thermodynamically the boron tubes and thereby makes themlower-lying isomers.

4. Concluding remarks

In the present theoretical study, we determined the geometriesand chemical bonding phenomena of some boron clusters Bn withn = 30, 36, 40 and 42 in their neutral state using density functionaltheory and molecular orbital computations. The most importantresults emerge as follows:

(i) The bowl-shaped clusters B30, B36 and the planar disk B42

were confirmed to have a disk aromatic character. This isstrong in bowl B30 with consistent diatropic ring current.The weaker disk aromaticity in planar B42 makes it thermo-dynamically less stable,

(ii) A fullerene type having two hexagons and four heptagonswas identified as the lowest-lying isomer of B40. The MOsof this B40 fullerene closely mimic those of the buckyballB80. The presence of both hexagonal and heptagonal facesin a boron fullerene represents a novel structural aspectwhich deserves to be investigated further, and

(iii) The B42 cluster exhibits a triple ring as its lowest-lying iso-mer, which is slightly energetically more stable than the pla-nar disk. The triple ring B42 is characterized by a consistentdiatropic response to external magnetic field. Such a charac-teristic tubular aromaticity contributes to its high thermo-dynamic stability.

Overall, it appears that stabilized fullerene-type and tubularcylinder are frequently occurred in boron clusters Bn in the sizerange of n P 40, and their typical formation and specific bondingforeshadow different growth paths leading to larger cages, fuller-enes and tubes. Further investigations on larger boron clusterscould allow these patterns to be established.

Acknowledgments

We are grateful to the Department of Science and Technology ofHo Chi Minh City, Vietnam, for granting major research projects atICST. MTN is indebted to KU Leuven Research Council (GOA andIDO programs) for continuing support. We appreciate the help ofRemco Havenith in plotting the ring current maps, and the valu-able discussion with Arnout Ceulemans on the boron conundrum.

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.2014.05.069.

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