A disk-aromatic bowl cluster B30: towardformation of boron buckyballs†
Truong Ba Tai,a Long Van Duong,b Hung Tan Pham,b Dang Thi Tuyet Maia andMinh Tho Nguyen*a
The B30 boron cluster has a bowl rather than a double-ring or a
triple-ring tubular structure. This bowl isomer exhibits disk-
aromaticity similar to that found for B202� and B19
� clusters. We
confirmed that the concept of disk-aromaticity can be applied to
both planar and non-planar systems.
Over the past decade, boron clusters have been of great interestowing to their intriguing geometrical characteristics and aro-maticity.1–12 Boron is a rare element whose atomic clusters Bn
retain either planar or quasi-planar geometries as the size goesbeyond 20 atoms. Combined theoretical and experimentalstudies showed that anionic boron clusters Bn
� with n valuesof up to 24 have either planar or quasi-planar structures.3 Theneutral and cationic boron clusters adopt three-dimensionalstructures at smaller sizes.3,4,7,10,11 The B20 cluster was theore-tically found to have a double-ring form in which the twoten-membered rings are connected together in an antiprismbonding motif,4 whereas most smaller species are planar.1–3,7–9
In addition, theory previously predicted that the clusters B22,B24, B32 and B36 also have double-ring geometries which con-tain two 11-, 12-, 16- and 18-membered rings, respectively.1
These observations lead to the popular thinking thatintermediate-sized boron clusters Bn with n Z 20 exhibitsimilar tubular geometries. Nevertheless, while experimentalinfrared (IR) spectroscopic studies recently confirmed theplanar geometries of B11, B16 and B17, there is no good agree-ment between theoretical and experimental spectra for B20.5
Although a satisfactory explanation for such a discrepancy isstill lacking, it casts a certain doubt about the double-ringcharacteristic of these intermediate-size boron clusters andtherefore motivates us to further investigate their structures.
In this context, we performed a careful search for low-lyingisomers of B30 that is just two atoms less than the double-ringB32. We found that the B30 cluster prefers a bowl-shapedstructure I as a convex cluster rather than a double-ring tubularshape III like B32. More interestingly, this bowl species containsa disk-aromatic character similar to those found for B19
� andB20
2�, and this is the main reason for the high stability of I. Ourfindings not only give new insights into the bonding motif andaromaticity of boron clusters, but also suggest a new patternfor the formation of fullerene-like boron buckyballs such asB80 and B92.
The unbiased search for possible B30 structures was per-formed using a stochastic search method that was recentlydeveloped by us.13 The local minima with relative energies of0.0–5.0 eV were further optimized using several density func-tionals, such as the TPSSh, PBE, PBE0 and PW91, with the6-311+G(d) basis set. Recent studies have shown that func-tionals such as TPSSh, PBE, PBE0 can reasonably provide therelative energies of the isomers of boron clusters and arecomparable to those obtained using CCSD(T) calculations.7e,14
Finally, the calculations of electronic single point energies forthree lowest-lying isomers of the B30 cluster are performed atthe CCSD(T)/6-311G(d) level of theory with the PBE/6-311+G(d)geometries. Our computed results at the CCSD(T)/6-311G(d)revealed that while the double-ring tubular isomer III (Ci,
1Ag)(Fig. 1a) is less stable with a relative energy of 18.7 kcal mol�1,a triple-ring tubular isomer II (D5d, 1A1g) in which threeten-membered rings are connected together in an antiprismbonding motif, is the second lowest-lying isomer. More inter-estingly, the global minimum of B30 is calculated to be a bowl-shaped structure I (C5v, 1A1), which is more stable than II by13.3 kcal mol�1.
Bowl I is composed of an inner five-membered ring, sur-rounded by two larger ten- and fifteen-membered rings (Fig. 1).The HOMO–LUMO gap of I is equal to 2.2 eV and much largeras compared to the values of 1.3 eV for II and 0.2 eV for III. ThePBE cohesive energy (Ec) of I amounts to 5.499 eV per atom andis close to the value of 5.640 eV per atom of the buckyball B80.15
a Department of Chemistry, University of Leuven, Celestijnenlaan 200F,
B-3001 Leuven, Belgium. E-mail: [email protected] Institute for Computational Science and Technology (ICST), Ho Chi Minh City,
Vietnam
† Electronic supplementary information (ESI) available: Computational methods;shapes of selected MOs and shapes and relative energies of the low-lying isomersof B30 at the TPSSh/6-311+G(d) level. See DOI: 10.1039/c3cc48392d
Received 1st November 2013,Accepted 2nd December 2013
While the isomer II has a relatively high Ec value of 5.485 eV peratom, that of isomer III is lower (Ec(III) = 5.461 eV per atom).
Since the neutral Bn with n = 20–24, 32 and 36 were reportedto feature double-ring tubular structures, the present predic-tion marks a breakthrough in the bonding motif and structuralcharacteristics of boron clusters. Moreover, the existence of abowl B30 cluster suggests a more consistent pattern for theformation of the buckyball B80 which is still not wellunderstood.
Although experimental results are yet to come, the buckyballB80 and its derivatives have attracted much attention.15–17 Withthe high stability of the bowl B30, formation of B80 becomesstraightforward. As seen in Fig. 1b, B80 can be effectivelyformed from B30 units. Connection of two bowl B30 species totwo up and down sides of double-ring B30 species, followed bythe removal of ten B-atoms from the centres of 10 five-membered rings (orange colored atoms, Fig. 1b), will form abuckyball B80. Additionally, a fullerene-like structure B92 will beformed as two excess B-atoms are added into the remainingholes of two five-membered rings. However, the latter specieswas theoretically reported to be less stable, and its cohesiveenergy is smaller than that of B80.15
In the same vein, another bowl structure is also located as astable local minimum on the potential energy surface of B45.The bowl B45 cluster has a cohesive energy of 5.513 eV per atom,which is consistently higher than the value of 5.499 eV per atomof the bowl B30 I mentioned above. It is noteworthy that asimilar structure was found for B46 by Boustani.2 Furtherstudies are necessary to confirm as to whether it is the B45
global minimum, but the presence of this stable bowl providesus with further evidence for the high stability of B80. It also
suggests a route for the formation of boron fullerenes: B30 -
B45 - B90 - B80 and/or B92.Aromaticity is another consistent characteristic of boron
clusters and was extensively discussed in the literature. Whilethere is now a consensus in the evaluation of aromaticityof small boron clusters Bn, the aromaticity of larger clustersremains a challenging issue.3,4,6–9 For instance, B19
� and B202�
are identified as aromatic species which have negative NICSvalues. However, each contains 12 valence p-electrons whichdoes not obey the classical Huckel rule of (4N + 2) electrons. Toaddress this contradiction, we recently proposed the concept ofdisk-aromaticity on the basis of a simple model of a particle in acircular box.6 Within its framework, a free particle is moved ona plane encircled by infinite walls. The radius of the disk isdenoted by r = R. In polar coordinates, the Schrodinger equa-tion for this problem is written as follows:
� �h2
2m@2
@r2þ 1
r
@
@rþ 1
r2@2
@j2
� �Cðj; rÞ ¼ ECðj; rÞ (1)
where �h is the Planck constant and m is mass of the particle.Because of the circular symmetry, C(j,r) can be written asR(r).F(j), with F(j) = exp(imj)/(2p)1/2.
The cyclic boundary condition requires the angular part tobe periodic. As a result, the cylindrical quantum number mustbe an integer: m = 0, �1, �2. . . Substitution into the Schrodin-ger equation gives us the radial part:
@2RðrÞ@r2
þ 1
r
@RðrÞ@rþ k2 �m2
r2
� �RðrÞ ¼ 0 (2)
with �h2k2 = 2mE. This equation is known as Bessel’s differentialequation, and its solutions are the integer Bessel functionsJm(kr).18 The potential wall at r = R requires the radial functionto vanish at the boundary of the box: Jm(kR) = 0. The radii thatcorrespond to the zeroes of the Bessel function are denoted as am,n.Here n is a radial quantum number that counts the zeroes.The am,n quantities are dimensionless. They give rise to aquantisation of the energy as:
E ¼�h2 am;n� �22mR2
with : n ¼ 1; 2; 3 . . .m ¼ 0;�1;�2;�3 (3)
The rotational quantum numbers are usually denoted by Greekletters as m = s, p, d, f, g. . . States with non-zero values for mwill be twofold degenerate. The lowest eigenstates in theascending order are 1s, 1p, 1d, 2s etc. . . (Fig. 2). We considerthat the systems containing 2, 6, 10, 12, 16, 20. . . electronswhich fully occupy degenerate eigenstates of the model willexhibit a disk-aromaticity. Contrarily, the systems containing 4,8, 14, 18. . . electrons which only singly occupy one of twohighest degenerate eigenstates will be disk-antiaromatic. Thisconcept rationalizes the intriguing aromaticity of B19
� andB20
2�, consistent with other indexes such as magnetic ringcurrent, nucleus independent chemical shift (NICS) calcula-tions and the ipsocentric model. We expect that this conceptcan be applied to related systems such as the bowl B30 cluster,even though they are not strictly planar.
Fig. 1 (a) Shapes of the lowest-lying isomers of B30 and (b) route forformation buckyball B80 and B92 from B30. Orange colored points areatoms located at the centres of five-membered rings.
Shapes of selected molecular orbitals (MOs) in Fig. 2 revealthat I contains 20 valence p-electrons that fully occupy theorbitals HOMO, HOMO – 1, HOMO – 4, HOMO – 5, HOMO –11 and HOMO – 13. Based on Huckel’s rule, this species must beantiaromatic (4N electrons, N = 5). However, based on the modelof a particle in a circular box, the isomer I is a disk-aromaticsystem whose valence p-electrons fully occupy the lowest eigen-states in ascending order of (1s)2(1p)4(1d)4(2s)2(2p)4(1f)4. Thesepredictions agree with NICS calculations.19 NICSzz values calcu-lated at various positions (see ESI†) show that I and II are highlyaromatic species with highly negative NICSzz values (NICSzz(1) (I) =�57.8 to �71.1 and NICSzz(1) (II) = �4.1 to �29.8). In contrast,the double-ring tubular structure III is antiaromatic with highlypositive NICSzz values. The aromatic character of isomers I and IIis presumably the main reason for their high stability, whereasthe double-ring III becomes less stable due to its antiaromaticcharacter.
Moreover, it is noteworthy that there is concurrence in thenumbers of electrons (x) between the Huckel (4N + 2) electronsand the model of a particle in a circular box as x r 10.The classical Huckel’s rule is mostly effective for monocycliccompounds with a limited number (x) of valence p-electrons.A monocyclic compound with x > 10 has not been observed yet.Thus, the proposed model of disk-aromaticity is promising forevaluation of aromaticity as it can be applied for both smallmonocyclic and larger polycyclic systems, irrespective of theirplanarity. Similar conclusions are also reached when the ipso-centric model20 is applied. As shown in the ESI,† both isomers Iand II are aromatic, whereas isomer III is antiaromatic in theradial stack, but aromatic in the tangential stack system.
In conclusion, we found that the B30 boron cluster has abowl-shaped structure I, rather than the double-ring or triple-ring tubular structure as previously suggested. This bowl isomer Iexhibits disk-aromaticity similar to that found for B20
2� and B19�
clusters. These findings not only mark an important breakthrough
in the understanding of bonding motif and structural character-istics of intermediate-sized boron clusters, but they also suggesta consistent route for the formation of boron buckyballs suchas B80 and B92, which arise from multiple B30 units. Furtherinvestigations need to be performed to determine the energybarriers of the formation pathways of these buckyballs, andthereby obtain more realistic molecular mechanisms. We alsoconfirmed that the concept of disk-aromaticity can be applied tonon-planar boron clusters. Further studies are highly desirableto apply this concept to other polycyclic compounds, and inparticular to different classes of atomic clusters.
We are indebted to the KU Leuven Research Council (GOAand IDO programs). TBT thanks the FWO-Vlaanderen for apostdoctoral fellowship.
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Fig. 2 Shapes of p-MOs of bowl B30 cluster and the lowest-lying wave-functions for a particle in a circular box.
