Electronic Structure and Chemical Bonding in the ... - ACS Publications

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Electronic Structure and Chemical Bonding in the Double Ring Tubular Boron Clusters Hung Tan Pham,† Long Van Duong,† and Minh Tho Nguyen*,‡ †

Institute for Computational Science and Technology (ICST), Quang Trung Software City, Ho Chi Minh City, Vietnam Department of Chemistry, KU Leuven, B-3001 Leuven, Belgium

‡

S Supporting Information *

ABSTRACT: We analyzed the chemical bonding phenomena of the boron B2n tubes (n = 10−14). The B2n tubes represent stable hollow cylinders each having a double ring (DR) among pure boron clusters. The shapes of the molecular orbitals of a B2n DR can be predicted by the eigenstates of a simple model of a particle on a hollow cylinder, which shows both radial and tangential components. In a DR tube, strong diatropic responses to external magnetic field occur in both radial and tangential types of electrons, and thus confer it a characteristic tubular aromaticity. The presence of a consistent aromatic character contributes to the high thermodynamic stability of a DR. The number of electrons in a hollow cylinder should attain (4N + 2M) with M = 0 and 1 for both series of radial and tangential electrons, depending on the number of nondegenerate MOs occupied, to properly fill the electron shells. In the case of B20, M = 1 for both radial and tangential electrons, and the classical Hückel counting rule is thus recovered.

1. INTRODUCTION Kiran and co-workers first reported in 20041 that the most stable equilibrium structure of the 20-atoms boron cluster has a double ring (DR) form, rather than a quasiplanar shape as in the smaller size boron clusters (Bn with n < 20).2,3 The B20 DR structure arises from a superposition of two B10 strings leading to a tube with a diameter of ∼5.5 Å, and this was considered as an embryo of the thinnest boron nanotubes. Subsequently, extensive searches on the larger sizes of boron clusters revealed that the DR tubular structure remains the lowest-energy isomer in the B2n neutral species up to the size B28. Recently,4 the B22 DR is confirmed to be the most stable isomer in the neutral, cationic, and dicationic states.5,6 For the B24 size, the DR tubular structure is also predicted to be the global minimum at the neutral and positively charged states.4,7 The chemical bonding and aromatic character of the tubular B20 were investigated in subsequent studies.1,8,9 The B20 DR can be formed by rolling up a planar B20 structure including two B10 strings, and then it is in turn stabilized by strong sp2 hybridized σ bonds connecting both B10 rings, and also by delocalized π bonds over both inner and outer surfaces. As a consequence, the aromatic feature of the B20 DR analogues was regarded as similar to quasiplanar boron clusters.1 In other theoretical investigations on B2n DR structures with n = 10−14, Johansson8 and Bean and Fowler9 demonstrated that the structures having an even (2n) number of B atoms are likely to keep high symmetry with a point group of Dnd, whereas a DR with an odd number of B atoms will be distorted into a lower Ci or even C1 point group, due to the open-shell electronic structure.8,9 The latter authors9 also explored the bonding of © XXXX American Chemical Society

B2n DRs using the ring current maps. Accordingly, the total ring current densities involving both radial and tangential electrons for B2n clusters turn out to be strongly diatropic, due to the contributions of both diatropic radial and tangential electron responses. In the corresponding odd electron number species, the total current density was calculated to be paratropic arising from a competition between both tangential and radial currents. Within a framework of a double aromatic character, it was suggested that both radial and tangential electrons of these DR tubes, called boron toroids, simply obey the classical (4N + 2) Hückel rule, as is the case of planar cyclic structures.9 A boron toroid basically exhibits the shape of a hollow cylinder. Therefore, a fundamental question is as to whether their electronic structure can be rationalized in terms of a model of a particle on a hollow cylinder. Recently, the model of a particle on a disk was shown to be able to determine the electron distribution and aromatic feature of planar and quasiplanar disk molecules in terms of an irreducible representation of MOs and electron counting.10−13 In this context, we set out to analyze the electronic distribution and aromatic character of the B2n DR clusters using, among others, a model of a particle in a hollow cylinder box.

2. METHODS All electronic structure calculations are carried out with the Gaussian 09 program.14 Geometries of the B2n DR structures Received: August 5, 2014 Revised: September 25, 2014

A

dx.doi.org/10.1021/jp507901n | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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are optimized by using density functional theory with the TPSSh functional15 and the 6-311+G(d) basis set. To confirm the identity of the B2n DR forms considered, their harmonic vibrational frequencies are also calculated at the same level. The shapes of the optimized structures of the DR considered are given in Figure S1 of the Supporting Information. To probe further the chemical bonding, we use the electron localizability indicator (ELI-D) technique whose map reveals the molecular basins. A basin is a region of the molecule that the electrons are likely to occupy. Accordingly, the basins usually correspond to the cores, valence bonds, and lone pairs.16 The aromatic character is approached by the ring current maps17 along with the ipsocentric model.18,19 Calculations of the magnetic ring currents are carried out with the SYSMO program,20 in conjunction with the Gamess-UK program.21

Θ(θ ) =

1 ilθ e 2π

The Z(z) function responds to the boundary conditions at two surfaces z = 0 and z = L, in such a way that the rational quantum number k must be a positive integer (1, 2, 3, ...). The periodic boundary condition for the Θ(θ) function implies that to the rotational quantum number l must also be an integer (0, ± 1, ± 2, ...). Solution of the P(ρ) function with the Dirichlet boundary conditions (vanishing of the wave function on the boundary) at inner and outer radii positions (ρ = R0 and ρ = R1) turns out to be a Bessel function (eq 3): Jl (klnR(1 + ε))Yl(klnR(1 − ε)) − Jl (klnR(1 − ε)) Yl(klnR(1 + ε))

3. RESULTS AND DISCUSSION 3.1. The Hollow Cylinder Model. Let us first briefly present the hollow cylinder model (HCM). Figure 1 displays

(3)

=0

in that Jl and Yl are Bessel functions of order l of the first and second kind. The value of ε = r/R is called the pseudoradius of the hollow cylinder. Determination of the klnR parameters helps us find the eigenvalues of eq 1 that can be written as (eq 4): E=

⎛ kπ ⎞2 ⎞ ℏ2 ⎛ 2 ⎜ ⎜ f ⎟ ⎟⎟ + k R ( ) ⎜ ln ⎝L ⎠ ⎠ 2mR2 ⎝

(4)

f

with L = L/R being called the pseudoheight of the hollow cylinder. The value of klnR is determined by solving f(klnR) = 0 with the certain pseudoradius ε and the rotational quantum number l. This gives raise to the third quantum number in the HCM, namely the radial quantum number n (n = 1, 2, 3, ...). Figure S2 in the Supporting Information shows a determination of the parameter kjnR. Overall, the eigenstates in the HCM are characterized by three quantum numbers including the rational (k = 1, 2, 3, ...), the rotational (l = 0, ±1, ±2, ...), and the radial (n = 1, 2, 3, ...). The eigenstates in ascending order of the corresponding eigenvalues will be changed by any change of the value of ε or Lf. The shape of these eigenstates will be plotted and compared in a following section. 3.2. The Cylinder Model for B2n Double Rings. After determining the eigenstates of the HCM, we now compare them with the molecular orbitals (MOs) actually computed using quantum chemical methods for DR tubes. In the case of a B2n DR form, with n = 10−14, the MOs that are lying closed to the HOMO can be classified into two distinct sets, namely the radial MOs (r-MOs) and the tangential MOs (t-MOs), on the basis of the orientation of the p-lobes.9 In the more conventional terminology, t-MOs correspond to σ-MOs, whereas r-MOs can be described as π-MOs. Another set of MOs can be called s-MOs that are constructed by major contributions from s-AOs. The densities of states (DOS) of the B20 DR displayed in Figure 2 indicate the participation of AOs in all MOs, whereas the partial DOS arise from each individual MO set, including the tangential, radial, and s-MOs, and compared to the total DOS. As seen in Figure 2, there is a major contribution of a typical AO in each peak of the DOS. Accordingly, the s-MOs set is an essential contributor in the low-energy region, whereas the tangential and radial sets have more significant contributions to MOs near the HOMO region. The DOS plots of B24, B28 tubes are given in Figures S3−S5, Supporting Information.

Figure 1. Parameters of a hollow cylinder: L is height, R is radius. The movement of the particle is restricted to the annular space from R0 to R1, with R0 = R − r, R1 = R + r. r is called the active radius of the hollow cylinder.

the parameters defining a cylinder. The Schrödinger equation for a particle moving in such a cylinder box is written as follows (eq 1): ⎤ ⎡ ℏ2 ⎛ ∂ 2 1 ∂ 1 ∂2 ∂2 ⎞ ⎢− + 2 2 + 2 ⎟ + V (ρ , z ) ⎥ Ψ ⎜ 2 + ρ ∂ρ ρ ∂θ ∂z ⎠ ⎦ ⎣ 2m ⎝ ∂ρ (1) = EΨ where ρ is the radial coordinate and z is the height of the box. Here, V(ρ,z) is a step potential confining the particle to the hollow cylinder, i.e.,

⎧ 0 if R 0 ≤ ρ ≤ R1 and 0 ≤ z ≤ L V (ρ , z ) = ⎨ ⎩∞ otherwise

(2)

22

Gravesen and co-workers solved eq 1 in which the wave function could be written as P(ρ)Θ(θ )Z(z) with Z(z) =

2 ⎛ kπ ⎞ sin⎜ z⎟ L ⎝L ⎠

and B



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Figure 2. Total (DOS) and partial (pDOS) densities of state of B20 DR.

the cylinder model. Likewise, the first doubly degenerate occupied HOMO−5,5′ are excellently described by the solution (1 ±1 2) of the cylinder model in terms of irreducible representation. The next doubly degenerate HOMO−1,1′ can be assigned to the (1 ±2 2) eigenstates. Again, most of the radial MOs are basically formed by the 2p or 2sp AOs with their axis perpendicular to the tube’s axis. Therefore, the numbers of the radial node have only one value n = 2. The number of rational nodes k could not be greater than the number of rings in the tube. The number of rotational nodes l could also not be greater than the number of atoms in each ring. In the B20 DR case, we thus have the quantum numbers n = 2, k = 1, 2, and l = 1, 2, ...10. Figure 4 also compares the MO energies of the r-MOs set obtained by actual DFT calculations to the spectrum of eigenvalues derived from the HCM. Accordingly, a good correlation for this set of orbitals in terms of MO energies can be obtained. Regarding the characteristics of the r-MOs set for a DR tube, the ascending ordering of MOs in this set again follows a certain rule. The (1 0 2) state turns out to be the lowest-lying state in this set, along with the eigenstates (2 0 2) also having a rotational number l = 0. Located between the (1 0 2) and (2 0 2) states, the states with l ≠ 0 are known as 2-fold degenerate states. The number of 2-fold degenerate states appearing in the above positions constitutes a parameter of the hollow cylinder size (pseudoradius and pseudoheight). In this context, the electron count will basically depend on the occupancy of nondegenerate states. Hence, the electrons involved will be (4N + 2) if only the (1 0 2) solution is filled, or (4N) if the (2 0 2) eigenstate is also fully occupied. 3.2.b. The Tangential MOs Set (t-MOs) of B20. Let us now apply the HCM to the tangential MOs set. Figure 5 displays the t-MOs (σ-MOs) set of the B20 DR actually calculated with DFT/TPSSh/6-311+G(d), and the corresponding eigenstates obtained by the HCM with the parameters ε = 0.177 and Lf = 1.803. Here, the HOMO−3 corresponds to the lowest-energy

It is clear that each type of AOs gives a significant contribution to a peak of the DOS. The electron pattern of the B20 DR cluster can thus be divided into three separate subshells. Because the energies of the s-MOs set are close to the core region, and the fact that this MOs set has no significant contribution to the current density, they will not be considered further in the treatment of the HCM. 3.2.a. The Radial MO Set (r-MOs) of B20. Figure 3 compares the shapes of the radial set involving the occupied MOs and the corresponding eigenstates (or solutions) derived by the HCM with the parameters ε = 0.694, Lf = 1.714. The HOMO−7 has the same nodal characteristic with the solution (1 0 1) given by

Figure 3. Shape of some delocalized MOs (including occupied and unoccupied) of B20 DR obtained with the DFT/TPSSh/6-311+G(d) method and classified according to the radial (π) shape of the MOs, and the corresponding wave functions obtained by solving the Schrödinger equation for the particle in a hollow cylinder model (HCM). Lower panel: The model solution with quantum numbers k, l, and n, respectively. The ± values stand for a doubly degenerate state. C



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= 1. The number of rational nodes k is consistently greater than the ring number of the tube. Finally, the number of rotational nodes cannot be larger than the number of atoms of each ring. As a result, we now have for this MO set the numbers n = 1, k = 3, 4, ..., and l = 1, 2, ...10. Figure 5 again illustrates a good correlation between the calculated t-MOs orbitals and the HCM solutions in terms of eigenvalues. The tangential MOs set for a DR (in an ascending ordering) should naturally obey a certain rule. For example, the solution (3 0 1) is thus the lowest-lying state of this MOs set. The (4 0 1) eigenstate is the next nondegenerate state, but it is an antibonding MO when applied to the boron DR structure. Thus, if the (4 0 1) level is occupied, the tube does not get much benefit in terms of stabilization. The doubly degenerate MOs corresponding to the value l ≠ 0 will be filled by 4 electrons. In such a case, the t-MOs set simply involves (4N + 2) electrons. 3.2.c. The Electron Count for a Hollow Cylinder. Table 1 summarizes the numbers of electrons involved in a DR form Table 1. Number of Electrons for Radial (π) and Tangential (σ) of the B20 DR Cylinder π-MOs

Figure 4. A correlation between computed orbital energies (DFT/ TPSSh/6-311+G(d)) of B20 and eigenvalues given by the particle in HCM.

σ-MOs

k

l

n

k

l

n

1 1 1 1 2

0 ±1 ±2 ±3 0 ⋮

2 2 2 2 2

3 3 3 3 4

0 ±1 ±2 ±3 0 ⋮

1 1 1 1 1

(such as the B20 DR) in both radial and tangential types. As stated above, in the HCM, each MO is described by three quantum numbers including k, l, and n. The value of n depends on the nature of MO, namely n = 1 for the t-MOs set and n = 2 for the r-MOs set. Consequently, the energy ordering and irreducible representation of MOs depend on both numbers k and l. Let us now apply the electron counting rule for B2n DR structures. As shown in Figures 3 and 5 the B20 DR has enough valence electrons to be filled up to the (1 ±2 2) state of the radial MOs set with 10 electrons. The tangential MOs set of B20 DR is also occupied by 10 electrons. Therefore, both t-MOs and r-MOs sets of B20 DR satisfy the (4N + 2) electron count with 10 electrons, and as a consequence the B20 DR keeps the highest point group of D10d. The t-MOs set of the B22 DR is fully occupied by 10 electrons, whereas 12 electrons are occupied in the r-MOs set of this structure. Each MO of the doubly degenerate (1 ±3 1) solution is occupied by two electrons. As a result, the B22 DR is subjected to a Jahn−Teller effect, and will be distorted into a Ci structure. The B24 DR comprising two B12 strings has sufficient electrons to fill both t-MOs and r-MOs sets with 10 and 14 electrons, respectively. This is apparently the driving force for the B24 DR structure to exist in high symmetry with a D12d point group. Similar to the B22 DR, the B26 DR formed by superposition of two B13 strings has a Ci point group. This DR has enough radial electrons to doubly occupy to the (1 ±3 2) state, but the number of tangential electrons are enough to fill only one of

Figure 5. Shape of some delocalized MOs (including occupied and unoccupied) of B20 DR obtained with the DFT/TPSSh/6-311+G(d) method, and classified according to the tangential (σ) shape of the MOs, and the corresponding wave functions obtained by solving the Schrödinger equation for the particle in a HCM. Lower panel: The model solution with quantum numbers k, l, and n, respectively. The ± values stand for a doubly degenerate state.

(3 0 1) eigenstate obtained by the cylinder model. The doubly degenerate MOs, namely HOMO−2,2′, are reproduced by the (3 ±1 1) eigenstates. Similarly, the (3 ±2 1) solutions describe quite well the doubly degenerate MOs comprising both HOMO and HOMO′, in terms of irreducible representation. The LUMO+3,3′ of B20 DR have the same irreducible representation as the (3 ±3 1) solution, but when applying to the B20 DR they are vacant MOs. Basically, the t-MOs result from a combination of 2p or 2sp hybridized AOs having their axis parallel to the tube’s axis. In this situation, the radial quantum number has only one value n D



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Figure 6. Ring currents maps of the radial (π), tangential (σ), and total (π + σ) electrons in the B20, B24, and B28 DR structures, using the ipsocentric model with an external magnetic field directed along the cylinder axis. The current maps are plotted in the median plane, with arrows representing the in-plane component and contours representing the modus. Circular symbols indicate the nuclear positions projected into the plotting plane. The σ and π electrons induce diatropic current densities, indicating that both of them contribute into the aromaticity of B2n.

value of l ≠ 0 is doubly degenerate, they give rise to 4N electrons when filled in both MOs sets. However, the MOs having l = 0 only contribute 2 valence electrons each into the total account of electrons when they are occupied. In this context, the classical (4N +2) Hückel rule is a special case of the HCM when only one MO having l = 0 is doubly occupied. Figure 7 displays the electron localizability indicator (ELI-D) maps of the B20 DR structure. To identify the molecular basins that radial and tangential electrons likely occupy, the decomposed ELI-Dπ and ELI-Dσ maps are also calculated for only radial (Figure 7a) and tangential (Figure 7b) electrons, respectively. The localization pattern of π-MOs comprises both

the doubly degenerate (3 ±3 1) states. Therefore, a Jahn− Teller distortion again occurs. The B28 DR constructed by combination of two B14 strings has again enough electrons to fill both t-MOs and r-MOs sets, with 14 electrons for each set. Overall, the electron count derived from the HCM can successfully explain the geometrical features of boron B2n DR cluster structures. In an attempt to probe the aromatic and antiaromatic features of B2n clusters, the current density maps for each MOs set and the total densities involving all π and σ electrons are performed by using the B3LYP/6-311G(d) densities and shown in Figure 6. Our calculations are consistent with those of Bean and Fowler.9 The B20, B24, and B28 DR structures are calculated to have diatropic current density in both t-MOs and r-MOs sets, and as a result, they have an aromatic character. For the B22 DR, the t-MOs induce a diatropic (clockwise) magnetic response, whereas its r-MOs set produces a strong paratropic (anticlockwise) ring current. In the B26 DR, the t-MOs set also produces strong paratropic current density, whereas the r-MOs counterpart raises a diatropic current. A certain correlation between the magnetic response and the number of valence electrons involved can be found for the B2n DR structures. Only the MOs set which satisfies the (4N + 2) electron counting rule induces a diatropic ring current in terms of magnetic response, and consequently the relevant B2n DR forms can be assigned to have a tubular aromaticity. However, we demonstrate here that the electron counting rule derived by the HCM appears to be more general than the classical Hückel (4N + 2) rule. Due to the fact that the MO corresponding to a

Figure 7. Maps of (a) ELI-Dπ (radial) and (b) ELI-Dσ (tangential) isosurfaces of the DR B20 at a bifurcation value of 1.3. E



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(2) Tai, T. B.; Tam, N. M.; Nguyen, M. T. The Boron Conundrum: The Case of Cationic Clusters Bn+ with n = 2−20. Theor. Chem. Acc. 2012, 131, 1241−1256. (3) Tai, T. B.; Tam, M. N.; Nguyen, M. T. Structure of Boron Clusters Revisited, Bn with n = 14−20. Chem. Phys. Lett. 2012, 530, 71−76. (4) Hung, P. T.; Long, D. V.; Buu, P. Q.; Nguyen, M. T. The 2D-to3D Geometry Hopping in Small Boron Clusters: The Charge Effect. Chem. Phys. Lett. 2013, 577, 32−37. (5) Hikmat, B. C.; Tunna, B.; Rajendra, R. Z. Low-lying Planar Isomers of Neutral and Charged B22 Clusters. J. Phys. B: At., Mol. Opt. Phys. 2012, 45, 225101−225107. (6) Sergeeva, A. P.; Piazza, Z. A.; Romanescu, C.; Li, W. L.; Boldyrev, A. I.; Wang, L. S. B22 and B23: All-Boron Analogues of Anthracene and Phenanthrene. J. Am. Chem. Soc. 2012, 134, 18065−18703. (7) Chacko, S.; Kanhere, D. G.; Boustani, I. Ab Initio Density Functional Investigation of B24 Clusters: Rings, Tubes, Planes, and Cages. Phys. Rev. B 2003, 68, 035414−035424. (8) Johansson, M. P. On The Strong Ring Currents in B20 and Neighboring Boron Toroids. J. Phys. Chem. C 2009, 113, 524−530. (9) Bean, D. E.; Fowler, P. W. Double Aromaticity in “Boron Toroids. J. Phys. Chem. C 2009, 113, 15569−15575. (10) Moreno, D.; Pan, S.; Zeonjuk, L. L.; Islas, R.; Osorio, E.; Martínez-Guajardo, G.; Chattaraj, P. K.; Heine, T.; Merino, G. B182−: A Quasi-Planar Bowl Member of The Wankel Motor Family. Chem. Commun. 2014, 50, 8140−8143. (11) Tai, T. B.; Long, D. V.; Hung, P. T.; Mai, D. T. T.; Nguyen, M. T. A Disk-Aromatic Bowl Cluster B30: Toward Formation of Boron Buckyballs. Chem. Commun. 2014, 50, 1558−1560. (12) Tai, T. B.; Ceulemans, A.; Nguyen, M. T. Disk Aromaticity of the Planar and Fluxional Anionic Boron Clusters B200/2−. Chem.Eur. J. 2012, 18, 4510−4512. (13) Tai, T. B.; Havenith, R. W. A.; Teunissen, J. L.; Dok, A. R.; Hallaert, S. D.; Nguyen, M. T.; Ceulemans, A. Particle on a Boron Disk: Ring Currents and Disk Aromaticity in B202−. Inorg. Chem. 2013, 52, 10595−10600. (14) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar,S. S. et al. Gaussian 09, revision B.01; Gaussian, Inc.: Wallingford, CT, 2009. (15) Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Climbing the Density Functional Ladder: Nonempirical Meta−Generalized Gradient Approximation Designed for Molecules and Solids. Phys. Rev. Lett. 2003, 91, 146401−146404. (16) Kohut, M.; Wagner, F. R.; Grin, Y. Atomic Shells from Electron Localizability in Momentum Space. Int. J. Quantum Chem. 2006, 106, 1499−1507. (17) Zanasi, Z. Coupled Hartree−Fock Calculations of Molecular Magnetic Properties Annihilating The Transverse Paramagnetic Current Density. J. Chem. Phys. 1996, 105, 1460−1469. (18) Steiner, E.; Fowler, P. W. Patterns of Ring Currents in Conjugated Molecules: A Few-Electron Model Based on Orbital Contributions. J. Phys. Chem. A 2001, 105, 9553−9562. (19) Keith, T.; Bader, R. F. W. Calculation of Magnetic Response Properties Using a Continuous Set of Gauge Transformations. Chem. Phys. Lett. 1993, 210, 223−231. (20) Lazzeretti, P.; Zanasi, R.; SYSMO package, University of Modena: Modena, Italy, 1980; with additional routines for evaluation and plotting of current densities by Steiner, E.; Fowler, P. W.; Havenith, R. W. A.; Soncini, A. (21) Guest, M. F.; Bush, I. J.; van Dam, H. J. J.; Sherwood, P.; Thomas, J. M. H.; van Lenthe, J. H.; Havenith, R. W. A.; Kendrick, J. The GAMESS-UK Electronic Structure Package: Algorithms, Developments and Applications. Mol. Phys. 2005, 103, 719−747. (22) Gravesen, J.; Willatzen, M.; Yan Voon, L. C. L. Schrödinger Problems for Surfaces of Revolution - The Finite Cylinder as a Test Example. J. Math. Phys. 2005, 46, 012107−012112.

inner and outer surfaces, in which the outer surface is more delocalized than the inner counterpart. The σ-electrons mainly occupy the tangential orbitals connecting B atoms on each string. The localization domains of σ-MOs involve three domains delocalized over the whole DR structure. The sum of contributions of both types of electrons results in the basins located between each boron−boron bond either within the same string or between two strings. Overall, the delocalization pattern of both sets of radial and tangential MOs suggests that the electrons occupying these MOs cylindrically move over the whole 2n tube.

4. CONCLUDING REMARKS In summary, we have proposed in this theoretical investigation a simple model based on the solutions of the particle on a hollow cylinder to address the electronic structure of the double ring tubular boron clusters B2n, with n = 10−14. We have demonstrated that the MO shapes of B2n DR can be predicted (or reproduced) excellently by the eigenstates of the hollow cylinder model. The electron counts for both r-MOs and t-MO sets of DR tubes are also established, which are either (4N + 2) or (4N) depending on the occupancy of a nondegenerate orbital. The classical Hückel (4N + 2) rule, when applied to the tubes, appears thus to be a special case of the electron counts derived by the hollow cylinder model. The latter is valid when only one nondegenerate orbital is double occupied. Combined with the current density maps, we have demonstrated that the B2n DR clusters can be regarded as species with tubular aromaticity. Such a behavior is similar to that of a triple ring, such as the B27+ cluster,23 whose MOs can also be predicted by a hollow cylinder model.

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ASSOCIATED CONTENT

S Supporting Information *

Table defining the parameters of a double ring cylinder and figures showing the B2n DR structures, determination of the k parameter, and total and partial densities of states (DOS) diagrams for some clusters. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +32-16327361. Fax: +32-16-327992. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to the Department of Science and Technology of Ho Chi Minh City, Vietnam, for granting major research projects at ICST. M.T.N. thanks ICST for making his stays in Vietnam enjoyable. We are indebted to KU Leuven Research Council (GOA and IDO programs) for continuing support. We thank Dr. Remco Havenith from Groningen University for assistance with the SYSMO package.

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REFERENCES

(1) Kiran, B.; Bulusu, S.; Zhai, H.-J.; Yoo, S.; Zeng, X. C.; Wang, L.-S. Planar-to-Tubular Structural Transition in Boron Clusters: B20 as The Embryo of Single-walled Boron Nanotubes. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 961−964. F



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(23) Duong, V. L.; Pham, H. T.; Tam, N. M.; Nguyen, M. T. A Particle on a Hollow Cylinder: The Triple Ring Tubular Cluster B27+. Phys. Chem. Chem. Phys. 2014, 16, 19470−19478.

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