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J. Phys. Chem. C 2009, 113, 524–530
On the Strong Ring Currents in B20 and Neighboring Boron Toroids Mikael P. Johansson† Lundbeck Foundation Center for Theoretical Chemistry, Aarhus UniVersity, DK-8000 Århus C, Denmark, and Laboratory for Instruction in Swedish, Department of Chemistry, UniVersity of Helsinki, FI-00014 Helsinki, Finland ReceiVed: June 5, 2008; ReVised Manuscript ReceiVed: NoVember 11, 2008
Magnetically induced ring currents in the toroidal, double ring forms of the boron clusters Bn (n ) 16, 18, 20, 24) are investigated by using quantum chemical methodology. B16, B20, and B24 are found to sustain high diamagnetic currents, fulfilling the magnetic criterion of aromaticity, while B18 and B22 appear to be strongly antiaromatic. Possible connections to the stabilities of these nanoclusters are discussed. 1. Introduction The neutral form of B20 was recently suggested to be a highly symmetric, beautiful double ring,1 marking the onset of true threedimensionality for small, neutral boron clusters. Interestingly, the tubular anion, B20-, is much less stable when compared to alternative, pseudoplanar isomers. This finding has been corroborated by later theoretical work by An et al.2 The cation, on the other hand, has been shown to be tubular in a joint experimental and theoretical work by Oger et al.3 Kiran and co-workers1 suggested B20 to be highly aromatic, in analogy with earlier findings for smaller boron clusters.4 The calculated high magnetic shielding in the ring center also hints at a presence of a strong induced ring current.2 This effect could perhaps account for the observed stability of the species. Here, the magnetically induced ring currents5-7 in B20 and its nearest neighbors are investigated. Strong ring currents are found in all species. 2. Methodology The quantum chemical calculations have been performed within the density functional theory8,9 (DFT) framework with Becke’s three-parameter hybrid exchange functional10 in connection with the Lee-Yang-Parr correlation,11 B3LYP; the correlation of the uniform electron gas was modeled with the Vosko-Wilk-Nusair VWN5 formulation.12 A doubly polarized basis set of triple-ζ quality, TZVPP,13 was used. The structures were optimized, followed by nuclear magnetic resonance (NMR) calculations. The ring currents themselves have been obtained with the Gauge Including Magnetically Induced Currents (GIMIC) methodology, developed by Juse´lius et al.14 GIMIC can provide a quantitative determination of the strength of ring currents. Earlier work, with the Continuous Transformation of the Origin of the Current Density (CTOCD) and later refinements15-20 leading the way, has so far not been used to obtain a definitive value for the current strength. In GIMIC, the magnetically induced current is calculated explicitly, from first principles, in points in space. By defining suitable cut planes, the net current in the ring, including diamagnetic and paramagnetic contributions, can be obtained by numerical quadrature. Figure 1 shows a schematic of a typical definition of the cut plane. The method has previously proven reliable in studies of ring currents in small heterocycles21 and metal clusters,22,23 as well as for three-dimensional sphere currents.24 † E-mail:
[email protected].
Figure 1. Schematic showing a definition of the cutplane (in green) used for integrating the induced currents. The thick blue arrows represent the current flow around the boron toroid. The z-axis is marked by a thin black arrow. The definitions of width and height of the integration plane are marked with “w” and “h”, respectively.
For comparison, the Nucleus-Independent Chemical Shift (NICS)25,26 has also been considered and calculated. To check the multireference character of the species, natural orbital occupations were calculated at the approximate second-order coupled cluster (CC2) level.27 The Turbomole program suite28-32 was used for all DFT and CC2 calculations. The NMR calculations were performed with the mpshift program33 modified to provide the perturbed densities necessary for the ring current calculations. Vibrational frequencies were calculated analytically.34 Standard convergence and threshold parameters were used, with the following, tighter, exceptions: For currents, the “m4” type DFT grid35 was used and energies were converged to 10-7 hartree; the geometry optimizations used a gradient norm convergence of 10-4; for the vibrational frequencies and binding energies, the very tight “reference” grid and an energy convergence of 10-8 hartree were used. The GIMIC calculations were performed with the GIMIC
10.1021/jp8087918 CCC: $40.75 2009 American Chemical Society Published on Web 12/19/2008
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J. Phys. Chem. C, Vol. 113, No. 2, 2009 525
TABLE 1: Calculated Structural Parameters for the Boron Clusters Studied: Symmetry, Spin-State (S), Lowest Vibration Frequency (ν1, cm-1), Binding Energy Per Atom (Eb, kJ/mol), HOMO-LUMO Gap (gap, eV), and HOMO and LUMO Orbital Symmetries (Occupation for HOMO)a sym
S
ν1
Eb
gap
HOMO
B16 B20 B24
D8d D10d D12d
0 0 0
176 115 79
458.2 473.9 478.9
2.36 2.47 1.96
e2 (4) e2 (4) e3 (4)
e2 e3 e3
B18
D9d C2h D11d Ci
1 0 1 0
143 17 92 17
464.6 464.3 474.3 474.5
1.41 1.30 1.10 1.18
R-e1g (2) bg (2) R-e2u (2) au (2)
β-e1g ag β-e2u au
B22
LUMO
a The z-axis is perpendicular to the ring face, as in Figure 1, except for B18/C2h, which has the x-axis going through the ring instead.
program v1.2.1. Vector and orbital plots were produced with gOpenMol.36-38 3. Results and Discussion 3.1. Molecular and Electronic Structure. The highest possible symmetry of each boron double ring B2n in the usual staggered form is Dnd. These were also the starting points for the geometry optimizations. Other, possibly less energetic structures have not been the focus of this work. B16, B20, and B24 were found to be stable closed-shell molecules in the highest symmetry group. B18 and B22, on the other hand, both relax to lower symmetries, C2h for B18 and Ci for B22. The hypothetical highsymmetry D9d and D11d structures are notably higher in energy, even when allowing the electronic symmetry to be lower than the geometric. For B18, the energy difference is 26 kJ/mol; when constraining electronic symmetry to equal the geometric, the D9d structure has a half-filled orbital of e-symmetry, and lies 63 kJ/mol higher in energy. For B22, the D11d structure with broken electronic symmetry lies 106 kJ/mol above the true Ci structure; with electronic symmetry constrained to D11d, the difference is 188 kJ/mol. Both B18 and B22 have triplet states that do possess true D9d and D11d symmetry, respectively. These are energetically very close to the closed-shell structures. It can also be noted that for B22, the Ci geometry is practically isoenergetic with a more symmetric C2h form. The lowest vibrational frequency (at B3LYP/TZVPP level) is, however, imaginary for the C2h isomer, being i8 cm-1. Table 1 summarizes the structure of the studied boron toroids. All coordinates are available as Supporting Information. From Table 1, one sees the expected increase in binding energy with increasing system size. The highly symmetric closed-shell B16, B20, and B24 all have large gaps between the highest occupied molecular orbitals (HOMO’s) and the lowest unoccupied orbitals (LUMO’s). For B18 and B22, the closed shell gaps are notably smaller, and similar to the gaps of the triplet states. The high-symmetry B16, B20, and B24 toroids are thus expected to be chemically more stable.39,40 It is worth noting that the low-symmetry B18 and B22 and the ideal high-symmetry structures are quite similar. The root-mean-square difference of the superimposed structures is only 0.04 and 0.08 B18 and B22, respectively. The difference thus lies in electronic, not molecular structure. 3.2. Ring Currents. The induced ring current in B20 on a plane dividing the two B10 “rings” is shown in Figure 2. Note that the quantity shown is the induced conventional current;
Figure 2. Vector plot of the induced ring currents in the xy-plane of B20. The magnetic field is directed out of the plane of the ring, along the z-axis as defined in Figure 1, so anticlockwise currents are diamagnetic. The lengths of the arrows represent the relative magnitudes of the currents in the specific points of space.
the electrons flow in the reverse direction. The current is seen to be diamagnetic throughout. The uniformity of the current is striking; the electrons traverse the molecule in near-circular paths. Not even crossing the B-B bonds, where local bond currents could prevail, disturbs the flow. Local currents just around the boron atoms are still present though (not shown). In the plane, a slight wavy nature of the currents is observed. Above and below the ring, the current flow is almost planar. For a more interactive view of the currents, the Reader is directed toward the Supporting Information, which includes vector maps of the currents. These can readily be visualized three-dimensionally with, for example, the gOpenMol package.38 From the picture, it can be concluded that ring currents are indeed induced in the molecule, hinting at aromaticity. However, the aromatic degree cannot be deduced from the picture alone, although the uniformity is telling. It should also be kept in mind that an authoritative designation of aromatic character, an inherently multidimensional problem,41 cannot be made based on ring current criteria alone; ring currents can be considered a necessary, but not necessarily sufficient criterion for aromaticity. To obtain a quantitative value for the current strengths in the toroids, the currents passing a cut-plane through the rings were calculated. The cut planes were perpendicular to the planes of the rings, and intersected the B-B bonds to avoid excess contributions from the local currents circling close to the atoms. Table 2 shows the strength of the induced ring currents in the studied clusters. To put the current strengths into perspective, a comparison to the quasiplanar, also highly stable B12 cluster, the spherically aromatic42 C6010+ fullerene cation, and the archaromatic benzene, is made. The induced currents in all high-symmetry species are very strong, ranging from 31 to 50 nA/T. Only the current in the larger C6010+ is stronger.24 Looking at the individual contributions from diamagnetic and paramagnetic currents to the net current emphasizes the stark uniformity of the currents; paramagnetism is all but absent. In practice, only the very local currents circling the atoms provide currents of opposite direction. The same degree of uniformity is present also for B12, with a
526 J. Phys. Chem. C, Vol. 113, No. 2, 2009
Johansson
TABLE 2: Integrated Induced Currents for Selected Molecules in nA/Ta sym
total current
diamagnetic
paramagnetic
NICS
31 42 50
33 43 51
-1 -1 -1
-33 -40 -35
C2h Ci
-117 -121
11 8
-128 -130
+62 +90
C3ν Ih D6h
25 60 12
25 81 17
0 -21 -5
-30 -82 -8
B16 B20 B24
D8d D10d D12d
B18 B22 B12 C6010+ b C6H6 a
Reference 24. B3LYP/SVP//TZVPP level. b The diamagnetic (positive) and paramagnetic (negative) contributions, as defined by their circulation direction through the integration plane, are reported together with the total current. The negative of the magnetic shieldings at the centers of the molecules (NICS, in ppm) are shown in the last column.
current twice as strong as in benzene. The difference to benzene, which sustains a fair amount of paramagnetic current, is noteworthy. For B18 and B22, the currents radically shift character, becoming highly paramagnetic. Traces of diamagnetic current can be found inside the rings, but also here, the currents are quite uniform. It can be noted that the choice of which B-B bonds are intersected by the cutplane (see Figure 1) does not change the total current (which naturally should stay constant) or the ratios of diamagnetic to paramagnetic currents. The negative of the magnetic shieldings at the centers of the molecules, that is, the NICS value,25 is reported. The values of this work agree with previous studies. For B20, An et al.2 report a value of -40. For B12, Zubarev and Boldyrev43 calculated a NICS value of -28. Although not a quantitative indicator of aromaticity,44 NICS is seen to agree qualitatively with the induced ring currents. The ratio between current strength and NICS value stays fairly constant when considering the highsymmetry structures. For the larger diamagnetic rings, the magnitude of the NICS value decreases compared to the magnitude of the current. This is understandable, as the center of the ring becomes increasingly more exposed from above and below, as the radius increases, weakening the ability of the ring to (de)shield its center. Aromaticity aside, the NICS values provide interesting information per se. Of the stable systems, B20 is found to shield its interior most efficiently. B18 and B22, on the other hand, strongly deshield their centers. For completeness, Figure 3 shows the Aromatic Ring Current Shielding45 (ARCS) curves for B20, B12, and benzene. The ARCS method is based on the observation that aromatic molecules exhibit strong shielding well outside the region of appreciable electron density; the ARCS curve simply plots the magnetic shielding on a line perpendicular to the ring plane. It is seen that all molecules induce shielding very far from the ring itself, the magnitude of the effect correlating with the ring current strength. The structure has a decisive effect on the (calculated) ring currents in the B18 and B22 clusters, underlining the importance of using proper geometries. The currents calculated by using the hypothetical high-symmetry D9d or D11d structures are actually diamagnetic. Combined with the fact that the triplet states (for which integrated currents cannot presently be obtained) are energetically very competitive with the closed shell singlets, the assignment of current strength for B18 and B22 should be considered somewhat preliminary.
Figure 3. The ARCS curves for B20 (whole line), B12 (dashed line), and benzene (dotted line), beginning from the centers of the molecular ring systems, going perpendicularly outward (along the z-axis). Note that for B20, the center is buried 0.75 Å inside the tube. The negative of the value at the origin corresponds to the NICS value.
A dissection of the current to the most significant orbital contributions proved to be difficult; all, except the core 1s orbitals, are highly delocalized. Figure 4 shows some representative examples, beginning with the deepest noncore orbital, 2a1. An et al.2 performed an analysis of the orbital contributions to the NICS value of B20. Although not all orbital contributions were reported, a discrepancy between the integrated current and NICS contributions is seen. At the B3LYP level, a positive contribution of at least +30 ppm was reported, compared to the total NICS value of -40, implying strongly competing diamagnetic and paramagnetic contributions. The direct GIMIC integration of the currents clearly contradicts this interpretation, as paramagnetic currents are shown to be very weak. A straightforward NICS dissection should thus be considered with care. The orbital dissection method of Steiner and Fowler46,47 could prove to be more appropriate, and future work will hopefully address this issue more conclusively. Simple electron counting rules for aromaticity appear inadequate faced with the boron double rings. Failures of electron counting have been noted previously also for lithium-aluminum clusters.48 The “even-odd” alteration in n of the currents in the B2n series does hint at an underlying structure, however. Although far from spherical, some sort of pseudoatom approach, similar to that for spherically aromatic species,49,50 could perchance prove fruitful. Again, an orbital contribution analysis could help clarify the situation. This could potentially also give insight into possible double aromaticity,51 recently discussed in connection with monocyclic boron rings.52 For the moment, this remains an open question, but as far as the ring current is concerned, the molecule behaves as one large entity. To spatially pinpoint the regions of strong current, integrations over partial planes were performed. Figure 5 shows the increase in current when going outward from the center of the B20 ring, i.e., with increasing width of the integration plane along the xy-plane as shown in Figure 1. As expected from Figure 2, most of the current originates from near the cage framework (indicated by the vertical line in the figure). A small dip in the general curvature of the derivative can be noted just beyond the borons. This should not be taken as an indication of p-contribution domination over s, or some other orbital structure of the molecule. The atomic cores and their local currents simply decrease the effective flow-area at this radius. The fact that the
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J. Phys. Chem. C, Vol. 113, No. 2, 2009 527
Figure 4. The merry-go-round, nondegenerate valence orbitals of B20. From left to right, 2a1; 2b2; 3a1; 4a1. Note that the spheres at the atomic positions in 2a1 and 2b2 represent orbital phases, not the atoms. The orbitals were plotted with the same isocontour value.
Figure 5. Accumulated current for increasing width of the integration plane, going out from the center of the B20 ring (whole line with markers). The derivative in arbitrary units is also shown (dotted line). The vertical line at 2.59 Å marks the radius of the ring, i.e., the position of the atoms.
Figure 6. Accumulated current for increasing height of the integration plane, going out from the center of the B20 ring (whole line with markers). The derivative in arbitrary units is also shown (dotted line). The vertical line at 0.75 Å marks the position of the B10 ring. The net current levels out at half of the total current as only one-half of B20 is included.
current between the two B10 rings is uniformly strong when passing the ring supports this notion. As seen in Figure 5, about two-thirds of the total current comes from the inside region, although the tail of the still increasing net current is rather long; 90% of the total current is reached when the integration radius is extended 1 Å beyond the boron atoms. The in-plane current is not concentrated to the location of the B10 rings. The strongest contribution actually comes from the middle plane of the ring. This is seen in Figure 6, which depicts how the current increases toward full strength as the height of the integration plane, as defined in Figure 1, grows from the center out. The derivative of the accumulated current is largest at the center, decreasing when going outward. A similar dip as in Figure 5 for the current strength near the region of the atoms occurs also here. 3.3. Connection to Stability. In this section, I will briefly discuss the connection between ring currents and the aromaticity they imply, and the stability of the toroidal forms of the boron clusters. While Zhai et al. showed that clusters up to 15 atoms stay quasiplanar,4 experimental data on the neutral boron clusters of 16 atoms or more are sparse. For the small region between 16 and 19 atoms, both anions and neutrals have been suggested to be planar, based on photoelectron spectroscopy data by Wang and co-workers,53,54 while neutral B20 was suggested to mark the cross-over to three-dimensionality.1 For the cations, the situation is more well defined, with the recent report of experimental collision cross-sections.3 Experimental data are of decisive importance, as the current affordable level of theory alone cannot be trusted to provide the right energy ordering between boron isomers.3 Let us first consider the well-defined high-symmetry series B16, B20, and B24: The current strength increases from 31 via
42 to 50 nA/T. In the least, this does not contradict the notion of a stabilizing effect of the ring currents. The current in B16 might be just a bit too weak to support the tubular form in favor of planar isomers. B24 should thus be quite stable too, which previous DFT calculations support.55,56 Compared to the cations,3 the same correlation between current strength and relative stability of the cylindrical forms over alternatives is seen. For B16+, a clear verdict could not be given, and it is possible that both quasiplanar and tubular forms are formed; the anion B16-, on the other hand, is quasiplanar.54 Both B20+ and B24+ were found to be positively tubular. For the antiaromatic B18, the lowest vibrational frequency (see Table 1) is very small, which does imply a quite unstable structure, consistent with a destabilizing effect of the paramagnetic ring currents. However, the cation B18+ was found to be tubular.3 The cation is of course not the same as the neutral. For example, the lowest vibrational frequency, calculated at the B3LYP/SVP57 level, is raised to 119 cm-1. Further, the electronic structure can change dramatically with small changes in geometry; removal or addition of electrons can thus qualitatively change the current flow as well as the stability of a cluster, as shown for the B20 anion.1 A similar effect is observed for the Au32 fullerene, which in its neutral form is predicted to exhibit strong spherical aromaticity.58 Its stability, as well as aromaticity, is highly dependent on charge. Both the cation and especially the anion are significantly destabilized compared to alternative isomers.59,60 Thus, neutral B18 should be nontubular, if there is a correlation between paramagnetic current flow and destabilization. It should be kept in mind, however, that the energetically close triplet state has not been considered here. Toroidal B22 also possesses strong paramagnetic currents, and should be destabilized. As in B18, the lowest vibrational
528 J. Phys. Chem. C, Vol. 113, No. 2, 2009 frequency of B22 is very low. In this case, the cation data agree, the tubular structure is not found experimentally. Several interesting questions remain for further experimental and theoretical investigations. For example, the underlying reason for the destabilization of the toroidal B20 anion compared to the planar isomers is worth additional attention. Does the aromaticity of the highly symmetrical isomer vanish or weaken, destabilizing the species? No unambiguous evidence for a stabilizing effect arising from the aromaticity of the neutral compared to the anion exists, yet. In fact, the computed optical spectra on B20 indicate no major changes in electronic delocalization upon reduction.61 Perhaps it is the other way around, that an added electron stabilizes the planar isomers anomalously much, as is observed for gold clusters.62-64 Very recently, methodologies for studying induced currents in open-shell systems were developed,65,66 which should be able to help address these questions in the future. 3.4. Brief Technical Notes. From a technical point of view, the boron toroids present something of a challenge, due to their very complex electronic structure. Special care has to be taken to ensure that the computed solutions are correct. The high sensitivity toward molecular structure was already discussed. But also the electronic structure needs to be correct, beyond just a possession of right symmetry. To check this, a stability analysis30 of the Kohn-Sham solution was performed. This was done in C1 symmetry, using the smaller polarized split-valence basis set, SVP.57 Indeed, some self-consistent field (SCF) calculations were found to have converged to a saddle point instead of a physically relevant minimum, also in nonobvious cases where the HOMO-LUMO gap was positive and of reasonable magnitude. Without a check, these unphysical solutions would easily go unnoticed, as other hints at failure are absent. As a remedy, reordering of orbitals and tuning of the level shift was necessary. By ensuring that the HOMO-LUMO gap and NICS values of the no-symmetry calculations agreed with the final values obtained with symmetry and the TZVPP basis, the sanity of the results is confirmed. From this point of view, recent methods that ensure convergence to an electronic minimum67 are very welcome. Details of the analysis are provided in the Supporting Information. The low-energy vibrational frequencies of B18 and B22 were also found to be very sensitive to the tightness of the grid size used for the DFT quadrature. Only by using the tightest “reference” type grid could a reliable assignment of the character of the lowest vibration, imaginary or real, be assigned. Importantly, numerical second derivatives provide very unreliable low-region spectra. The lowest frequencies are much too high, 127 and 88 cm-1 for B18 and B22, respectively, compared to the analytical value of 17 cm-1. Further examples are discussed in the Supporting Information. Another complication might arise from the possible multireference character of the boron double rings. The coupled cluster CC2/TZVPP natural orbital occupation numbers corresponding to the Hartree-Fock68,69 frontier orbitals differ from unity and zero by a moderate degree: For the closed-shell clusters, the formally HOMO occupation lies between 1.77 and 1.82, and the formally LUMO occupation lies between 0.15 and 0.20 (the calculations were performed in abelian symmetry, so the ideal closed shell occupations are 2/0 for all species, even when the HOMO/LUMO actually is of degenerate e-symmetry, as in the high-symmetry Dnd systems). For none of the toroids were negative occupation numbers70 encountered, however. Further, density functionals encumbered by electron selfinteraction error are known to be able to treat mild multirefer-
Johansson ence satisfactorily, also when lower order coupled cluster fails.71-75 At least the high-symmetric closed shell singlets should thus be well described at the DFT level; for B18 and B22 where the singlet and triplet states are energetically very close, the description could possibly be of lower quality. 4. Conclusions and Outlook Magnetic properties of the neutral, closed-shell boron double rings between 16 and 24 atom were elucidated. The experimentally known B20 was found to possess very strong ring currents, 3.5 times higher than in, e.g., benzene. It is thus tempting to consider it a superaromatic species, naturally with the implicit assumption of a strong correlation between ring current strength and aromaticity. Also B16 and B24 were shown to sustain induced diamagnetic currents, slightly weaker and stronger, respectively. The currents of B18 and B22 were found to be even stronger, but paramagnetic. Thus, they are expected to be destabilized compared to competing isomers. The strong currents, both diamagnetic and paramagnetic, and the resulting high shielding at the centers of the double rings, combined with their geometry makes for an inspiring combination. An especially interesting situation would arise if the magnetic properties remain after attachment of the structures onto a substrate or surface. Add a possible tuning of the currents and shielding by either changing the electron count or spin state, and several applications can be imagined. From this point of view, the preparation of toroids with strong paramagnetic currents would only be welcome. This highly interesting class of nanostructures deserves further attention, both theoretical and experimental. An immediate extension of this work would be to study the open-shell species in detail. Acknowledgment. I thank Doc. Dage Sundholm and Prof. Lai-Sheng Wang for inspiring discussions. Dr. Jonas Juse´lius is acknowledged for a copy of the GIMIC analysis program. This work was supported by Svenska Tekniska Vetenskapsakademien i Finland and the Lundbeck Foundation. CSC, The Finnish IT Center for Science, hosted part of the calculations. Supporting Information Available: Atomic coordinates of the boron toroids, selected vector maps, and details on the technical analyses. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (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. 2005, 102, 961–964. (2) An, W.; Bulusu, S.; Gao, Y.; Zeng, X. C. Relative stability of planar versus double-ring tubular isomers of neutral and anionic boron cluster B20 and B20-. J. Chem. Phys. 2006, 124, 154310. (3) Oger, E.; Crawford, N. R. M.; Kelting, R.; Weis, P.; Kappes, M. M.; Ahlrichs, R. Boron Cluster Cations: Transition from Planar to Cylindrical Structures. Angew. Chem., Int. Ed. 2007, 46, 8503–8506. (4) Zhai, H.-J.; Kiran, B.; Li, J.; Wang, L.-S. Hydrocarbon analogues of boron clusterssplanarity, aromaticity and antiaromaticity. Nat. Mater. 2003, 2, 827–833. (5) Lazzeretti, P. Ring Currents. Prog. Nucl. Magn. Reson. Spectrosc. 2000, 36, 1–88. (6) Gomes, J. A. N. F.; Mallion, R. B. Aromaticity and Ring Currents. Chem. ReV. 2001, 101, 1349–1383. (7) Heine, T.; Corminboeuf, C.; Seifert, G. The Magnetic Shielding Function of Molecules and π-Electron Delocalization. Chem. ReV. 2005, 105, 3889–3910. (8) Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. ReV. 1964, 136, B864-B871.
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