J. Phys. Chem. C 2009, 113, 15569–15575
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Double Aromaticity in “Boron Toroids” David E. Bean and Patrick W. Fowler* Department of Chemistry, The UniVersity of Sheffield, S3 7HF, United Kingdom ReceiVed: June 24, 2009; ReVised Manuscript ReceiVed: July 17, 2009
Ring current maps for the toroidal boron clusters B2n (n ) 6-14) are computed within the ipsocentric approach. They reveal double aromaticity for the clusters with even n but mixed aromaticity for those with odd n, consistent with angular momentum selection rules extended to the separate radial and tangential manifolds of molecular orbitals. Introduction The molecules informally known as boron toroids,1 with formula B2n, consist of two boron rings of n atoms each arranged in an antiprism geometry to create a cylindrical structure (see Figure 1). Kiran and coworkers2 have shown, using photoelectron spectroscopy (PES) in combination with geometry optimization, that, although small boron clusters favor planar 2-dimensional structures, those with 20 atoms or more adopt 3-dimensional tubular structures of the toroidal type. They liken this structural transition to the ring-to-fullerene transition in carbon clusters. Ion-mobility studies on boron cluster cations3 are also consistent with a change from quasiplanar to cylindrical/ toroidal structures in the same size regime. This structural trend has been ascribed to variation in aromatic character: the π systems of planar boron clusters are thought to be delocalized and aromatic for small clusters but to become localized with increasing molecular size.4 On the other hand, strong aromaticity of a delocalized π system covering the inner and outer surfaces of the molecule appears to contribute to the stability of cylindrical B20.2 A further indication5 of aromaticity in B20 is provided by the large negative nucleus-independent chemical shift (NICS)6 value (-40 ppm) at the center of the cylinder. A recent theoretical study of B20 and neighboring toroids1 uses the gauge including magnetically induced currents (GIMIC) method of Juse´lius et al.,7 and NICS6 calculations, to investigate ring currents in these molecules. In this study,1 Johansson finds that B16, B20, and B24 adopt geometries with maximum Dnd symmetry and are strongly aromatic on the ring-current criterion. In contrast, he finds that the character of B18 and B22 is equivocal, with antiaromaticity displayed by low-symmetry closed-shell singlet states, although the same molecules show diatropic (aromatic) currents in hypothetical Dnd geometries and might be expected to be aromatic in the nearby triplet states,8 for which currents are not available. The GIMIC method allows calculation of the total first-order induced current density and investigation of the spatial distribution of current. NICS values are integrals over the spatial distribution of this total first-order induced current density. As implemented so far, GIMIC does not give a framework for attributing current to components of the electronic structure. Johansson notes some difficulties in reconciling currents and reported orbital breakdowns of NICS.1 He suggests that these could possibly be resolved by an orbital decomposition of the induced current (as they have been in other cases9). In the present paper, the well-established ipsocentric10 * To whom correspondence should be addressed.
Figure 1. B20 boron toroid. Bond lengths within the large rings are 1.601 Å, and those connecting the large rings are 1.704 Å (calculated at the B3LYP/TZVPP level1).
continuous transformation of the origin of current densitydiamagnetic zero (CTOCD-DZ)11,12 method is used to plot and analyze current-density maps for a range of boron toroids. By examining contributions from individual molecular orbitals, we are able to provide a systematic picture of aromaticity in these molecules. A close analogy with the more familiar double aromaticity13 of all-carbon rings is revealed.14-16 Method Systems B2n with 6 e n e 14 are treated here. Geometries for B16, B18, B20, B22, and B24 were taken from ref 1 (Supporting Information), and for additional molecules (B12, B14, B26, and B28) the geometries were optimized at the B3LYP/6-31G* level using Gaussian03.17 In all the calculated geometries, the average B-B separation indicates some degree of multiple bonding, with the smallest nearest-neighbor distances being those for atoms in the same n-sided ring (see Table S1 of the Supporting Information). All systems B2n, where n is even, retain maximum Dnd symmetry. The precise geometries of the systems with odd values of n are more sensitive to the level of theory: B18 and B22 have C2h and Ci symmetries, respectively, when a larger basis set and the VWN5 variant of B3LYP are used1 but drop to Ci and C1 on optimization in the 6-31G* basis set with the VWN3 variant as used in Gaussian03.17 B14 and B26 also show low symmetries at the B3LYP/6-31G* level. In practice, the computed properties turn out to be insensitive to these small geometric differences. Induced currents are calculated for the lowest closed-shell singlet state of each system using the ipsocentric method at the B3LYP/6-31G* level with GAMESS-UK18 and additional
10.1021/jp905926j CCC: $40.75 2009 American Chemical Society Published on Web 08/11/2009
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Figure 2. Current density induced in B20 by an external magnetic field directed along the cylinder axis. The current is plotted in the median plane, with arrows representing the in-plane component and contours representing the modulus. Circular symbols indicate the nuclear positions projected into the plotting plane. 19,20
SYSMO routines for plotting current density. The ipsocentric method is a distributed-origin approach in which current density at a point is calculated with that point taken as the origin.10-12 Usefully, it allows breakdown of the total current density into nonredundant contributions from individual occupied molecular orbitals10 and further decomposition into contributions from specific occupied-to-virtual orbital excitations.21 Global character of a ring current identified within a given current-density pattern can also be tested by comparing contributions from canonical and localized molecular orbitals.22 On the magnetic criterion, an aromatic (antiaromatic) molecule is one that is able to support a diatropic (paratropic) ring current,23 and, hence, the interest here is in maps of the current density induced by an external magnetic field directed along the long axis of the boron cylinder. These maps are plotted in the plane midway between the two boron rings (the “median plane”) and give a direct visual distinction between aromaticity and antiaromaticity, as anticlockwise (clockwise) circulation indicates diatropic (paratropic) current. Further supporting evidence for the interpretation of the maps is given by the components of magnetizability and central shielding (equal to NICS(0) values but of opposite sign), calculated in the PZ24 allocentric25 distributed-origin approach, using SYSMO.19,20 Results Currents in Systems B2n with Even n. Figure 2 shows the total current density in the median plane of the B20 molecule induced by an external magnetic field directed perpendicular to that plane. The intense diatropic ring current clearly demonstrates the aromaticity of this system, in agreement with the calculations of Johansson.1 The current in B20 is strong: the maximum value in the median plane is 1.4 times the benzene “standard” π current (calculated for a height of 1 a0 above the benzene ring using the same level of theory). The total current in B20 is at its strongest in the median plane,1 but a closer analysis using partitioning of the current into contributions from individual molecular orbitals reveals that it consists of two distinct currents that have different spatial distributions with respect to the median plane. Comparison of contributions from all 50 doubly occupied canonical molecular orbitals shows that 90% of the total current in the median plane (as judged from the maximum) arises from just four of these: the HOMO and HOMO-1 degenerate pairs.
Figure 3. Contributions to the B20 current from (a) the HOMO and (b) the HOMO-1. The top row shows currents in the median plane, and the bottom row shows side-on views, plotted in a plane positioned outside the ring at a distance of 1 a0 from the nearest boron atom. The magnetic field is directed along the principal axis in both cases.
Figure 3 shows top-down and sideways-on views of the separate HOMO and HOMO-1 contributions to current. The HOMO contribution (Figure 3(a)) consists of three “doughnutlike” circulations of current flowing above, between, and below the rings of boron nuclei. A layered spatial distribution of this type is expected from the summation of π currents of two rings stacked face-to-face, each current arising from p orbitals perpendicular to the ring. These contributing atomic orbitals lie in the surface of the notional cylinder in which the nuclei are embedded and hence are tangential in the sense of the tensor surface harmonic (TSH) theory of cluster bonding.26-28 Currents of this type will be referred to as “tangential” in the remainder of this paper. It is notable that, in the ipsocentric description, the whole of this 3-dimensional current arises from the response of only four electrons, just as the entire π current of benzene arises from its four HOMO electrons.10 As with the π current of benzene, the ring current here is intrinsically nonlocalizable.22 For B12, for example, partition of the total current according to contributions from the maximally localized molecular orbitals produced by the Pipek-Mezey29 procedure leads to maps with open flowlines, and global circulation is recovered only on summation over equivalent sets (see Figure S1 of the Supporting Information). The second current, arising from the doubly degenerate HOMO-1, has a strikingly different distribution in space (see Figure 3(b)). Viewed down the principal axis, the current consists of concentric circulations separated by a radial node, and as the side view shows, these circulations have a cylindrical character. The existence of a cylindrical nodal surface is compatible with a current arising from p orbitals that are locally perpendicular to the cylinder and hence radial in the TSH sense. Currents of this type will be referred to as “radial” in the remainder of the paper. Again, within the ipsocentric description, it is the response of four electrons that determines the whole of this 3-dimensional current. Analogous tangential and radial currents are seen in all the B2n molecules investigated here (see the top-down views in Figures 4 and 5). In the whole set, B12, B16, B20, B24, and B28, both tangential and radial currents are diatropic. The presence of two orthogonal, cyclically delocalized systems in the same
Double Aromaticity in “Boron Toroids”
Figure 4. (a) Tangential and (b) radial currents in B12, B16, B20, B24, and B28.
molecule is defined by Chandrasekhar et al.13 as double aromaticity. In this sense, the B2n molecules with even values of n are not only aromatic but doubly aromatic. Currents in Systems B2n with Odd n. In contrast to the systems with even n, the B2n toroids with odd n show strong paratropic total currents in their lowest closed-shell singlet states. Orbital analysis shows that these net paratropic circulations all arise from competition between tangential and radial currents (see Figure 5). The systems B18 and B22, both found by Johansson1 to be antiaromatic in their singlet-state geometries, illustrate the two different possibilities. In B18, a strong paratropic tangential current arises from the two electrons in the nondegenerate HOMO and is opposed by the radial current
J. Phys. Chem. C, Vol. 113, No. 35, 2009 15571 arising from the four electrons in the doubly degenerate HOMO1. In B22, it is the radial current that is paratropic and arises from response of two electrons and the tangential current that is diatropic and four-electron in character. In the framework of double aromaticity, the systems with odd n investigated here would be classified as displaying mixed aromaticity; from consideration of total current alone, they would simply be labeled “antiaromatic.” The various diatropic and paratropic ring currents in the B2n systems can be compared in strength with conventional π ring currents. As the values in Table 1 illustrate, the diatropic currents in the boron systems are similar in strength to the current in benzene, ranging from 50 to 125% of the current density evaluated at 1 a0 above the benzene ring (jmax ) 0.080 au), where π charge and current densities are close to maximum. Paratropic currents are typically stronger, ranging from four to nine times the benzene standard for the B2n systems studied here. This qualitative difference between paratropic and diatropic currents is to be expected from their origins in excitations within and between angular momentum shells, respectively; paratropic currents typically arise from virtual excitations with small energy denominators. Global Response Properties. Table 2 shows the results of computations of the magnetizability and central shielding for the series of boron tori. The magnetizability tensor is strongly anisotropic, with small diamagnetic (negative) components in the directions perpendicular to the principal axis, and a large component along the principal axis itself. The sign of the large component is negative (diamagnetic) for B2n with even n and positive (paramagnetic) for B2n with odd n. These signs are compatible with the diatropic sense of both tangential and radial currents in the first case, and with the paratropic sense of the dominant current in the second, confirming the visual interpretation of the current-density maps in terms of double and mixed aromaticity, respectively. Precise values of computed magnetizabilities are likely to be dependent on basis set and level of theory, particularly in the case paratropic systems with small HOMO-LUMO gaps, but it is interesting to note at a qualitative level that the total mean magnetizability is large and positive for B18, B22, and B26, indicating that these “mixed aromaticity” systems are essentially antiaromatic, as would be expected from the dominance of the paratropic ring current in the maps of total current density (Figure 5(c)). The same trends are apparent in the computed NICS values. The boron toroids B2n with even n all show similar values indicative of aromaticity. The boron toroids B2n with odd n show increasingly large positive values, indicative of antiaromaticity arising from stronger and stronger paratropic currents. Similar results, with the small differences attributable to basis set, were found by Johansson1 for NICS values of the tori B16, B18, B20, B22, and B24. Discussion Table 3 summarizes the computed aromaticity characteristics of the B2n series. The doubly aromatic nature of B2n with even n is apparent from the reinforcing radial and tangential diatropic currents, and the mixed nature of the B2n systems with odd n is shown by the alternation of paratropic and diatropic contributions. In addition to the maps themselves, the ipsocentric approach to calculation of magnetic properties provides extra information that allows a general explanation of these systematic patterns, starting from the electronic structure. Inspection of the molecular orbitals shows a common qualitative picture for all the systems studied here. In the general
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Figure 5. (a) Tangential and (b) radial currents in B14, B18, B22, and B26. Column (c) shows the total current obtained by summing contributions from all 5n occupied orbitals for each B2n torus illustrating the dominance of the paratropic contribution in every case.
TABLE 1: Signed Maximum Current Strengths (jmax) of the Dominant Contributions from the Tangential and Radial Systemsa jmax/au B12 B14 B16 B18 B20 B22 B24 B26 B28
tangential
radial
-0.044 -0.046 -0.038 0.378 -0.105 -0.074 -0.068 0.734 -0.100
-0.051 0.317 -0.079 -0.083 -0.023 0.642 -0.095 -0.097 -0.081
a These may be compared with the “standard” value of -0.080 au for the π ring current in benzene calculated for a height of 1 a0 and using the same level of theory. Diatropic (paratropic) currents are indicated by notional negative (positive) signs attached to jmax.
closed-shell boron toroid, B2n, which belongs to symmetry group Dnd or a subgroup, there are 10n electrons, occupying 5n
TABLE 2: Computed Molecular Magnetisabilities (χ) and NICS(0) Values for the B2n Toroidsa B12 B14 B16 B18 B20 B22 B24 B26 B28
χ/au
NICS(0)/ppm
-24.64 36.47 -59.44 186.47 -110.56 379.62 -195.29 802.99 -303.00
-32 29 -32 61 -38 88 -34 102 -37
a All values are calculated at the CTOCD-PZ/B3LYP/6-31G* level. χ| is the component of the magnetisability tensor along the axis perpendicular to the median inertial plane. Shielding constants are calculated at the center of the molecule, and NICS(0) values are obtained from them by changing the sign.
molecular orbitals. 4n electrons are in 1s2 cores occupying 2n orbitals spanning the reducible representation Γ ) A1 + B2 + E1 + E2 + ... + En-1 of Dnd and well separated in energy from
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TABLE 3: Aromaticity Characteristics of the B2n Toroidsa sense of current point group B12 B14 B16 B18 B20 B22 B24 B26 B28
D6d C1 D8d C2h/Cib D10d Ci/C1b D12d Ci D14d
tangential diatropic diatropic diatropic paratropic diatropic diatropic diatropic paratropic diatropic
radial diatropic paratropic diatropic diatropic diatropic paratropic diatropic diatropic diatropic
electron occupancy tangential 6 6 6 8 10 10 10 12 14
active occupied/virtual orbitals
radial 6 8 10 10 10 12 14 14 14
tangential c
1 e1/2 e2 1d (a + a)/2 (a + a) 1d e1/2 e2 2c bg/2 ag 2c e2/3 e3 2d (ag + ag)/3 (au + au) 2d e2/3 e3 3c au/3 au 3c e3/4 e4
radial d
1 e1/2 e2 2c a/2 a 2c e2/3 e3 2d (bg + ag)/3 (bu + au) 2d e2/3 e3 3c au/3 au 3c e3/4 e4 3d (au + au)/4 (ag + ag) 3d e3/4 e4
a Point group symmetries, senses of ring current, partition of electrons between the tangential and radial subsystems, and angular momentum (Λ) and symmetry labels of the occupied/virtual orbitals involved in the excitations responsible for the tangential and radial currents. b B3LYP/TZVPP gives a structure belonging to the higher group and B3LYP/6-31G* to the lower. c HOMO. d HOMO-1.
Figure 6. Occupied molecular orbitals in B12. (a) Shows the orbitals containing the 24 1s2 core electrons; (b) shows the orbitals containing the 24 electrons involved in the 3-center, 2-electron bonds; (c) shows the orbitals in the tangential system; (d) shows the orbitals in the radial system. Within each set, the orbitals are listed in order of energy, but there is some overlap between the energy ranges for the different sets.
the remainder of the occupied orbitals. The remaining valence orbitals fall naturally into three sets albeit with some overlap of energy ranges. The first set, of 2n filled molecular orbitals, is framework-bonding and for book-keeping purposes can be described as a collection of 2-electron, 3-center bonds on all triangular faces of the boron cylinder. These are equisymmetric with the core orbitals. The remaining valence orbitals separate naturally into two partially occupied sets of qualitatively different types. One set, which can be described as combinations of tangential p orbitals, is characterized by the feature that each n ring of boron nuclei lies in its own nodal plane. In addition to these two “horizontal” nodes, the orbitals have increasing numbers of “vertical” nodes with increasing orbital energy and can be labeled by their component of orbital angular momentum, Λ, about the cylinder axis. The complete set of tangential orbitals spans the product representation Γ × Γz (where Γz is the symmetry of a translation along the cylinder axis), which is also equal to Γ in Dnd. The other partially occupied set, which can be described as combinations of radial p orbitals, is characterized by a cylindrical nodal surface, containing all boron nuclei. These orbitals too
have increasing numbers of vertical nodes with increasing energy, and hence, Λ is also a good quantum number for the orbitals within this set. In common with the other three sets, the complete set of radial orbitals spans Γ. Figure 6 illustrates the full set of occupied molecular orbitals of D6d B12, comprising 12 cores (Figure 6(a)), 12 3-center bonds (Figure 6(b)), 3 tangential orbitals (Figure 6(c)), and 3 radial orbitals (Figure 6(d)). Calculated electronic configurations for the toroids follow a regular pattern (Table 3). In all cases, the core and low-lying framework bonding orbitals account for 8n electrons. For even n, the remaining 2n electrons achieve a 4m + 2 count in each of the radial and tangential subsystems; where mathematically possible, they are split equally, and otherwise, four more electrons occupy radial orbitals. For odd n, the split between subsystems is necessarily unequal and leads to two more electrons in radial orbitals, implying a count of 4m for radial or tangential subsystems, alternately. Pure electron counting is therefore consistent with the observed trends in ring current in that systems with even n have both radial and tangential Hu¨ckel aromaticity and systems with odd n have either radial aromaticity
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Figure 7. Virtual excitations responsible for (a) tangential and (b) radial ring currents in B20. Both obey the translational selection rule and so produce diatropic current.
and tangential antiaromaticity or tangential aromaticity and radial antiaromaticity. A more detailed picture is given by considering the specific symmetries and nodal properties of the orbitals. In the ipsocentric model, a virtual excitation from an occupied to an empty molecular orbital can result in a contribution to the overall current that is paratropic, diatropic, null, or both.10 The effect of the excitation is determined by simple selection rules. If the product of the symmetries of starting and target orbitals contains the symmetry of a translation perpendicular to the magnetic field (Tx or Ty), the excitation is translationally allowed and gives a diatropic contribution. If the product contains the symmetry of the in-plane rotation (Rz), the excitation is rotationally allowed and gives a paratropic contribution. In structures with cylindrical symmetry, such as the boron toroids, these rules simplify: if ∆Λ, the difference in orbital angular momentum quantum number between starting and target orbitals, is 1, the contribution is diatropic, and if ∆Λ ) 0, it is paratropic. Orbital contributions will be large when the energy gap between the starting and target orbitals is small, and hence, currents are typically dominated by the frontier orbitals. It turns out that this analysis accounts for all the observed features of the computed current-density maps. Within the ipsocentric model, the only excitations that are allowed are between occupied and empty orbitals. Low-lying and completely filled angular momentum manifolds therefore make no significant contribution to global ring currents. On the other hand, the highest occupied orbitals within the radial and tangential stacks are separated by smaller energy gaps from unoccupied partners with similar spatial distributions and accessible by one or other of the angular momentum selection rules. As an example of a doubly aromatic toroid, Figure 7 shows the starting and target orbitals for the two dominant virtual excitations responsible for the tangential and radial currents in the B20 molecule. For this system, the highest occupied molecular orbital pairs of the radial and tangential stacks both have Λ ) 2 (two vertical nodes) and have nearby lowest unoccupied partners in the same stack with Λ ) 3. The selection rule predicts separate diatropic currents arising from the translationally allowed virtual excitation within each stack. The distinct spatial distributions of these currents (Figure 3) are compatible with the other nodal features of the stacks, a cylindrical node for the radial, and two layered horizontal nodes for the tangential. Calculation of maps for the current density arising from the relevant excitations (see Figure S2 of the Supporting Information) shows that this frontier-orbital analysis is not only qualitative but in fact accounts for essentially the whole of the orbital current in each case. Analysis proceeds along the same lines for all the systems B2n with even n, using the values of Λ listed in Table 3.
Bean and Fowler
Figure 8. Virtual excitations responsible for (a) tangential paratropic and (b) radial diatropic ring currents in B18.
For the toroids of mixed aromaticity, the analysis invokes both translational and rotational selection rules. Recall that the systems B2n with odd n have an occupation of 4m in one stack and 4m′ + 2 in the other. Qualitative theory predicts symmetry breaking in this case. In full Dnd symmetry, occupation of an angular momentum stack by 4m electrons will lead to an open shell; this will close under Jahn-Teller distortion, and in the lowered symmetry, the HOMO and LUMO of the stack will be an angular momentum pair separated by a small energy gap. Symmetry lowering also produces splittings in the 4m′ + 2 stack, but in this case, the HOMO and LUMO will be members of pairs with angular momentum differing by 1 unit. As Jahn-Teller splittings are small, the selection rules therefore predict an intense paratropic current arising from the HOMOLUMO transition in the 4m stack and a less intense diatropic current from the 4m′ + 2 stack. As an example, Figure 8 shows starting and target orbitals for the dominant virtual excitations for tangential and radial currents in B18. For this system, the highest occupied molecular orbitals of the radial and tangential stacks both have Λ ) 2, with nearby lowest unoccupied partners having Λ ) 2 in the tangential and Λ ) 3 in the radial stack. As predicted by the selection rules, there is a strong paratropic current from the highest occupied tangential orbital and a diatropic current from the highest occupied pair of radial orbitals. Spectral decomposition of these currents confirms that the frontier orbital excitation is indeed responsible for the large paratropic current. Calculation of current density for triplet Dnd states of B2n toroids with odd n have not been carried out here, but the qualitative prediction is that both stacks of orbitals will contribute diatropic currents, as in these states the ∆Λ ) 0 excitation is spin-forbidden whereas diatropic ∆Λ ) 1 excitations from the open shell are allowed, as in the analogous discussion for triplet [4n]annulenes.30 The toroids in these states are therefore predicted to be doubly aromatic. Conclusion Computation and mapping of current density, relying on the special features of the ipsocentric method, has allowed construction of a systematic picture of the magnetic properties of a proposed1 class of boron toroids. Cylindrical B2n turns out to be the analogue of an all-carbon ring, warranting the ascription of doubly aromatic character to these novel 3-dimensional structures. In the carbocycles, the two (competing or reinforcing) ring currents arise from the orthogonality of in-plane and outof-plane π systems;16 in the boron systems, they arise from the existence of part-filled radial and tangential manifolds with separate electron-counting rules. The two global ring currents in both cases share common features with the conventional π currents of annulenes. They have the same 2-electron/4-electron nature and obey the symmetry selection rules that are ultimately
Double Aromaticity in “Boron Toroids” based on angular momentum. Like conventional ring currents, they are intrinsically nonlocalizable.22 Finally, it can be noted that the ipsocentric picture gives a perspective on the properties of hypothetical extended systems in which more rings of boron atoms are added to the toroid, in antiprismatic fashion, to make taller cylindrical towers. It is not known if such structures are stable. However, trial calculations on B18 and B24 towers, constrained to maximum symmetry, indicate continuation of the double aromaticity of the B12 parent, with tangential and radial currents fitting the framework of selection rules. Supporting Information Available: Nearest neighbor separations in the boron toroids, figures of the localized orbital contributions to the overall B12 current density and the current arising from virtual excitations between molecular orbitals within the tangential and radial subsystems of B20, and boron toroid geometries. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Johansson, M. P. J. Phys. Chem. C 2009, 113, 524–530. (2) Kiran, B.; Bulusu, S.; Zhai, H.-J.; Yoo, S.; Zeng, X. C.; Wang, L.-S. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 961–964. (3) Oger, E.; Crawford, N. R. M.; Kelting, R.; Weis, P.; Kappes, M. M.; Ahlrichs, R. Angew. Chem., Int. Ed. 2007, 46, 8503–8506. (4) Zhai, H. J.; Kiran, B.; Li, J.; Wang, L. S. Nat. Mater. 2003, 2, 827–833. (5) An, W.; Bulusu, S.; Gao, Y.; Zeng, X. C. J. Chem. Phys. 2006, 124, 154310. (6) Schleyer, P. v. R.; Maerker, C.; Dransfeld, A.; Jiao, H.; van Eikema Hommes, N. J. R. J. Am. Chem. Soc. 1996, 118, 6317. (7) Juse´lius, J.; Sundholm, D.; Gauss, J. J. Chem. Phys. 2004, 121, 3952–3963. (8) Baird, N. C. J. Am. Chem. Soc. 1972, 94, 4941–4948. (9) Steiner, E.; Fowler, P. W. Phys. Chem. Chem. Phys. 2004, 6, 261– 272. (10) Steiner, E.; Fowler, P. W. J. Phys. Chem. A 2001, 105, 9553–9562. (11) Keith, T. A.; Bader, R. F. W. Chem. Phys. Lett. 1993, 210, 223– 231. (12) Coriani, S.; Lazzeretti, P.; Malagoli, M.; Zanasi, R. Theor. Chem. Acta. 1994, 89, 181–192.
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