Magic Electron Counts and Bonding in Tubular Boranes Musiri M. Balakrishnarajan, † Roald Hoffmann,* ,† Pattath D. Pancharatna, ‡ and Eluvathingal D. Jemmis* ,‡ Department of Chemistry and Biochemistry, Cornell UniVersity, Ithaca, New York 14853, and School of Chemistry, UniVersity of Hyderabad, Hyderabad, India 500 046 Received December 6, 2002 Ring stacking in some closo-borane dianions and the hypothetical capped borane nanotubes, predicted to be stable earlier, is analyzed in a perturbation theoretic way. A “staggered” building up of rings to form nanotubes is explored for four- and five-membered B n H n rings. Arguments are given for the stacking of B 5 H 5 rings being energetically more favorable than the stacking of B 4 H 4 rings. Elongated B-B distances in the central rings are predicted for some nanotubes, and the necessity to optimize ring-cap bonding is found to be responsible for this elongation. This effect reaches a maximum in B 17 H 17 2- ; the insertion of additional rings will reduce this elongation. These closo-borane nanotubes obey Wade’s n + 1 rule, but the traditional explanation based on a partitioning into radial/ tangential molecular orbitals is wanting. 1. Introduction Boron exhibits a wide variety of polyhedral frameworks in boranes and boron-rich solids, ranging from highly symmetric octahedra and icosahedra to quite open and unsymmetrical skeletons. Following a variety of theoretical approaches including resonance theory 1 and molecular orbital theory, 2 and a general formulation by Lipscomb and others 3 of electron deficient multicenter bonding for boranes, a major breakthrough in our qualitative understanding of these systems was achieved in the early 1970s by the formulation of Wade’s rule. 4 This rule provides the electronic require- ments of a given closo polyhedral skeleton as n + 1 electron pairs, where n is the total number of vertices. Wade’s rule is derived from empirical observations and preceding theo- retical calculations 2,3 for n-vertex closo polyhedra, and has some well-known exceptions. 5 With its ability for extension to nido and the more open arachno systems, Wade’s rule has enjoyed huge success in rationalizing the diverse structural patterns exhibited by polyhedral boranes, and in predicting new structures. While the origin of the rule was empirical, Wade gave a justification based on molecular orbital theory as follows: The boron atoms in a generalized n-vertex polyhedron are assumed to exhibit sp hybridization, with an sp hybrid radiating away from the center of the cage used for a 2c- 2e bond to a hydrogen or an external ligand. This leaves one radial sp hybrid and two unhybridized tangential p orbitals for skeletal bonding. Taking octahedral B 6 H 6 2- for illustration, 6 Wade reasoned that from the set of n radial sp hybrids that point toward the center only one strongly bonding molecular orbital is formed; the rest of the radial MOs were taken as antibonding. The 2n tangential p orbitals form n bonding and n antibonding skeletal orbitals, resulting in a total of n + 1 bonding molecular orbitals (BMO). This generalization is assumed to hold for all closo polyhedra. Some stabilization due to the mixing of the higher lying out- of-phase combinations of radial MOs with the tangential BMOs is anticipated, but this is not expected to affect the general conclusion. Initial attempts to justify Wade’s rule employed graph theory; 7 the presence of the single bonding radial orbital was correlated with the unique positive eigenvalue of the * Authors to whom correspondence should be addressed. E-mail: rh34@cornell.edu (R.H.); jemmis@uohyd.ernet.in (E.D.J.). ² Cornell University. ‡ University of Hyderabad. (1) Lipscomb, W. N. Boron Hydrides; Benjamin: New York, 1963. (2) (a) Longuet-Higgins, H. C.; Roberts, M. de V. Proc. R. Soc. London, Ser. A 1955, 230, 110. (b) Longuet-Higgins, H. C.; Roberts, M. de V. Proc. R. Soc. London, Ser. A 1954, 224, 336. (c) Longuet-Higgins, H. C. Q. ReV., Chem. Soc. 1957, 11, 121. (3) Hoffmann, R.; Lipscomb, W. N. J. Chem. Phys. 1962, 36, 2179. (4) (a) Wade, K. J. Chem. Soc. D 1971, 792. (b) Wade, K. AdV. Inorg. Chem. Radiochem. 1976, 18, 1. (5) (a) Stone, A. J.; Alderton, M. J. Inorg. Chem. 1982, 21, 2297. (b) Fowler, P. W. Polyhedron 1985, 4, 2051. (6) (a) Albright, T. A.; Burdett, J. K.; Whangbo, M. Orbital Interactions in Chemistry; John Wiley & Sons, Inc.: New York, 1985; Chapter 22. (b) Fox, M. A.; Wade, K. In The Borane, Carborane, Carbocation Continuum; Casanova, J., Ed.; John Wiley & Sons, Inc.: New York, 1998; p 57. Inorg. Chem. 2003, 42, 4650-4659 4650 Inorganic Chemistry, Vol. 42, No. 15, 2003 10.1021/ic0262435 CCC: $25.00 © 2003 American Chemical Society Published on Web 06/20/2003