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17, 224 (1965); 18, 321 (1965). Advance in Chemical Physics, VolumeXVI Edited by I. Prigogine, S. A. Rice Copyright © 1969 by John Wiley & Sons Ltd. A Free-Electron Study of Common Boron Frameworks* JOEL A. HOUTMAN AND GEORGE H. DUFFEY Department of Physics, South Dakota State University, Brookings Abstract In many boron compounds, the boron atoms form a fragment of the regular icosahedron, or a combination of such fragments. We have carried out free-electron calculations on such structures. TOexplain the relative stabilities of various alternate structures, one has to make the delta-function potential at each nucleus positive.

G. Hall, Proc. Roy. (London), A213, 102, I 1 3 (1952). 10. J. 0. Hirschfelder and C. N. Weygandt, J . Chein. , 6, 806 (1938). 11. R. Hoffmann, J . Chem. , 39, 1397 (1963). 12. R. Hoffmann and W. N. Lipscomb, J . Chem. , 36,2179, 3489 (1962). 13. R. HOffmaM and W. N. Lipscomb, J . Chem. , 37, 2872 (1962). 14. C. K. Jmgensen, Chem. Phys. Letfers, 1 , 1 1 ( I 967). 15. R. Lefkbvre, Modern Quantum ChemiJtry, 0. , Part I , Academic, New York, 1965, p. 125. 16. L. L. Lohr, Jr. and W. N. Lipscomb, J .

77 A-l. Again, parameter C must be positive to explain the bonding of the two C,, B,H,,’s in octadecaborane-22. I7 kl. 77 k1 is comparable to that needed to break the same number of bonds in splitting octadecaborane-22. In each case the calcuIated value is around 12 eV. The free-electron energies of some observed boron cages are compared 22 A FREE-ELECTRON STUDY OF COMMON BORON FRAMEWORKS Table XII. Description of the Configurations Treated and the Governing Secular Equations Configuration Tetraborane-10 Tetragonal pyramid Pentaborane-11 Hexaborane-10 Triangular dodecahedron Octaborane-12 Enneaborane ion Enneaborane-15 Decaborane-16 Decaborane-14 Octadecaborane-22 Geometric Figure Bent rhombus Square-based pyramid Fragment of icosahedron Pentagonal pyramid Distortion of tetragonal antiprism Fragment of icosahedron Fragment of icosahedron Fragment of icosahedron Square pyramids linked apex to apex Fragment of icosahedron Condensed decaborane-14 cages Degree of Point Secular Example Group Equation czu B4HIO B5Ha =,u B5H11 ‘ 1 , 5 5 B6H10 ‘5, 6 B8C18 4, 8 BEHI2 =, BBH14- ‘3, BBH15 ‘1” B10H16 D4h B10H14 2‘, s2 B18H22 4 8 9 9 lo 10 18 with those of possible alternatives in Table XIV.

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