Barclay Kamb

The discussion of tetragonal boron by the resonating-valence-bond theory of electron-deficient substances*

An analysis of the structure of tetragonal boron, as determined by H O A R D , H U G H E S and SANDS, has been made in terms of the resonating-valence-bond theory. The observed interatomic distances show tha t each icosahedral boron atom forms five bonds with bond number 0.41 to 0.55 within the icosahedron (icosahedral edges) and one bond with bond number 0.41 to 1.03 (average 0.75) extending outward. There is some electron transfer, such as to increase the valence of the tetrahedral interstitial boron atoms to 4 and to permit relief of strain caused by geometrical constraints. A simple theory is proposed to account for the greater strength of the bonds extending out from the icosahedra (average bond number 0.75) than of those along icosahedral edges (average 0.49). * Contribution No. 2469 from the Gates and Crellin Laboratories of Chemistry, California Insti tute of Technology, Pasadena, California. The discussion of tetragonal boron 473 Boron can be described as an electron-deficient substance. Electrondeficient substances are substances in which the atoms have more stable orbitals than electrons in the valence shell, or in which, through pairing of some of the valence electrons to form unshared pairs, one orbital (or more than one) may be liberated. The boron atom has four orbitals in its valence shell, and only three valence electrons. For most electron-deficient substances the atoms have ligancy that is not only greater than the number of valence electrons but is even greater than the number of stable orbitals. For example, in the tetragonal form of crystalline boron most of the boron atoms have ligancy 6. Also, lithium and beryllium, with four stable orbitals and only two valence electrons, respectively, have structures in which the atoms have ligancy 8 or 12. All metals can be considered to be electron-deficient substances. A careful and apparently thoroughly reliable determination of the structure of the tetragonal form of crystalline boron has been reported1. There are fifty boron atoms in the tetragonal unit of structure. These atoms form four groups of twelve, each group consisting of atoms at the corners of a nearly regular icosahedron. In the B ^ icosahedron each boron atom forms five bonda with adjacent atoms. The icosahedra and the two additional boron atoms per unit are arranged in such relative positions that each icosahedral boron atom also forms one more bond, extending in the direction pointing out from the icosahedral center, as shown in Fig. 1. Thus each of the icosahedral boron atoms has ligancy 6. The two interstitial atoms have ligancy 4. There are all together 148 boron-boron bonds per unit. Eight of these bonds are formed by the interstitial atoms with adjacent icosahedral atoms, and the others, 140 per unit, are bonds between icosahedral atoms. There are only 150 valence electrons per unit cell; that is, only 75 electron pairs, so that the 148 boron-boron bonds cannot be ordinary single bonds (shared-electron-pair bonds). The suggestion was made many years ago that in electron-deficient compounds, such as the boranes, there are one-electron bonds, rather than the ordinary electron-pair bonds. GILBERT NEWTON LEWIS then suggested that the structures of the substances involves resonance of electron-pair bonds among alternative positions, in such a way as to correspond to fractional bonds. It now seems likely that both of these suggestions have some significance. 1 J . L . H O A B D , R. E. H U G H E S and D. E. S A N D S , The structure of tetragonal boron. J . Amer. Chem. Soc. 80 (1958) 4507—4545. 474 L . PAUXING a n d B . K A M B We shall first discuss the structure of tetragonal boron in terms of the resonance of shared electron pairs among the alternative positions. There are 75 pairs of valence electrons per unit cell, and 148 bond positions. If the electron pairs were distributed uniformly among the interatomic positions, each boron-boron bond would have bond number 0.506. The boron-boron distance for a single bond is about 1.62 A. A correction of 0.18 A is needed to obtain the expected bond length for a bond with bond number 0.506, with use of the equation D(n) = Z>(1) — 0.60 Ä log n. Accordingly the expected bond length in crystalline boron is 1.80 Ä. This value agrees with the average of the observed values for the bonds. However, improved agreement is obtained with a somewhat refined theory. We may first ask why the structure is the complex one observed, rather than the simple one in which boron atoms are at the points of a simple cubic lattice; each boron atom would than have ligancy 6, with the six bonds (extending toward the corners of an octahedron) at 90 ° (or 180 °) with one another. The answer to this question has been given by Professor V . S C H O M A K E R (private communication). It is that with this octahedral arrangement of bonds the bonds in the crystal would form squares, the faces of the unit cube. With a square arrangement of bonds there is a strong non-bonding repulsion between atoms separated by the diagonals of the squares. This non-bonding repulsion across the diagonal of the square introduces a significant destabilization of the structure. For cyclobutane, for example, it results in an increase in the bond length to 1.56 A (as compared with the normal value 1.54 A). With the icosahedral structure, however, the distances between atoms that are not bonded to one another are much greater, and the structure can accordingly be expected to be more stable. The principal structural element of tetragonal boron is the icosahedron of twelve boron atoms. Each boron atom forms five bonds with neighboring atoms in the isocahedral group or with one of the interstitial boron atoms. Its ligancy is accordingly 6. We may now ask whether the bonds within an icosahedron and the bonds extending out from the icosahedron would be expected to have the same strength. It seems likely that these bonds would not have the 2 L I N U S PAULING, Atomic radii and interatomic distances in metals. J . Amer. Chem. Soc. 69 (1947) 542—553. The discussion of tetragonal boron 475 same strength, for the following reason. The four orbitals of the boron atom can be considered to be directed toward the four corners of a tetrahedron. If one of these orbitals were directed in the outward direction, the other three lie at an angle with it (121.8°) that roughly approximates the tetrahedral angle. Hence the other three orbitals are directed approximately in the direction of the five intra-icosahedral bonds, and we might accordingly expect the intra-icosahedral bonds to be only about 60 percent as strong as the extra-icosahedral bond. The extra-icosahedral bonds then have bond number 0.75, and the intra-icosahedral bonds 0.45. Fig. 1. Structure of tetragonal boron as viewed in the direction of the c axis. One unit cell is shown. Two of the icoeabedral groups (light lines) are centered at ζ = J, and the other two (heavy lines) at ζ = f . The interstitial boron atoms (open circles) are at 0, 0, 0 and i, Numbers identify the various structurally non-equivalent boron atoms. All of the extra-icosahedral bonds are shown with the exception of Bt—Bt, which is formed parallel to the c axis from each icosahedron to the icosahedra in cells directly above and below Another way of making an estimate of the relative strengths of these orbitals is by dividing the solid angle 4 π about the boron atom into solid angles associated with the six bonds, by passing bisecting planes between each pair of bond directions. When this calculation is made it is found that the extra-icosahedral bond has the fraction 0.268 of the total solid angle associated with it, and each of the others the fraction 0.146. If the total valence 3 of the boron atom is divided in these ratios, the extra-icosahedral bond would be expected to have bond number 0.80 and the intra-icosahedral bonds bond number 0.44. 476 L . P A U L I N G a n d B . K A M B Let us assume that the bond numbers have the values about 0.75 for extra-icosahedral bonds and 0.45 for intra-icosahedral bonds. The packing of icosahedra reported by HOARD, HUGHES , and SANDS is based upon the body-centered arrangement. With a body-centered cubic arrangement of icosahedra each icosahedron can form bonds with its eight nearest neighboring icosahedra. By decreasing somewhat the axial ratio c/α the two icosahedra above and below along the c axis can also be brought into contact with each icosahedron. In this way each icosahedron forms bonds with ten neighboring icosahedra. The two remaining boron atoms in each icosahedron then form their extra-icosahedral bonds with the interstitial boron atoms, as illustrated in Fig. 1. If the assumption is made that the icosahedra are regular, with boron-boron bond length 1.81 A, and that the mean-squared deviation of the extra-icosahedral bond lengths from the expected length 1.70 Ä is a minimum, then the axial lengths of the tetragonal unit can be predicted. The values obtained are a = 8.60 Ä and c = 5.18 Ä. Although the average extra-icosahedral bond length is 1.70 A in this structure, the geometrical constraints imposed by the way in which the icosahedra are stacked are such that all of the extra-icosahedral bond lengths deviate from the average value. I t would be expected that some distortion of the icosahedra would occur, such as to bring the extra-icosahedral bond lengths toward the expected values, and this is in fact observed. The predicted and observed extra-icosahedral bond lengths are as follows: Bond B4—B4 B3—B5 B ^ B 1 B2—B2 Predicted 1.74 1.56 1.88 1.64 Observed 1.70 1.61 1.85 1.67 showing that in each case the adjustments within the icosahedra are such as to allow the extra-icosahedral bonds to approach the expected value. (The observed values quoted are averages of those given by HOABD, HUGHES , and SANDS for their separate parameter determinations for needles and for plates, with a weight of three for the needles and one for the plates

Linus Pauling : selected scientific papers

Volume 1 - Physical Sciences: The Chemical Bond: Metallic Bonding Hydrogen Bonding Crystal and Molecular Structure and Properties: Ionic Crystals and X-Ray Difraction Molecules in the Gas Phase and Electron Diffraction Entropy and Molecular Rotation in Crystals and Liquids Volume 2 - Biomolecular Sciences: Biological Macromolecules: Antibodies: Structure and Function The Alpha Helix and the Structure of Proteins Health and Medicine: Molecular Disease Orthomolecular Medicine Summary of Linus Paulings Life and Scientific Work: Biographical Memoir (J D Dunitz) and other papers.