11 July 1955

Professor H. C. Longuet-Higgins

The Chemical Laboratory

University of Cambridge

Cambridge, England

Dear Longuet-Higgins:

I have read with interest your paper by Roberts, in the last issue of the Proceedings of the Royal Society, on the electronic structure of an icosahedron of boron atoms. I am in general pleased with your results, but there is one point that I should like to discuss with you. Some years ago I began work along similar lines, and the work is in fact being continued now by Dr. Hoerni, who will remain here for another year.

You have reached the conclusion that the tetragonal structure for boron reported by Hoard and his co-workers cannot be of the closed-shell type. I think that there is a flaw in your argument. First, I am not sure that each of the two tetrahedral boron atoms in the unit must bear an extra electron, and form four single bonds with its neighbors. Let us, however, assume that this is so, and that 16 electrons of the total of 150 in the unit are tied up in this way. There remain 134 electrons. Let us assume that 26 electrons per icosahedron are then involved in filling the closed shells. There then remain 30 electrons per unit, which are to be assigned to the bonds connecting boron icosahedra. There are 20 bonds of this sort per unit, which would require 40 electrons if the bonds were single bonds. I suggest, however, that the bonds are not single bonds, but are three-quarters bonds, requiring only one and one half electrons per bond - that is, I suggest that a treatment of the entire crystal, with the icosahedra conjugated together, would show that this number of electrons corresponds to a closed shell.

I might point out that your assumption that the outward-pointing orbitals of the icosahedron are all involved in forming single bonds, rather than fractional bonds, does not correspond to the assumption made for the inward-pointing orbitals - namely, that only those with a positive value of the energy parameter (corresponding to stabilization) are to be occupied. In your Table 5 four of the outward-pointing orbitals, rather than 12, are given positive values of the energy parameter. By including the five orbitals with symmetry V, which have a small negative value of this parameter, a total of 9 would be obtained, corresponding to an average of three-quarters occupancy. This is not exactly right, of course, because it includes the two boron atoms bonded to tetrahedral borons, as well as the other ten. But, of course, the icosahedron of boron atoms does not have true icosahedral symmetry in this tetragonal structure, in which two of the twelve boron atoms are bonded to tetrahedral boron atoms, and the other ten are not. I do not think that the argument that you have given is justification for throwing doubt on the structure described by Hoard and his co-workers.

There is another point that I may make. If your assumption were correct, the boron-boron distance between icosahedra should be significantly less than that within an icosahedron. I do not think that the x-ray evidence supports this. It is true that the structure determinations for B₄C and tetragonal boron are not precise ones. However, the problem is the same in CaB₆, which is the substance which we have also been treating theoretically. In CaB₆ each boron atom forms four bonds within an octahedron, and a fifth bond pointing out from the octahedron. You would probably treat this problem in an analogous way to that in which you have treated the problem of ocosahedral structures, whereas we have been considering the problem of the entire crystal, involving conjugated octahedra. Your assumption would in this case mean that the bonds pointing out from an octahedron are single bonds, involving two electrons per bond, and leaving only 1.2 electrons per bond within the octahedron. Twenty years ago Weinbaum and I made a very careful study of CaB₆, and we were able to show that the boron-boron distances between octahedra are identical, to within 0.01 Å, with those within an octahedron. It is evident that these bonds are of essentially the same type, involving about 1.33 electrons per bond.

Sincerely yours,

[Linus Pauling]

Linus Pauling:W