Equivalent Chemical Bonds Formed bys,p, anddEigenfunctions

Abstract

1. The work of Pauling on the linear combinations of eigenfunctions to form bond functions has been extended.2. A theorem concerning the maximum strength in a given direction of a bond function orthogonal to existing bond functions has been derived. It is found that the strongest bond functions must lie at such angles to one another that the maximum of each coincides with a node of each of the others.3. A general expression for equivalent Pauling bond functions formed from linear combinations of $s$, $p$, and $d$ eigenfunctions has been derived.4. By assuming the bond functions to have cylindrical symmetry, like the best possible bond functions, the equations can be readily solved. It is found that equivalent cylindrical bond functions will be orthogonal if the angles between every pair have no more than two values. The angles must be greater than 54\ifmmode^\circ\else\textdegree\fi{}44' and their sum must be greater than 180\ifmmode^\circ\else\textdegree\fi{}; otherwise they are unrestricted. No more than six such bond functions may be formed. Solutions for six and less equivalent bond functions are given.5. There are only four possible configurations for six equivalent cylindrical bond functions. Two of these are much weaker than the other two and so are improbable. The configurations for the stronger bond functions are in striking agreement with the only two structures, the octahedron and the trigonal prism, which are found experimentally.6. A discussion of the factors influencing bond energies is made and examples of chemical compounds cited. It is predicted that the structure of ${P}_{4}$ is that of a tetrahedron with a $P$ atom at each corner.