Geometrical and algebraical invariances and general angular dependences of sets of s-p hybrid orbitals. Their angular overlap model relevance

Abstract

An s-p hybrid orbital can be conceived as a linear combination of a scalar and a vector and can be written as h p ( θ, φ) = h(sp p , θ, g4) = h( χ, θ, ϕ)= s sin χ + p σ ( θ, φ) cos χ where ρ = cot 2 χ determines the shape of the hybrid and p σ ( θ, φ) = p x 〈p x |p σ ( θ, φ)〉 ang+p y 〈p y |p σ ( θ, φ) 〉 ang+p z 〈p z |p σ ( θ, φ)〉 ang determines its direction. The coefficients to the unit vectors along the three Cartesian axes, written as p functions, are the angular overlaps between these unit vectors and the unit vector along the direction of the hybrid. The situation of four mutually orthogonal s-p hybrids has been analyzed by using these concepts and it has been found that these hybrids invariably lie pairwise in perpendicular planes. Moreover, if each hybrid h i with shape parameter χ i is replaced by a vector with the direction of the hybrid and the length sin2 χ i , then their vector sum is vanishing. The four-hybrid problem has three degrees of freedom determining shapes and relative directions. Formulas are given for a general analysis of the situation, and particularly for the analysis — on the basis of three experimentally determined angles — of the shape and direction of a single lone-pair, which is necessary in an angular overlap model (AOM) context. The formalism of the AOM is shown to function on the basis of hybrid orbitals, and this property of the AOM gives the clue to its handling of non-linearly ligating ligands.