Sigma-pi and equivalent-orbital models

The σ-π model and equivalent-orbital model refer to two possible representations of molecules in valence bond theory. The σ-π model differentiates bonds and lone pairs of σ symmetry from those of π symmetry, while the equivalent-orbital model hybridizes them. The σ-π treatment takes into account molecular symmetry and is better suited to interpretation of aromatic molecules (Hückel's rule), although computational calculations of certain molecules tend to optimize better under the equivalent-orbital treatment. The two representations produce the same total electron density and are related by a unitary transformation of the occupied molecular orbitals; different localization procedures yield either of the two. Two equivalent orbitals h and h' can be constructed by taking linear combinations h = c₁σ + c₂π and h' = c₁σ – c₂π for an appropriate choice of coefficients c₁ and c₂.

In a 1996 review, Kenneth B. Wiberg concluded that "although a conclusive statement cannot be made on the basis of the currently available information, it seems likely that we can continue to consider the σ/π and bent-bond descriptions of ethylene to be equivalent. Ian Fleming goes further in a 2010 textbook, noting that "the overall distribution of electrons [...] is exactly the same" in the two models. Nevertheless, as pointed out in Carroll's textbook, at lower levels of theory, the two models make different quantitative and qualitative predictions, and there has been considerable debate as to which model is most useful conceptually and pedagogically.