Application of a quadratic free energy relationship to non-additive substituent effects

Abstract

A quadratic free energy relationship is formulated in which the reaction constant dependes linearly upon substituent constant, e.g. in Hammett's formalism ρ=ρ₀+ 2mσ. The relationship is applied to dual substitution at equivalent positions (X and Y) in the form log k_XY/k_HH=ρ₀(σ_X+σ_Y)+m(σ_X+σ_Y)², using data from the literature for the bromination of diphenylethylenes, solvolytic and equilibrium formation of diphenylmethyl carbonium ions, and the ionisation of 2,7-substituted fluorenes. Non-additivity of substituent effects is quantitatively described and quadratic co-efficients 2m/ρ₀² can be derived without specifying substituent constants. Values of 2m/ρ₀² measure the sensitivity of ρ and the selectivity of the reaction to changes in reactivity. Relative magnitudes are interpreted in terms of variations in transition state structure, resonance saturation, and steric inhibition of resonance. The relationship is compared with Miller's equation, log k_XY/k_HH=ρ₀(σ_X+σ_Y)+qσ_Xσ_Y, with respect to effectiveness of correlations, chemical interpretation, desirability of preserving linear relationships for monosubstitution, and practical application to multiple substitution at non-equivalent positions.