Hammett Equation - Estimation of Hammett Sigma Constants

Estimation of Hammett Sigma Constants

Core-electron binding energy (CEBE) shifts correlate linearly with the Hammett substituent constants (σ) in substituted benzene derivatives.

ΔCEBE ≈ κσ_p (1)

Consider para-disubstituted benzene p-F-C₆H₄-Z, where Z is a substituent such as NH₂, NO₂, etc. The fluorine atom is para with respect to the substituent Z in the benzene ring. The image on the right shows four distinguished ring carbon atoms, C1(ipso), C2(ortho), C3(meta), C4(para) in p-F-C₆H₄-Z molecule. The carbon with Z is defined as C1(ipso) and fluorinated carbon as C4(para). This definition is followed even for Z = H. The left-hand side of is called CEBE shift or ΔCEBE, and is defined as the difference between the CEBE of the fluorinated carbon atom in p-F-C₆H₄-Z and that of the fluorinated carbon in the reference molecule FC₆H₅.

ΔCEBE ≡ CEBE(C4 in p-F-C₆H₄-Z) – CEBE(C4 in p-F-C₆H₅) (2)

The right-hand side of Eq. 1 is a product of a parameter κ and a Hammett substituent constant at the para position, σp. The parameter κ is defined by eq. 3:

κ = 2.3kT(ρ - ρ*) (3)

where ρ and ρ* are the Hammett reaction constants for the reaction of the neutral molecule and core ionized molecule, respectively. ΔCEBEs of ring carbons in p-F-C6H4-Z were calculated with density functional theory to see how they correlate with Hammett σ-constants. Linear plots were obtained when the calculated CEBE shifts at the ortho, meta and para Carbon were plotted against Hammett σ_o, σ_m and σ_p constants respectively.

κ value calculated ≈ 1. Hence the approximate agreement in numerical value and in sign between the CEBE shifts and their corresponding Hammett σ constant.

• Plot of calculated CEBE shift (eV) against sigma-para

• Table of CEBE shifts (eV) and sigma-para

• Plot of calculated CEBE shift (eV) against sigma-meta

• Table of CEBE shifts (eV) and sigma-meta

• Plot of calculated CEBE shift (eV) against sigma-o

• Table of CEBE shifts (eV) and sigma-ortho