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Fluctuation formulas for the external potential v(r) are introduced in a modified Legendre-transformed representation of the density functional theory of electronic structure (isomorphic ensemble). A new (nuclear/geometric) reactivity index h(r), having the same status as the electronic Fukui function in the canonical ensemble, is thereby identified, h(r)=(1/N)[δμ/δσ(r)]N,T=(1/kT) [〈μ⋅v(r)〉−〈μ〉〈v(r)〉], where μ is the electronic chemical potential, σ is the shape factor of the electron density distribution, N is the number of electrons, 〈...〉 denotes the ensemble average of a quantity, and 〈v(r)〉 is the ensemble averaged external potential. This new local quantity is shown to be an inverse of the local softness, and to provide a useful definition of a local hardness.

The Elcctronegativity Equalization Method (EEM), a semi-empirical approach rooted in density functional theory, enables the direct computation of the first and second order derivatives of the energy with respect to the number of electrons (N) and the external potential (v). Within this framework, various response properties of a system, as defined in the sensitivity analysis, can be evaluated. The method and its extensions are outlined, compared with other (theoretical and empirical) approaches, and applied to model systems. Applications are mainly sought in the field of structural chemistry and heterogeneous catalysis. From the theory, general rules addressing hardness/softness-structure-reactivity relationships are formulated. These provide the experimental chemist with guidelines to understand and predict the properties of materials and their role in perturbing and activating adsorbed molecules.

The recently introduced nuclear Fukui function φα is formally identified as a reactivity index of the density functional theory (according to the postulated criterion of |dμ|) and is shown to constitute the conformational contribution to a change in the molecular electronic chemical potential μ, through the relation dμ|N=∫ f(r)dν(r)dr=−∑αφαdRga, with φα=(∂Fα/∂N)ν=−(δμ/δRα)N, where N is the number of electrons, f(r) the electronic Fukui function, ν(r) the external potential at point r, Rα the space coordinate of nucleus α, and Fα the force on nucleus α. Scaling of the nuclear coordinates with a factor λ, as a particular conformational change, links the nuclear Fukui function with Berlin’s binding function B(r) for polyatomic molecules, dμλ|N=dλ∫ f(r)B(r)dr=−∑αφα. This relation is instructing for interpretative purposes: changes in electron density are weighted by the binding function, which, according to Berlin’s theorem, separates the system in binding and antibinding regions.

1The average local electrostatic potential function, V ( r ) / p (~) , is calculated for 87 atoms, Li -Ac, in the ground state using the nonrelativistic average-over-configuration numerical Hartree-Fock density. It is found empirically that in a given atom the shell boundaries are expressed as the successively increasing maxima in V(r)/p(r) and the outermost maximum presents good approximate estimates of the core-valence separation in atoms. The likeness in behavior of V ( r > / p ( r ) at each shell boundary with the maximum hardness principle is discussed. The single-exponent-fit parameters for the electron density in the valency region are provided for all atoms. 0

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The H-exchange of methane in Faujasite is analyzed with the electronegativity equalization method (EEM). The polarization channels and their hardnesses are presented as promising tools for estimating the reactivity of acid zeolite-catalyzed reactions along the reaction coordinate. The influence of the Si/Al ratio and the Al and H distribution on the reaction mode hardness is investigated. The results are in agreement with the next nearest neighbors (NNN) principle, which stresses the importance of the number of Al atoms in the second coordination sphere as a key reactivity parameter.