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A quantitative relation (q = V2(K/a)l13) between electric dipole polarizability a and hardness 9 is developed.In this relation K is a constant and r and a are expressed in au. With K = 0.792, using the available a data, the above equation is found to reproduce the experimental r values for a number of open shell atoms and monoatomic cations of various charges very satisfactorily. This equation applies to the clusters of nonmagic numbers of sodium atoms also. However, for closed shell species, the above equation gives hardness values much lower than the experimental data. This is probably because the equation fails to reveal the "extra" amount of stability imparted to a chemical species by a closed shell configuration. Similar lower values of 7 are produced by this equation for clusters of magic numbers of sodium atoms.In 1963, using the idea of polarizability, Pearson classified the Lewis acids and bases in terms of "hardness".' A less polarizable species is hard and a more polarizable one soft, softness being the reciprocal of hardness. This hard-soft classification proved to be quite successful in rationalizing a variety of chemical observations, especially the direction of exchange reaction^.^,^ By polarizability, he meant the ease of deforming the valence electron cloud of a chemical species. A closely related experimental quantity is electric dipole polarizability, a which actually describes the response of the electron cloud of a chemical species to an external electric field much lower than what would be needed to ionize the system. The proportionality constant for the dipole moment p induced by the applied electric field F is a: p = aF.4 A connection between Pearson's hardness and a was immediately sought by several worker^.^ However, because of the lack of a quantitative definition of hardness, no definite conclusions could be drawn then. In this context, Jorgensen even felt in 1967 that a is "a far more physical than chemical q~a n t i t y " .~ Pearson's idea of polarizability originally rested upon the nature of the charge on and size of a chemical species-for ions of similar charge hardness decreases with the increase in its radius and for cations of the same radius hardness increases with the amount of positive charge. Fajans' rules are helpful in this regard.6 In 1983, Pearson together with Parr gave a quantitative definition of hardness (r) in the form of eq 1, where Eel is the electronic (1) 1 2 2 11 = -( S~E , , I S N~) = +scl/slv, energy of the system having N number of electrons and p is the chemical potential of the electron cloud.' Equation 1 is based upon density functional theory. Its finite difference approximation allows one to calculate 17 in terms of the ionization potential I and electron affinity A of a chemical species-eq 2. It should be noted that eq 1 yields eq 2 only ( 2 ) when the total energy of a system is assumed to be a quadratic function of its charge q (= Z -N , where Z is the nuclear charge). With a working formula for r being available (eq 2), currently there has been a r...

We present the first large-scale empirical examination of the relation of molecular chemical potentials, μ(0)(mol) = -½(I(0) + A(0))(mol), to the geometric mean (GM) of atomic electronegativities, <χ(0)(at)>(GM) = <½(I(0) + A(0))(at)>(GM), and demonstrate that μ(0)(mol) ≠ -<χ(0)(at)>(GM). Out of 210 molecular μ(0)(mol)values considered more than 150 are not even in the range min{μ(0)(at)} < μ(0)(mol) < max{μ(0)(at)} spanned by the μ(0)(at) = -χ(0)(at) of the constituent atoms. Thus the chemical potentials of the large majority of our molecules cannot be obtained by any electronegativity equalization scheme, including the "geometric mean equalization principle", ½(I(0) + A(0))(mol) = <½(I(0) + A(0))(at)>(GM). For this equation the root-mean-square of relative errors amounts to SE = 71%. Our results are at strong variance with Sanderson's electronegativity equalization principle and present a challenge to some popular practice in conceptual density functional theory (DFT). The influences of the "external" potential and charge dependent covalent and ionic binding contributions are discussed and provide the theoretical rationalization for the empirical facts. Support is given to the warnings by Hinze, Bader et al., Allen, and Politzer et al. that equating the chemical potential to the negative of electronegativity may lead to misconceptions.

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