Absolute hardness: companion parameter to absolute electronegativity

For neutral and charged species, atomic and molecular, a property called absolute hardness eta is defined. Let E(N) be a ground-state electronic energy as a function of the number of electrons N. As is well-known, the derivative of E(N) with respect to N, keeping nuclear charges Z fixed, is the chemical potential ..mu.. or the negative of the absolute electronegativity chi: ..mu.. = (deltaE/deltaN)/sub Z/ = /sup -/chi. The corresponding second derivative is hardness: 2eta = (delta..mu../deltaN)/sub Z/ = (deltachi/deltaN)/sub Z/ = (delta/sup 2/E/deltaN/sup 2/)/sub Z/. Operational definitions of chi and eta are provided by the finite difference formulas (the first due to Mulliken) chi = 1/2(I+A), eta = 1/2(I-A), where I and A are the ionization potential and electron affinity of the species in question. Softness is the opposite of hardness: a low value of eta means high softness. The principle of hard and soft acids and bases is derived theoretically by making use of the hypothesis that extra stability attends bonding of A to B when the ionization potentials of A and B in the molecule (after charge transfer) are the same. For bases B, hardness is identified as the hardness of the species B/sup +/. Tables of absolute hardness are given for a number of free atoms, Lewis acids, and Lewis bases, and the values are found to agree well with chemical facts. 1 figure, 3 tables.