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Additional condition for transferability in pseudopotentials.

The chemical property of hardness is extremely important in the correct prediction of electron transfer between reacting chemical systems. Hardness is defined to be one-half the second derivative of the total energy of a chemical system with respect to the number of electrons. The requirement that the total energies of atom and pseudoatom match to second order with arbitrary changes in valence-state occupancy yields major decreases in the errors made when using pseudopotentials. The concept also clarifies the role of core density in pseudopotential application. A practical prescription is presented for generating such pseudopotentials and their corresponding core densities, and several ab initio results are compared with those of high-quality norm-conserving pseudopotentials. Significant improvements are found.