Orbital hardness tensors from hydrogen through xenon from kohn-sham perturbed orbitals

A systematic study of the orbital hardness tensor and total hardness for atoms ranging from H to Xe is presented. Results are obtained by the use of an efficient algorithm for the computation of density functional-based orbital reactivity indices exploring the concept of fractional occupations. So, the orbital reactivity indices are defined within the space spanned by the orbital occupation numbers and the Kohn-Sham oneelectron energies. The explicit treatment of degenerate orbitals within the algorithm makes it particularly suitable for resolving orbital hardness elements for atoms. Very good numerical stability toward basis sets and exchange-correlation functionals has been achieved. The symmetry of the hardness tensor is maintained, even though its elements are computed differently, using either the left side (occupied) or right side (unoccupied) derivative of the one-particle energies with respect to the orbital occupation numbers. The diagonal elements of the atomic hardness tensors are used as parameters in semi-empirical and tight-binding methods, and the total atomic hardnesses could be used to study reactivity indices for very large systems within the electronegativity equalization scheme.