Formulas governing fixed orbital hardnesses and their relation to the hardness kernel are derived. It is shown how the orbital hardness matrix and its inverse matrix, the orbital softness matrix, may thus be directly calculated, and then the total chemical hardness, softness, and electronegativity of a molecular species. These quantities are calculated for the molecule HCN, using Dirac exchange and von Barth-Hedin correlation in the local density form of Kohn-Sham theory. The result complies with the frontier orbital theory. As quantitative indicators of orbital reactivity, the frontier orbital softness and Fukui indices generally have larger values than inner electron orbitals. The relation of orbital hardness matrix elements to the two-electron orbital integrals in a typical molecular orbital calculation is discussed, and it is demonstrated that diagonalization of the orbital hardness matrix leads to orbitals more localized than conventional Kohn-Sham orbitals.