A new concept, the strength of bond, and a new form expressing the hardness of covalent and ionic crystals are presented. Hardness is expressed by means of quantities inherently coupled to the atomistic structure of matter, and, therefore, hardness can be determined by first-principles calculations. Good agreement between theory and experiment is observed in the range of 2 orders of magnitude. It is shown that a lower coordination number of atoms results in higher hardness, contrary to common opinion presented in general literature.

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The influence of cluster size and of cluster-substrate interaction on the magnetic properties of Co clusters of 1-10 atoms on Pt(111) and Au(111) is studied by fully relativistic ab initio calculations. The focus is on systematic trends of the spin and orbital magnetic moments, the exchange coupling, and the crossover temperature. The spin magnetic moments of Co clusters are larger for the Pt substrate than for the Au substrate, while the reverse is true for the orbital magnetic moments. The local magnetic moments of Co atoms generally increase if the number of Co neighbours decreases. The exchange coupling constants J i j depend on the cluster size and on the location of respective atoms. The crossover temperature increases monotonically with cluster size and is larger for clusters on Pt than for clusters on Au.