The energy of metallic (Ni, Cu, Pd, Ag, Pt, and Au) nanoparticles up to 5000 atoms are studied by equivalent crystal theory (ECT), a quantum approximate method (QAM) that describes the ground state structure and the surface properties of metals and semiconductors at zero temperature. ECT relies on the universal binding energy relation to predict with precision and speed the energy of a crystal in a specific configuration. For each pure metallic nanoparticle of each chosen motif (icosahedron, octahedron, and decahedron), the energy variation with the number of atoms $N_{\mathit{at}}$ is studied. Crossover and minimum energy values are calculated and/or estimated and compared with the results obtained by molecular dynamics (MD). Our results confirm the qualitative behavior (i.e., icosahedron shapes are less energetic for small sizes, decahedron for medium sizes, and octahedron for bigger sizes) predicted by MD, but the calculated crossover and minimum energy values are, in general, larger for all metals and geometries examined. Also, we studied the trends in relaxation between layers and the behavior of the average radius $R_{\mathit{av}}$ of each relaxed nanoparticle as $N_{\mathit{at}}$ was increased. For each motif, the most stable structures (i.e., with the best truncation) follow a simple law of $R_{\mathit{av}}$ in terms of $N_{\mathit{at}}$. This simple law is unchanged for the four different motifs and can be extended for all six metals after a simple parametrization is performed.

• Received 31 August 2007

DOI:https://doi.org/10.1103/PhysRevB.76.205429