The knowledge of the atomic structure of clusters composed by few atoms is a basic prerequisite to obtain insights into the mechanisms that determine their chemical and physical properties as a function of diameter, shape, surface termination, as well as to understand the mechanism of bulk formation. Due to the wide use of metal systems in our modern life, the accurate determination of the properties of $3d$, $4d$, and $5d$ metal clusters poses a huge problem for nanoscience. In this work, we report a density functional theory study of the atomic structure, binding energies, effective coordination numbers, average bond lengths, and magnetic properties of the $3d$, $4d$, and $5d$ metal (30 elements) clusters containing 13 atoms, $M_{13}$. First, a set of lowest-energy local minimum structures (as supported by vibrational analysis) were obtained by combining high-temperature first-principles molecular-dynamics simulation, structure crossover, and the selection of five well-known $M_{13}$ structures. Several new lower energy configurations were identified, e.g., ${\text{Pd}}_{13}$, ${\text{W}}_{13}$, ${\text{Pt}}_{13}$, etc., and previous known structures were confirmed by our calculations. Furthermore, the following trends were identified: (i) compact icosahedral-like forms at the beginning of each metal series, more opened structures such as hexagonal bilayerlike and double simple-cubic layers at the middle of each metal series, and structures with an increasing effective coordination number occur for large $d$ states occupation. (ii) For ${\text{Au}}_{13}$, we found that spin-orbit coupling favors the three-dimensional (3D) structures, i.e., a 3D structure is about 0.10 eV lower in energy than the lowest energy known two-dimensional configuration. (iii) The magnetic exchange interactions play an important role for particular systems such as Fe, Cr, and Mn. (iv) The analysis of the binding energy and average bond lengths show a paraboliclike shape as a function of the occupation of the $d$ states and hence, most of the properties can be explained by the chemistry picture of occupation of the bonding and antibonding states.

• Received 2 October 2009

DOI:https://doi.org/10.1103/PhysRevB.81.155446