Trends in the structure and bonding of noble metal clusters

We present a systematic study of the electronic properties and the geometric structure of noble metal clusters $X_n^{\nu }$ ($X=\mathrm{Cu}$, Ag, Au; $\nu =-1,0,+1$; $n⩽13$ and $n=20$), obtained from first-principles generalized gradient approximation density functional calculations based on norm-conserving pseudopotentials and numerical atomic basis sets. We obtain planar structures for the ground state of anionic $(\nu =-1)$, neutral $(\nu =0)$, and cationic $(\nu =1)$ species of gold clusters with up to 12, 11, and 7 atoms, respectively. In contrast, the maximum size of planar clusters with $\nu =-1,0,+1$ are $n=(5,6,5)$ for silver and (5,6,4) for copper. For $X_{20}$ we find a $T_d$ symmetry for gold and a compact $C_s$ structure for silver and copper. Our results for the cluster geometries agree partially with previous first-principles calculations, and they are in good agreement with recent experimental results for anionic and cationic gold clusters. The tendency to planarity of gold clusters, which is much larger than in copper and silver, is strongly favored by relativistic effects, which decrease the $s\text{−}d$ promotion energy and lead to hybridization of the half-filled $6s$ orbital with the fully occupied $5d_{z^2}$ orbital. That picture is substantiated by analyzing our calculated density matrix for planar and three-dimensional clusters of gold and copper. The trends for the cohesive energy, ionization potentials, electron affinities, and highest accupied and lowest unoccupied molecular orbital gap, as the cluster size increases, are studied in detail for each noble metal and rationalized in terms of two- and three-dimensional electronic shell models. The most probable fragmentation channels for $X_n^{\nu }$ clusters are in very good agreement with available experiments.