Size-Dependent Structural Motifs in Ag$_n$Mo (n = 2-13) Clusters: From Planar to Icosahedral Architectures

We present a comprehensive density functional theory (DFT) study of Mo-doped silver clusters Ag$_n$Mo ($n=1$-14), focusing on their structural, electronic, and bonding properties. Global optimization reveals an evolution from planar and low-symmetry isomers in small clusters to compact three-dimensional geometries with higher symmetry, culminating in a highly stable icosahedral structure at $n=12$. Binding energy and second-order energy difference analyses identify $n=12$ as a "magic number" cluster exhibiting enhanced thermodynamic stability and a pronounced HOMO-LUMO gap, indicative of electronic shell closure. Bond length analysis shows relatively constant Ag-Mo distances alongside a size-dependent increase in Ag-Ag bond lengths, reflecting the growth of metallic bonding networks. Hirshfeld charge analysis reveals significant charge transfer from Ag to Mo in small clusters, which decreases with size as the system transitions toward delocalized metallic bonding. These findings provide detailed insights into the size-dependent interplay of geometry, bonding, and electronic structure in Ag$_n$Mo clusters, with implications for their catalytic and material applications.