Electronic accommodation versus geometric frustration in doped icosahedral titanium nanoclusters

The stability of finite atomic systems is commonly rationalized in terms of geometric coordination or cohesive-energy arguments. In the sub-nanometer regime, however, where structural closure imposes intrinsic geometric frustration, such descriptors often fail to account for stability trends, symmetry breaking, and site selectivity. Herein, we present a physically transparent framework for analyzing stability in finite systems with coupled structural and electronic degrees of freedom, formulated as a competition between geometric frustration and local electronic accommodation. Using doped sub-nanometer metallic clusters as a minimal model system, we show that stability is not dictated by coordination alone, but by the balance between frustration-induced distortion energies and dopant-dependent electronic relaxation. Electronically flexible species mitigate geometric frustration through charge redistribution and adaptive hybridization, which stabilizes highly constrained environments, whereas electronically rigid species amplify frustration and favor lower-coordination configurations. This framework provides a quantitative interpretation of stability trends in finite systems, complementing coordination-based and system-specific descriptions.