The effect of different energy portions on the 2D/3D stability swapping for 13-atom metal clusters

The complexity of Cu₁₃, Ag₁₃, and Au₁₃ coinage-metal clusters was investigated through their energy contributions via a density functional theory study, considering improvements in the PBE functional, such as van der Waals (vdW) corrections, spin-orbit coupling (SOC), Hubbard term (+U), and their combinations. Investigating two-dimensional (planar 2D) and three-dimensional (distorted 3D, CUB - cuboctahedral, and ICO - icosahedral) configurations, we found that vdW corrections are dominant in modulating the stability swapping between 2D and ICO (3D) for Ag₁₃ (Au₁₃), whereas for Cu₁₃ its role is increasing the relative stability between 2D (least stable) and 3D (most stable), setting ICO as the reference. Among the energy portions that constitute the relative total energy, the dimensionality difference correlates with the magnitude of the relative dispersion energy (large for 2D/ICO and small for 3D/ICO) as the causal factor responsible for an eventual stability swapping. For instance, empirical vdW corrections may favor Ag₁₃ as ICO, while semi empirical ones tend to swap the stability by favoring 2D. The same tendency is observed for Au₁₃, except when SOC is included, which enlarges the stability of 3D over 2D. Energy decomposition analysis combined with the natural orbitals for the chemical valence approach confirmed the correlations between the dimensionality difference and the magnitude of the relative dispersion energies. Our structural analysis protocol was able to capture the local distortion effects (or even their absence) through the quantification of the Hausdorff chirality measure. Here, ICO, CUB, and 2D are achiral configurations for all coinage-metal clusters, whereas Cu₁₃ as 3D presents a slight chirality when vdW correction based on many body dispersion is used, at the same time Ag₁₃ as 3D turned out to be chiral for all calculation protocols as evidence of the role of the chemical composition.