The 2-D Monte Carlo (MC) simulations were based on DFT-calculated cluster structures and energies. For Pt 2 to Pt 8 , the energies of the clusters within 0.4 eV of the lowest energy structure were included, and the clusters were approximately represented as circles. The radii were set as the smallest circles in the xy plane encompassing the given cluster isomer, allowing for not only cluster sizes, but also the isomers to be distinguished. Note that Pt 8