Shell models of two-dimensional Coulomb crystals: assessment and comparison with the three-dimensional case

Three shell models, differing in accuracy and computational cost, are formulated for two-dimensional Coulomb crystals. Offering a new means of predicting and analyzing properties of these species, the new models also provide new insights into their previously derived three-dimensional counterparts. In particular, analysis of the individual components of the energy error points out to the neglect of the positional relaxation as the main source of the differences between the approximate and exact energies. Within the realm of shell models, the two-dimensional case turns out to be somewhat more challenging than the three-dimensional one. Due to the lack of exact closed-form expressions for the optimal shell radii, it is computationally more expensive and the energy predictions at the same level of approximation are less accurate (as indicated by the maximum relative energy error of 0.15% vs. that of 0.03% found for spherical Coulomb crystals).