Two-dimensional electrostatic analog of the March model of C60 with a semiquantitative application to planar ring clusters

The so-called March model of fullerene uses the one-center approximation of quantum chemistry to calculate the spherical pi-electron density rho by the Thomas-Fermi (TF) statistical method. Then the variational principle of the TF theory is employed by combining the electronic energy determined entirely by rho with the discrete C-60 nuclear potential energy to predict, semiquantitatively, the equilibrium radius. Here the same approach is worked out, but now largely analytically, in two dimensions, as a possible model of planar ring clusters. Again the equilibrium radii are calculated and consistency with realistic clusters of hydrogen and carbon rings is discussed in detail. Finally, an estimate of the "breathing" frequency of C-14, C-18, and C-22 is given.