the outer polygon and the inner polygon be denoted as G$_{opt}$. $U_{opt}$ and  G$_{opt}$ are given in Table 1.

  Table 1,  The optimal values $U_{opt}$ and the associated G$_{opt}$ of the (local)minima of $U$ , each is associated with a N$_{in}-$N$_{out}$ core-ring configuration.

   N$_{in}-$N$_{out}$    1-12     2-11     3-10     4-9      5-8      6-7      7-6      8-5
  -------------------- -------- -------- -------- -------- -------- -------- -------- --------
    $U_{opt}$ (meV)     281.41   278.35   274.83   274.22   274.93   276.31   275.73   278.76
       G$_{opt}$                  3.73     2.83     2.41     2.16     2.02     1.77     1.70

             Evidently, a too small or too large N$_{out}$ (say, N$_{out}\leq
7$ or N$_{out}\geq 11)$ is not advantageous to binding. Furthermore, the outer polygon should be neither too close to nor too far away from the core.

In what follows, when the wave function of a state is distributed surrounding a N$_{in}-$N$_{out}$ core-ring configuration, then the state is said to have a N$_{in}-$N$_{out}$ structure. If the configuration has an electron at the center, then the structure is further denoted as (N$%
_{in})_{c}-$N$_{out}$ ,  otherwise as (N$_{in})_{h}-$N$_{out}$ . The subscript $h$ implies a hollow structure.

It is shown in the table that the $U_{opt}$ of a number of configurations are quite close to each other. At a first glance, one might expect that 