\cdot d{\bf r}_{13}$        (7)   and

   $\ \ I_{out}$ $=M\int_{a}^{\infty }\rho _{1}(r_{1})r_{1}^{2}d{\bf r}%
_{1}$                     (8)  where $M=Nm^{\ast }$ is the total mass. Similarly, the $l_{in}$ and $%
I_{in}$ contributed from the inner region can also be defined.  Although these quantities are not good quantum numbers, they can help us to understand better the physics as shown later.

3, DYNAMICAL AND SYMMETRY BACKGROUND

Quantum mechanic systems are affected by both dynamical reasons and symmetry consideration . The following points are noticeable:

      (i) [**Core-ring structures**]{}.

The spatial wave functions of low-lying states are mainly distributed in an area where the total potential energy $U$ (eq.(1.3)) is lower.  In particular, they would like to be distributed surrounding the (local)minima of $U$. In order to find out the (local)minima, let  N$_{in}$ electrons be contained inside to form a core, and N$_{out}$ electrons be contained outside to form a ring, N$_{in}$+N$_{out}=$N. When the relative locations of the electrons are appropriately adjusted (e.g., they form two homocentric regular polygons with or without an electron at the center) $U$ will be optimized and arrives at its (local) minimum $U_{opt}$,  the associated configuration is called an N$_{in}-$N$_{out}$ core-ring configuration. In this configuration, let the  ratio of the radii of 