\qquad \qquad (1.2)$

$\qquad U=\sum\limits_{j=1}^{N}\frac{1}{2}m^{\ast }\omega _{o}^{2}r_{j}^{2}+%
\frac{e^{2}}{4\pi \varepsilon _{r}\varepsilon _{0}}\sum\limits_{j<k}^{N}%
\frac{1}{r_{jk}}$                             (1.3)

where m\* is the effective electron mass, $\varepsilon _{r}$ is the dielectric constant, and $\hbar \omega _{o}$measures the strength of the parabolic confinement ( $\omega _{o}$ arises mainly from a magnetic field $B$. This field leads also to a term linearly proportional to $B$ . This term has been neglected because it does not at all affect the eigenwavefunctions, and therefore not affect the electronic structures).

 In order to diagonalize the Hamiltonian, a set of orthonormalized single-particle harmonic oscillation (h.o.)states $\phi _{mk}$  are introduced.  Here,  $\phi _{mk}$  is an eigenstate of a pure h.o. Hamiltonian

    $h=-\frac{\hbar ^{2}}{2m^{\ast }}\nabla ^{2}+\frac{1}{2}%
m^{\ast }\Omega _{0}^{2}r^{2}$                                                                        (2)

where $\Omega _{0}$ is an adjustable parameter in general not equal to $%
\omega _{o}$, This eigenstate has eigenenergy $(m+k+1)\hbar \Omega _{0}$  and angular momentum $(m-k)\hbar $   . From them the many-body basis functions (BFs)

 $\ \psi _{\alpha }(1,2,\cdot \cdot \cdot ,N)=\frac{1}{\sqrt{N!}}%
\left|
\begin{array}{cccc}
\phi _{m_{1}k_{1}}(\vec{r}_{1}) & \phi _{m_{1}k_{1}}(\vec{r}_{2}) & \cdot
\cdot \cdot & \phi _{m_{1}k_{1}}(\vec{r}_{N}) \\
\phi _{m_{2}k_{2}}(\vec{r}_{1}) & \phi _{m_{2}k_{2}}(\vec{r}_{2}) & \cdot
\cdot \cdot & \phi _{m_{2}k_{2}}(\vec{r}_{N}) \\
\cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot 