Our trial wave function for the ground state with \( N \) atoms is given by
$$ \begin{equation} \Psi_T(\mathbf{R})=\Psi_T(\mathbf{r}_1, \mathbf{r}_2, \dots \mathbf{r}_N,\alpha,\beta)=\prod_i g(\alpha,\beta,\mathbf{r}_i)\prod_{i < j}f(a,|\mathbf{r}_i-\mathbf{r}_j|), \tag{7} \end{equation} $$where \( \alpha \) and \( \beta \) are variational parameters. The single-particle wave function is proportional to the harmonic oscillator function for the ground state
$$ \begin{equation} g(\alpha,\beta,\mathbf{r}_i)= \exp{[-\alpha(x_i^2+y_i^2+\beta z_i^2)]}. \tag{8} \end{equation} $$