The Hamiltonian of the quantum dot is given by
$$ \hat{H} = \hat{H}_0 + \hat{V}, $$where \( \hat{H}_0 \) is the many-body HO Hamiltonian, and \( \hat{V} \) is the inter-electron Coulomb interactions. In dimensionless units,
$$ \hat{V}= \sum_{i < j}^N \frac{1}{r_{ij}}, $$with \( r_{ij}=\sqrt{\mathbf{r}_i^2 - \mathbf{r}_j^2} \).