As we are only studying a model comprised of two electrons restricted to move in a one-dimensional external potential we have employed the configuration-interaction theory to compute the steady-state properties of the system. We have used a static, one-dimensional, grid-based basis set for the single-particle functions. This allows for flexibility in the choice of the external potential, and fits the interpolated potential particularly well.
The Hamiltonian of \( N \) interacting electrons confined by some potential \( v(r) \) can be written on general form $$ \begin{equation} \hat{H} = \sum_{i=1}^N \left(-\frac{1}{2}\nabla_i^2 + v(r_i) \right) + \sum_{i < j} \hat{u}(r_i, r_j), \tag{1} \end{equation} $$ where \( \hat{u}(i,j) \) is the electron-electron (Coulomb) interaction.