Hamiltonian:
$$ \begin{align} \hat{H} &= \frac{\hat{p}_1^2}{2} + v^L[\hat{x}_1] + \frac{\hat{p}_2^2}{2} + v^R[\hat{x}_2] + \frac{\kappa}{\sqrt{(\hat{x}_1-\hat{x}_2)^2 + a^2}} \tag{8}\\ &\equiv h^L[\hat{p}_1,\hat{x}_1] + h^R[\hat{p}_2,\hat{x}_2] + u[\hat{x}_1,\hat{x}_2] \tag{9} \end{align} $$Energy states:
$$ \begin{equation} \vert\Phi_k\rangle = \sum_{i} \sum_{j} C_{ij, k}\vert \varphi^L_i \varphi^R_j\rangle, \tag{10} \end{equation} $$(product basis)