## Definitions and notations

The single-particle function $$\psi_{\alpha}(x_i)$$ are eigenfunctions of the onebody Hamiltonian $$h_i$$, that is $$\hat{h}_0(x_i)=\hat{t}(x_i) + \hat{u}_{\mathrm{ext}}(x_i),$$ with eigenvalues $$\hat{h}_0(x_i) \psi_{\alpha}(x_i)=\left(\hat{t}(x_i) + \hat{u}_{\mathrm{ext}}(x_i)\right)\psi_{\alpha}(x_i)=\varepsilon_{\alpha}\psi_{\alpha}(x_i).$$ The energies $$\varepsilon_{\alpha}$$ are the so-called non-interacting single-particle energies, or unperturbed energies. The total energy is in this case the sum over all single-particle energies, if no two-body or more complicated many-body interactions are present.