Coulomb interaction in a quantum dot

One approximation is made to describe a M+1 electron many-body wavefunction by a M electron many-body wavefunction and a single electron wavefunction. Under this approximation, we have derived the Coulomb energy which relates the exciton energy $E_{exc}$ in a quantum dot with the quasiparticle band gap (defined as the difference between the ionization energy and the electron affinity), and the Coulomb energy which relates $E_{exc}$ with the single particle eigen values. We found that these two Coulomb energies are different. We have compared our results with the formulae used in different groups, which are proposed either {\it ad hov}, or from the classical electrostatic point of view. We found important difference between our results and the classical formula. Finally, under the above approximation, we provided an effective single particle Hamiltonian, which gives the quasiparticle band structure in a bulk system.