Excitation energies can be calculated within Hartree-Fock (HF) theory via two methods. Koopmans' theorem applies if the number of electrons in the system is large. Then adding or removing a single electron from the system will not affect the orbitals of the other electrons and they can be assumed fixed. The energy required to remove the k electron from the system is then just . Hence the energy to transfer an electron from the i to the j state is . Therefore, within Koopmans' theorem, the single-particle energies, from the Hartree-Fock equations, Eq.(), can be interpreted as the excitation energies of the system.
Koopmans' theorem is only valid if the one-electron wavefunctions in the N-electron and the -electron Slater determinants are the same, i.e. the single-particle orbitals do not relax when an electron is added to or removed from the system. In a finite supercell calculation, such as those performed in chapters to of this thesis, it is not clear to what extent Koopmans' theorem still holds. In this case it is more appropriate to use the alternative method of performing total energy calculations for both the N and the 1 electron systems and then subtracting to find the ionisation energy and electron affinity. It has been shown  that for core levels in atoms there are significant differences between the results obtained using Koopmans' theorem and the method of performing two total energy calculations. There are many examples[71, 72, 73] of applications of these techniques to calculations of the HF bandstructure of semiconductors.