Excitation energies can be calculated within Hartree-Fock (HF) theory
via two methods. Koopmans' theorem[83] 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 [84] 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.