The local density approximation (LDA) is presently the most successful method for the determination of the ground state properties of solids. The eigenvalues of the LDA equation, though being a priori of no physical meaning are nevertheless commonly interpreted as single-particle energies. The energy gaps obtained from these single-particle energies are generally too small in comparison with experiment. The deviation of the LDA gap from experiment can be anywhere between 100% in the case of germanium (in which the LDA predicts a negative gap) to fairly small percentage errors in wide gap insulators. Since modern bandstructure codes have reached a stage where the calculations are well converged, it is clear that this error indicates either a shortcoming of the exchange-correlation functionals currently used in the LDA or a more fundamental incapability of Kohn-Sham (KS) Density Functional theory (DFT) to calculate excitation energies. It has been shown[86, 87] that the exchange-correlation potentials for an N-particle and N+1-particle system differ by a finite quantity , known as the discontinuity of the exchange-correlation potential. As this discontinuity in the exchange-correlation potential is a feature of exact KS-DFT, it is not immediately clear to what extent the errors in the energy gaps obtained from single-particle energies are due to limitations in the LDA or to the presence of this discontinuity. Godby, Schlüter and Sham[88, 89] have calculated an exchange-correlation potential for several semiconductors using the GW approximation for the self-energy, which can be expected to agree very closely with the exact exchange-correlation potential. This potential and the resulting KS-DFT bandstructure turned out to be in remarkably close agreement with the local density approximation. It therefore appears likely that the discontinuity in the exchange-correlation potential is responsible for over 80% of the errors observed in these gaps. It has also been demonstrated that for a two-band semiconductor model, every state-independent exchange-correlation potential is bound to fail to describe excitation energies because some essential features are missing. However, the opposite situation (i.e. a very small discontinuity in the exchange-correlation functional) was found in simple one-dimensional Hubbard-like model for semiconductors[91, 92] where the exchange-correlation potential and its discontinuity can be calculated exactly.
A possible method for overcoming the problem of calculating excited states within DFT was proposed by Kohn. He suggested determining excitation energies by calculating the ground state energies of the N and the electron systems and then subtracting to find the ionisation energy. This is exactly the method outlined above for use in HF calculations and employed in QMC calculations in section .