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[85]. 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[90] 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[93]. 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 
.