  We use projector operators to correct the Kohn-Sham Hamiltonian of density
functional theory (KS-DFT) so that the resulting mean-field scheme yields, in
finite systems, virtual orbitals and energy gaps in better agreement with those
predicted by quasiparticle theory. The proposed correction term is a
scissors-like operator of the form $(\hat{I}-\hat{\rho})\delta
\hat{H}(\hat{I}-\hat{\rho})$, where $\hat{I}$ is the identity operator,
$\hat{\rho}$ the density matrix of the N-particle system and $\delta \hat{H}$
is either the difference between the N+1- and N-particle Kohn-Sham Hamiltonians
or a non-self-consistent approximation to it. Such a term replaces the
Kohn-Sham virtual orbitals of the $N$-particle system by the HOMO and virtual
orbitals of the system with $N+1$ particles in an attempt to mimic a true
quasiparticle spectrum. Using a local density approximation (LDA) we compute
the gaps of a variety of small molecules finding good agreement with experiment
and computationally more demanding methods. For these systems we examine the
physical origin of this gap correction and show that so-called band gap
discontinuity, $\Delta_{xc}$, contains electrostatic contributions that do not
originate from the discontinuity in the exchange-correlation potential. The
similarity between the corrected and Hartree-Fock virtual orbitals is
illustrated and the extent to which the bare LDA virtual orbitals are improved
is considered. The lack of band-gap discontinuity and the presence of
self-interaction errors in the proposed correction are also discussed.
