  It is necessary to employ quasi-particle calculations to correct band gap
problems in LDA. As an expedient way for the reliable but massive GWA,
Quasi-Particle-LDA (QPLDA) is proposed by several authors, where the total
computational cost scales with N. Historically, Sham and Kohn introduced the
idea of the local mass operator based on local wavenumber similar to WKB, but
did not execute actual numerical calculations. They took into account exchange
interaction only, from which we could not expect precise treatments anyway.
Later, Pickett and Wang had proposed a more qualitative method and had shown
its potentiality in examples of semiconductors, such as silicon, diamond and
GaP. They used a model analytic formula for the dialectic function and adopted
a model energy dispersion, which is free-electron like one, except that, being
accompanied with an artificial band-gap discontinuity. The latter method has
two shortcomings. First, to execute the calculation, several parameters are
needed, such as the macroscopic dielectric constant and the band gap, which
should be evaluated by the calculation itself, rather than being prepared as
parameters. Second, in their computation of the mass operator, two or three
dimensional numerical integrations are needed on each point of the direct space
mesh, which demand large computational costs in iterative procedures in
determining local wavenumbers and quasi-particle energies. This may be the
reason why QPLDA has not been widely used until now. The present work proposes
a qualitative way of QPLDA which does not need any experimental parameters,
being much speedier than the formalism of Picket and Wang, and more precise
than that of Sham and Kohn. The computational method adopted by the present
work is a modification of the original proposal by Sham and Kohn, rather than
that of Picket and Wang.
