  Fully self-consistent GW (sc-GW) methods are now available to evaluate
quasiparticle and spectral properties of various molecular and bulk systems.
However, such techniques based on the full matrix of G and W are
computationally demanding. The routinely used single-shot GW-approximation
(G0W0) has an undesirable dependency on the choice of initial
exchange-correlation functional. In the literature, many so-called
self-consistent GW methods are based on diagonal approximation of G and
low-ranking approximation of W. It is thus worth to check how good such
approximations are in comparison with the full matrix method. In this work, we
consider AlAs, AlP, GaP, and ZnS as the prototype systems to perform sc-GW
calculations by expressing the full G matrix using a plane-wave basis set. We
compared our sc-GW results with the diagonal G and subspace W approximated
sc-GW results (sc-GW-diagG and sc-GW-subW methods). In the sc-GW-diagG method,
interacting G is expanded in the eigenvectors of non-interacting G such that
only diagonal elements are retained, whereas, the number of eigenmodes is
truncated in sc-GW-subW calculations. A systematic analysis of the results
obtained from the above techniques is presented. The differences in the
quasiparticle bandgap between the approximated and the full matrix sc-GW
approaches are mostly less than 1.7% that validates such widely adopted
approximations, and also shows how such low-ranking approximation can be used
to include higher-order terms like the vertex correction.
