  We consider strongly correlated limit of variants of the Hubbard model (HM)
in which on parts of the system it is energetically favourable to project out
doublons from the low energy Hilbert space while on other sites of the system
it is favourable to project out holes while still allowing for doublons. As an
effect the low energy Hilbert space itself varies with sites of the system.
Though the formalism is well developed for the case of doublon projection in
the literature, case of hole projection has not been explored in detail so far.
We derive basic framework by defining creation and annihilation operators for
electrons in a restricted Hilbert space where holes are projected out but which
still allows for doublons. We generalise the idea of Gutzwiller approximation
for case of hole projection which has been done in literature for the case of
doublon projection. To be specific, we provide detailed analysis of strongly
correlated limit of the ionic Hubbard model (IHM) which has a staggered
potential $\Delta$ on two sublattices of a bipartite lattice and the correlated
binary alloys which have binary disorder $\pm V/2$ randomly distributed on
sites of the lattice. In both the cases, for $\Delta \sim U \gg t$ and for
$V\sim U \gg t$, where $U$ is the Hubbard energy cost for having a doublon at a
site, there are sites on which doublons are allowed while holes are the maximum
energy states. We do a systematic generalization of similarity transformation
for both these cases and obtain the effective low energy Hamiltonian. We
further derive Gutzwiller approximation factors which provide renormalization
of various terms in the effective low energy Hamiltonian due to the Gutzwiller
projection operators, excluding holes on some sites and doublons on the
remaining sites.
