  Motivated by significant discrepancies between experimental observations of
electron-doped cuprates and numerical results of the Hubbard and $t$-$J$
models, we investigate the role of inter-site interactions $V$ by studying the
$t$-$J$-$V$ model on square lattices. Based on large-scale density matrix
renormalization group simulations, we identify the ground-state phase diagram
across varying inter-site interactions $V$ and doping concentration $\delta$.
We find that the phase diagram with finite inter-site interactions $2\lesssim
V/J\lesssim3$ offers a more accurate description of electron-doped cuprates
than the conventional Hubbard and $t$-$J$ models. Moreover, we reveal the role
of inter-site interactions $V$ at varying doping levels: at light doping,
inter-site interactions favor N\'{e}el antiferromagnetic order, and suppress
both superconductivity and charge density wave; around optimal doping, these
interactions support a pseudogap-like phase while suppressing
superconductivity, and we further perform the slave boson mean-field analysis
to understand the numerical results microscopically; at higher doping, the
effects of inter-site interactions become insignificant, with our numerical
predictions suggesting the emergence of incommensurate spin density wave phase.
Our specific focus around optimal doping with various inter-site interactions
identifies successive phases including phase separation, uniform $d$-wave SC
and a pseudogap-like phase, and reveals a relative insensitivity of charge
density wave to superconductivity. Our study suggests the $t$-$J$-$V$ model as
the minimal model to capture the essential physics of the electron-doped
cuprates.
