  We investigate resonant inelastic X-ray scattering (RIXS) at the O $K$-edge
in insulating cuprates by means of three methods: cluster perturbation theory
(CPT), Hartree--Fock approximation (HFA) and exact diagonalization (ED) method.
We consider the three-band Hubbard model and show the overall
momentum-dependence of the Zhang--Rice singlet (ZRS) excitation and
charge-transfer excitation by the CPT combining with the perturbation scheme. A
comparison of the RIXS spectra calculated using CPT and HFA reveals different
momentum-dependencies through the changes in the properties of the upper
Hubbard band and the ZRS band. These findings are supported by analyses using
the ED method on the RIXS spectra and dynamical charge structure factor.
