  We study the collective excitations of Na$_2$IrO$_3$ in an itinerant electron
approach. We consider a multi-orbital tight-binding model with the electron
transfer between the Ir $5d$ states mediated via oxygen $2p$ states and the
direct $d$-$d$ transfer on a honeycomb lattice. The one-electron energy as well
as the ground state energy are investigated within the Hartree-Fock
approximation. When the direct $d$-$d$ transfer is weak, we obtain nearly flat
energy bands due to the formation of quasimolecular orbitals, and the ground
state exhibits the zigzag spin order. The evaluation of the density-density
correlation function within the random phase approximation shows that the
collective excitations emerge as bound states. For an appropriate value of the
direct $d$-$d$ transfer, some of them are concentrated in the energy region
$\omega <$ 50 meV (magnetic excitations) while the others lie in the energy
region $\omega >$ 350 meV (excitonic excitations). This behaviour is consistent
with the resonant inelastic x-ray scattering spectra. We also show that the
larger values of the direct $d$-$d$ transfer are unfavourable in order to
explain the observed aspects of Na$_2$IrO$_3$ such as the ordering pattern of
the ground state and the excitation spectrum. These findings may indicate that
the direct $d$-$d$ transfer is suppressed by the structural distortions in the
view of excitation spectroscopy, as having been pointed out in the \it{ab
initio} calculation.

Strongly Correlated Electrons