  We report a combined experimental and theoretical x-ray magnetic circular
dichroism (XMCD) spectroscopy study at the Ir-$L_{2,3}$ edges on the Ir$^{5+}$
ions of the layered hybrid solid state oxide Sr$_2$Co$_{0.5}$Ir$_{0.5}$O$_4$
with the K$_2$NiF$_4$ structure. From theoretical simulation of the
experimental Ir-$L_{2,3}$ XMCD spectrum, we found a deviation from a pure
$J_{eff}=0$ ground state with an anisotropic orbital-to-spin moment ratio
($L_x/2S_x$ = 0.43 and $L_z/2S_z$ = 0.78). This deviation is mainly due to
multiplet interactions being not small compared to the cubic crystal field and
due to the presence of a large tetragonal crystal field associated with the
crystal structure. Nevertheless, our calculations show that the energy gap
between the singlet ground state and the triplet excited state is still large
and that the magnetic properties of the Ir$^{5+}$ ions can be well described in
terms of singlet Van Vleck paramagnetism.
