  Spin-orbit interaction in Sr$_2$IrO$_4$ leads to the realization of the
$J_{\mathrm{eff}}$ = 1/2 state and also induces an insulating behavior. Using
large-shift Raman spectroscopy, we found two high-energy excitations of the
d-shell multipletat at 690 meV and 680 meV with $A_{1g}$ and $B_{1g}$ symmetry
respectively. As temperature decreases, the $A_{1g}$ and $B_{1g}$ peaks narrow,
and the $A_{1g}$ peak shifts to higher energy while the energy of the $B_{1g}$
peak remains the same. When 25$\%$ of Ir is substituted with Rh the $A_{1g}$
peak softens by 10$\%$ but the $B_{1g}$ peak does not. We show that both
pseudospin-flip and non-pseudosin-flip dd electronic transitions are Raman
active, but only the latter are observed.
