  Motivated by studies of coexisting electron correlation and spin-orbit
coupling effect in Na$_2$IrO$_3$ and a recent experiment of its 4d analogue
Li$_2$RhO$_3$, we performed first-principles calculations of the rhodium oxide
compound. The experimentally observed ground state of Li$_2$RhO$_3$ can be
recovered only if both spin-orbit coupling and on-site Coulomb interaction are
taken into consideration. Within the proper $U$ range for 4d-orbitals
($2\leqslant U\leqslant 4$ eV), the ground state of Li$_2$RhO$_3$ could be
either zigzag-AFM or stripy-AFM, both yielding energy gap close to experimental
observation. Furthermore, the total energy differences between the competing
magnetic phases are $\leqslant 3$ meV/Rh within $2\leqslant U\leqslant 4$ eV,
manifesting strong magnetic frustration in the compound. Finally, the phase
energy of Li$_2$RhO$_3$ cannot be fitted with the two-dimensional
Heisenberg-Kitaev model involving only the nearest neighbor interactions, and
we propose that inter-layer interactions may be responsible for the
discrepancy.
