  We present an analytical study of the exchange interactions between
pseudospin one-half $d^{5}$ ions in honeycomb lattices with edge-shared
octahedra. Various exchange channels involving Hubbard U, charge-transfer
excitations, and cyclic exchange are considered. Hoppings within $t_{2g}$
orbitals as well as between $t_{2g}$ and $e_g$ orbitals are included. Special
attention is paid to the trigonal crystal field $\Delta$ effects on the
exchange parameters. The obtained exchange Hamiltonian is dominated by
ferromagnetic Kitaev interaction K within a wide range of $\Delta$. It is found
that a parameter region close to the charge-transfer insulator regime and with
a small $\Delta$ is most promising to realize the Kitaev spin liquid phase. Two
representative honeycomb materials Na$_2$IrO$_3$ and $\alpha$-RuCl$_3$ are
discussed based on our theory. We have found that both materials share dominant
ferromagnetic K and positive non-diagonal $\Gamma$ values. However, their
Heisenberg J terms have opposite signs: AFM J>0 in Na$_2$IrO$_3$ and FM J<0 in
$\alpha$-RuCl$_3$. This brings different magnetic fluctuations and results in
their different magnetization behaviors and spin excitation spectra. Proximity
to FM state due to the large FM J is emphasized in $\alpha$-RuCl$_3$. The
differences between the exchange couplings of these two materials originate
from the opposite $\Delta$ values, indicating that the crystal field can serve
as an efficient control parameter to tune the magnetic properties of $d^{5}$
spin-orbit Mott insulators.
