  Recent observations of novel spin-orbit coupled states have generated
tremendous interest in $4d/5d$ transition metal systems. A prime example is the
$J_{\text{eff}}=\frac{1}{2}$ state in iridate materials and $\alpha$-RuCl$_{3}$
that drives Kitaev interactions. Here, by tuning the competition between
spin-orbit interaction ($\lambda_{\text{SOC}}$) and trigonal crystal field
splitting ($\Delta_\text{T}$), we restructure the spin-orbital wave functions
into a novel $\mu=\frac{1}{2}$ state that drives Ising interactions. This is
done via a topochemical reaction that converts Li$_{2}$RhO$_{3}$ to
Ag$_{3}$LiRh$_{2}$O$_{6}$, leading to an enhanced trigonal distortion and a
diminished spin-orbit coupling in the latter compound. Using perturbation
theory, we present an explicit expression for the new $\mu=\frac{1}{2}$ state
in the limit $\Delta_\text{T}\gg \lambda_{\text{SOC}}$ realized in
Ag$_{3}$LiRh$_{2}$O$_{6}$, different from the conventional
$J_\text{eff}=\frac{1}{2}$ state in the limit $\lambda_{\text{SOC}}\gg
\Delta_\text{T}$ realized in Li$_{2}$RhO$_{3}$. The change of ground state is
followed by a dramatic change of magnetism from a 6 K spin-glass in
Li$_{2}$RhO$_{3}$ to a 94 K antiferromagnet in Ag$_{3}$LiRh$_{2}$O$_{6}$. These
results open a pathway for tuning materials between the two limits and creating
a rich magnetic phase diagram.
