  Using exact diagonalization, we study the spin-orbit coupling and
interaction-induced mixing between $t_{2g}$ and $e_g$ $d$-orbital states in a
cubic crystalline environment, as commonly occurs in transition metal oxides.
We make a direct comparison with the widely used $t_{2g}$ only or $e_g$ only
model, depending on electronic filling. We consider all electron fillings of
the $d$-shell and compute the total magnetic moment, the spin, the occupancy of
each orbital, and the effective spin-orbit coupling strength (renormalized
through interaction effects) in terms of the bare interaction parameters,
spin-orbit coupling, and crystal field splitting, focusing on the parameter
ranges relevant to 4d and 5d transition metal oxides. In various limits we
provide perturbative results consistent with our numerical calculations. We
find that the $t_{2g}$-$e_g$ mixing can be large, with up to 20\% occupation of
orbitals that are nominally "empty", which has experimental implications for
the interpretation of the branching ratio in experiments, and can impact the
effective local moment Hamiltonian used to study magnetic phases and magnetic
excitations in transition metal oxides. Our results can aid the theoretical
interpretation of experiments on these materials, which often fall in a regime
of intermediate coupling with respect to electron-electron interactions.
