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The energy splitting of the rovibronic levels of the $\mathrm{<mrow\; data-mjx-texclass="ORD">3</mrow>}II$ state of a linear triatomic molecule has been studied theoretically. These splittings are due to vibronic interaction (Renner effect), rotational interaction, and fine structure interaction. The latter includes electron spin-orbit-interaction electron spin-other-orbit interaction and electron spin-spin interaction. The operator of electron spin-spin interaction (also that of nuclear electric quadrupole interaction which causes hypertine structure splittings) is a second order tensor that can mix states of $\Delta A=\pm 2$ and $\Delta l=0$. Such mixing, which differs from those due to other interactions, contributes unique features to the energy splittings. In this study, the vibronic interaction is assumed to be larger than the rotational interaction which is in turn larger than the fine structure interaction. The rovibronic wave functions for Hund’s case “b” coupling are therefore used as the basis set and the fine structure interactions are considered as perturbation. As a result of the perturbation treatment, the energy of a fine structure level, J. $Mj$, is expressed explicitly in terms of all the quantum numbers and the coupling constants. This energy expression, (though not as rigorous as that obtained through matrix diagonalization) will allow direct comparison with the precision experimental results and thus will lead to determination of the coupling constants experimentally.