The band structures of $\text{In}\text{ }X$ ($X=\text{P}$, As, and Sb) are calculated using the hybrid HSE06 functional and $GW$ with spin-orbit coupling effects included up to second order. Conventional local or semilocal density functionals predict an incorrect band ordering for InAs and InSb when spin-orbit coupling is included. We show that inclusion of one quarter of the exact exchange allows us to predict very accurate band gaps for InP, InAs, and InSb, i.e., 1.48, 0.42, and 0.28 eV, respectively, in good agreement with recent experiments. Furthermore, calculated effective masses for the conduction band (electron) and the valence band (heavy-hole, light-hole, and split-off band) are in fairly good agreement, whereas the values obtained using semilocal functionals deviate significantly from experiment even for InP. The calculated Luttinger parameters are also in reasonable agreement with experiment, although a tendency toward underestimation is observed with increasing anion mass. This underestimation is shown to be partially related to a tendency to overestimate the band gaps for the heavier elements using the particular hybrid functional used in this study. By adjusting the screening parameter in the hybrid functional, agreement with the experimental band gap can be achieved, but even then the Luttinger parameters are 15% smaller than the experimental values for InAs and InSb.
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