Quantum dielectric theory has been used to predict the direct E0 energy gaps of ternary and quaternary III–V semiconductor alloys containing up to ∼10 mol % InBi. The calculated composition dependence of E0 for InSb1-xBix is in good agreement with experimental results with a predicted semiconductor–semimetal transition (77 K) at x=0.124. E0 (77 K) values corresponding to infrared wavelengths of 8 to 12 μm are obtained at x=0.043–0.070, respectively, corresponding to lattice mismatches with InSb substrates in the range from 0.11% to 0.17%. The quaternary alloys InAs1-x-ySbxBiy and In1-xGaxSb1-yBiy also yield E0 values in this range with similar InBi contents while maintaining exact lattice match to InSb substrates. InAs1-xBix is proposed as a new alloy for spanning the wavelength range from 3 to 5 μm. The required x values range from 0 to 0.079 (measuring E0 at 77 K) and the maximum lattice mismatch with either InAs or GaSb substrates for these compositions is ≪0.31%. The related quaternary alloy In1-xGaxAs1-yBiy also spans the 3–5 μm wavelength range while maintaining lattice match with InAs or GaSb substrates.