Predicting the band structure and optical absorption spectra of narrow band gap semiconductors is challenging for electronic structure methods. Here, it is shown shown that density functional theory can yield accurate band structures and time-dependent density functional theory (TDDFT) can yield accurate optical absorption spectra for these systems. This is achieved by using a screened range-separated hybrid (SRSH) functional with a single empirical parameter, fit to reproduce the experimental band gap. By comparing TDDFT results based on the SRSH approach with those obtained based on the Heyd-Scuseria-Ernzerhof functional it is shown that screened long-range exact exchange improves the accuracy of the TDDFT spectra for these systems. The optical absorption spectra of narrow band gap semiconductors (i.e., those with a band gap smaller than ≈ 0.8 eV) are often challenging to predict. A standard technique to calculate optical absorption spectra is to use many-body perturbation theory with the GW approximation [1-6] and the Bethe-Salpeter equation (BSE) [7-10] approach. While the GW-BSE approach is known for its accuracy, it can exhibit sensitivity to the density functional theory (DFT) starting point. For narrow band gap semiconductors, a challenge is that semilocal functionals, often used to generate a starting point for GW calculations, spuriously predict a metallic ground state [11-14] and even use of a self-consistent scheme does not result in sufficiently accurate band gaps. [15-18]