Stochastic time-dependent DFT with optimally tuned range-separated hybrids: Application to excitonic effects in large phosphorene sheets

We develop a stochastic approach to time-dependent density functional theory with optimally tuned range-separated hybrids containing nonlocal exchange, for calculating optical spectra. The attractive electron-hole interaction, which leads to the formation of excitons, is included through a time-dependent linear-response technique with a nonlocal exchange interaction which is computed very efficiently through a stochastic scheme. The method is inexpensive and scales quadratically with the number of electrons, at almost the same (low) cost of time dependent Kohn-Sham with local functionals. Our results are in excellent agreement with experimental data, and the efficiency of the approach is demonstrated on large finite phosphorene sheets containing up to 1958 valence electrons.