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We develop a stochastic approach to time-dependent DFT with optimally-tunedrange-separated hybrids containing non-local exchange, for calculating opticalspectra. The attractive electron-hole interaction, which leads to the formationof excitons, is included through a time-dependent linear-response techniquewith a non-local exchange interaction which is computed very efficientlythrough a stochastic scheme. The method is inexpensive and scales quadraticallywith the number of electrons, at almost the same (low) cost of time dependentKohn-Sham (TDKS) with local functionals. Our results are in excellent agreementwith experimental data and the efficiency of the approach is demonstrated onlarge finite phosphorene sheets containing up to 1958 valence electrons.

Stochastic Time-Dependent DFT with Optimally Tuned Range-Separated Hybrids: Application to Excitonic Effects in Large Phosphorene Sheets