Expeditious Stochastic Calculation of Random-Phase Approximation Energies for Thousands of Electrons in 3 Dimensions

A fast method is developed for calculating the Random-Phase-Approximation (RPA) correlation energy for density functional theory. The correlation energy is given by a trace over a projected RPA response matrix and the trace is taken by a stochastic approach using random perturbation vectors. The method scales, at most, quadratically with the system size but in practice, due to self-averaging, requires less statistical sampling as the system grows and the performance is close to linear scaling. We demonstrate the method by calculating the RPA correlation energy for cadmium selenide and silicon nanocrystals with over 1500 electrons. In contrast to 2nd order M{\o}ller-Plesset correlation energies, we find that the RPA correlation energies per electron are largely independent on the nanocrystal size.