Expeditious stochastic calculation of random-phase approximation energies for thousands of electrons in three 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. For a fixed statistical error in the total energy per electron, the method scales, at most, quadratically with the system size; however, in practice, due to self-averaging, it 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. We find that the RPA correlation energies per electron are largely independent of the nanocrystal size. In addition, we show that a correlated sampling technique enables calculation of the energy difference between two slightly distorted configurations with scaling and a statistical error similar to that of the total energy per electron.