Low scaling algorithms for the random phase and $GW$ approximation*

The computationally most expensive step in conventional RPA implementations is the calculation of the independent particle polarizability $\chi_0$. We present an algorithm that calculates $\chi_0$ using the Green's function in real space and imaginary time. In combination with optimized non-uniform frequency and time grids the correlation energy on the random phase approximation level can be calculated efficiently with a computational cost that grows only cubically with system size [1,2]. We apply this approach to calculate RPA defect energies of silicon using unit cells with up to $250$ atoms and $128$ CPU cores. Furthermore, we show how to extent the algorithm to the $GW$ framework of Hedin and solve the Dyson equation for the Green's function with the same computational effort. \\[4pt] [1] M. Kaltak, J. Klime\v{s}, and G. Kresse, Journal of Chemical Theory and Computation 10, 2498-2507 (2014). \newline [2] M. Kaltak, J. Klime\v{s}, and G. Kresse, Phys. Rev. B 90, 054115 (2014).