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Breaking the theoretical scaling limit for predicting quasi-particle energies: The stochastic GW approach

We develop a formalism to calculate the quasi-particle energy within the GW many-body perturbation correction to the density functional theory (DFT). The occupied and virtual orbitals of the Kohn-Sham (KS) Hamiltonian are replaced by stochastic orbitals used to evaluate the Green function, the polarization potential, and thereby the GW self-energy. The stochastic GW (sGW) relies on novel theoretical concepts such as stochastic time-dependent Hartree propagation, stochastic matrix compression and spatial/temporal stochastic decoupling techniques. Beyond the theoretical interest, the formalism enables linear scaling GW calculations breaking the theoretical scaling limit for GW as well as circumventing the need for energy cutoff approximations. We illustrate the method for silicon nanocrystals of varying sizes with over 3000 electrons.