We develop a formalism to calculate the quasiparticle energy within the $GW$ many-body perturbation correction to the density functional theory. The occupied and virtual orbitals of the Kohn-Sham Hamiltonian are replaced by stochastic orbitals used to evaluate the Green function $G$, the polarization potential $W$, and, thereby, the $GW$ self-energy. The stochastic $GW$ ($\mathrm{s}GW$) formalism relies on novel theoretical concepts such as stochastic time-dependent Hartree propagation, stochastic matrix compression, and spatial or 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 $N_e>3000$ electrons.

• Received 19 February 2014

DOI:https://doi.org/10.1103/PhysRevLett.113.076402