Within the framework of the full potential projector-augmented wavemethodology, we present a promising low-scaling $GW$ implementation. It allowsfor quasiparticle calculations with a scaling that is cubic in the system sizeand linear in the number of $k$ points used to sample the Brillouin zone. Thisis achieved by calculating the polarizability and self-energy in the real spaceand imaginary time domain. The transformation from the imaginary time to thefrequency domain is done by an efficient discrete Fourier transformation withonly a few nonuniform grid points. Fast Fourier transformations are used to gofrom real space to reciprocal space and vice versa. The analytic continuationfrom the imaginary to the real frequency axis is performed by exploitingThiele's reciprocal difference approach. Finally, the method is appliedsuccessfully to predict the quasiparticle energies and spectral functions oftypical semiconductors (Si, GaAs, SiC, and ZnO), insulators (C, BN, MgO, andLiF), and metals (Cu and SrVO$_3$). The results are compared with conventional$GW$ calculations. Good agreement is achieved, highlighting the strength of thepresent method.
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