Cubic scaling<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:math>: Towards fast quasiparticle calculations

The $G\phantom{\rule{0}{0ex}}W$ method is the standard method for predicting quasiparticle energies as measured in experimental photoemission spectroscopy. Until recently, routine $G\phantom{\rule{0}{0ex}}W$ calculations have been restricted to fairly small systems and few $k$ points for sampling the Brillouin zone, since the computational demand usually scales quartic with the number of atoms and quadratic with the number of $k$ points. The original work of Lars Hedin suggests that one can do better by calculating the self-energy as $\mathrm{\ensuremath{\Sigma}}$(1,2)=$i\phantom{\rule{0}{0ex}}G$(1,2)$W$(1,2) -- the equation that coined the phrase ''$G\phantom{\rule{0}{0ex}}W$''. Here, $G$ is the one-particle Green's function and $W$ the screened interaction between the electrons. Using this relation directly in computations brings the scaling down to linear in the number of $k$ points and cubic in the number of atoms. Although this approach has been used in the past by others, in this work its full potential is unveiled by implementing it in a massively parallel code and adopting it to the projector augmented wave method. This makes $G\phantom{\rule{0}{0ex}}W$ calculations as convenient as calculations based on local density functionals and allows for calculations for hundred atoms in few hours.