Low-Scaling with Benchmark Accuracy and Application to Phosphorene Nanosheets.

is an accurate method for computing electron addition and removal energies of molecules and solids. In a conventional implementation, however, its computational cost is () in the system size , which prohibits its application to many systems of interest. We present a low-scaling algorithm with notably improved accuracy compared to our previous algorithm [ , , 306-312]. This is demonstrated for frontier orbitals using the 100 benchmark set, for which our algorithm yields a mean absolute deviation of only 6 meV with respect to canonical implementations. We show that also excitations of deep valence, semicore, and unbound states match conventional schemes within 0.1 eV. The high accuracy is achieved by using minimax grids with 30 grid points and the resolution of the identity with the truncated Coulomb metric. We apply the low-scaling algorithm with improved accuracy to phosphorene nanosheets of increasing size. We find that their fundamental gap is strongly size-dependent varying from 4.0 eV (1.8 nm  ×  1.3 nm, 88 atoms) to 2.4 eV (6.9 nm  ×  4.8 nm, 990 atoms) at the ev@PBE level.