Random Green’s Function Method for Large-Scale Electronic Structure Calculation

We report a linear-scaling random Green’s function (rGF) method for large-scale electronic structure calculation. In this method, the rGF is defined on a set of random states and is efficiently calculated by projecting onto Krylov subspace. With the rGF method, the Fermi–Dirac operator can be obtained directly, avoiding the polynomial expansion to Fermi–Dirac function. To demonstrate the applicability, we implement the rGF method with the density-functional tight-binding method. It is shown that the Krylov subspace can maintain at small size for materials with different gaps at zero temperature, including H₂O and Si clusters. We find with a simple deflation technique that the rGF self-consistent calculation of H₂O clusters at T = 0 K can reach an error of ∼ 1 meV per H₂O molecule in total energy, compared to deterministic calculations. The rGF method provides an effective stochastic method for large-scale electronic structure simulation.