On spectral quadrature for linear-scaling Density Functional Theory

We provide a unified description of the Fermi operator expansion and recursion methods within the technique of spectral quadrature. Through rigorous error estimates, we prove that this approach is linear-scaling, stable and exponentially convergent. We use this analysis to determine the influence of smearing, band-gap, position of Fermi energy, and spectral width of the Hamiltonian on the convergence rates obtained in practical calculations. Additionally, we establish that super-geometric convergence can be achieved when the erfc function is used for smearing. We validate the spectral quadrature method and the accuracy of our analysis by means of selected examples.