Comparison of the convergence properties of linear-scaling electronic-structure schemes for nonorthogonal bases

This paper presents a detailed comparison of the convergence properties of density-matrix and localized-orbital $O(N)$ functionals within 512-atom cells of amorphous carbon using a first-principles local-orbital Hamiltonian. The functionals were minimized by means of the conventional but tensorially incorrect covariant derivatives as well as the correct contravariant derivatives. While the correct derivatives result in a much faster minimization, the energies obtained in this case are somewhat higher compared to using the covariant derivatives. However, we present a representation of the density-matrix functional which requires shorter minimization times and yet returns more accurate energies for practical sizes of the localization regions. Furthermore, while the density-matrix functional is superior in efficiency to the orbital-based functional when using the incorrect derivatives, both functionals exhibit similar decay properties in terms of conjugate-gradient iterations for the correct derivatives. This makes the orbital-based functional faster, especially when minimal sets of Wannier-like functions and projected initial functions can be used.