It is well known that realistic treatment of material defects requires at least tens of thousands of atoms in the simulation box, which is often beyond the capability of the current ab initio method within the framework of Kohn-Sham density functional theory. Despite that alternative methods ranging from simple extrapolation method to quasi-continuum method exist, it is nevertheless desirable to have the capability of direct and accurate modeling of large scale disordered systems, including material defects using ab initio calculation. The standard method for solving KSDFT requires solving N eigenvectors for an O(N) * O(N) Kohn-Sham Hamiltonian matrix, with N being the number of electrons in the system. The computational cost for such procedure is expensive and scales as O(N^3). We have developed pole expansion and selected inversion (PEXSI) method, in which KSDFT is solved by evaluating the selected elements of the inverse of a series of sparse symmetric matrices, and the overall algorithm scales at most O(N^2) for all materials including metallic and insulating systems. The electron density, total energy, Helmholtz free energy and atomic force calculated simultaneously and accurately. Combined with atomic orbital basis functions, we show that the new method can already be applied to study the electronic structure of boron nitride nanotube and carbon nanotube with more than 10,000 atoms (coarse basis for one SCF step), and to perform geometry optimization with 1,024 atoms (dense basis with many SCF steps) on a single processor.
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