Plane wave methods for quantum eigenvalue problems of incommensurate systems

We propose a novel numerical algorithm for computing the electronic structure related eigenvalue problems of incommensurate systems. Unlike the conventional practice that approximates the system by a large commensurate supercell, our algorithm directly discretizes the eigenvalue problems under the framework of a plane wave method. The emerging ergodicity and the interpretation from higher dimensions give rise to many unique features compared to what we have been familiar with in the periodic systems. The numerical results of 1D and 2D quantum eigenvalue problems are presented to show the reliability and efficiency of our algorithm. Furthermore, the extension of our algorithm to full Kohn-Sham density functional theory calculations is discussed.