Real-space density functional theory at large length and time scales

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Xu, Qimen
Suryanarayana, Phanish
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Over the course of the past few decades, quantum mechanical calculations based on Kohn-Sham density functional theory (DFT) have become a cornerstone of materials research by virtue of the predictive power and fundamental insights they provide. However, while less expensive than wavefunction based methods, the solution of the Kohn-Sham equations remains a formidable task. In particular, the computational cost scales cubically with the number of atoms, severely limiting the range of physical systems accessible to such first principles investigation. The planewave pseudopotential method has been among the most widely used techniques for solving the Kohn-Sham problem. The underlying Fourier basis is complete, orthonormal, diagonalizes the Laplacian, and provides spectral convergence for smooth problems. However, the Fourier basis restricts the method to periodic boundary conditions, whereby finite systems, as well as semi-infinite systems, require the introduction of artificial periodicity with large vacuum regions. Moreover, the global nature of the Fourier basis hampers scalability on parallel computing platforms, limiting the system sizes and time scales relevant to phenomena of interest. This dissertation has the following main contributions. (i) We present an open-source software package for the accurate, efficient, and scalable solution of the Kohn-Sham equations using real-space methods, referred to as SPARC. The package is straightforward to install/use and highly competitive with state-of-the-art planewave codes, demonstrating comparable performance on a small number of processors and order-of-magnitude advantages as the number of processors increases. (ii) We have developed a discrete discontinuous basis projection (DDBP) method to accelerate real-space electronic structure methods several fold, without loss of accuracy, by systematically reducing the dimension of the discrete eigenproblem that must be solved, via projection in a highly efficient discontinuous basis. In calculations of quasi-1D, quasi-2D, and bulk metallic systems, we find that accurate energies and forces are obtained with 8–25 projection basis functions per atom, reducing the dimension of full-matrix eigenproblems by 1–3 orders of magnitude. (iii) We present the SQ3 method, a density matrix based method for Kohn-Sham calculations at high temperature that eliminates the need for diagonalization, thus reducing the cost of such calculations significantly relative to conventional diagonalization based approaches. Upon implementation of the method in the SPARC code, we found systematic convergence to exact diagonalization results and significant speedups relative to conventional diagonalization based methods of up to ∼2x, with increasing advantages as the temperature and/or number of processors is increased.
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