# Fast algorithms for electronic structure analysis

# Fast algorithms for electronic structure analysis

Kohn-Sham density functional theory (KSDFT) is the most widely used electronic structure theory for molecules and condensed matter systems. For a system with N electrons, the standard method for solving KSDFT requires solving N eigenvectors for an O(N) * O(N) Kohn-Sham Hamiltonian matrix. The computational cost for such procedure is expensive and scales as O(N^3), and limits routine KSDFT calculations to hundreds of atoms. In recent years, we have developed an alternative procedure called the pole expansion and selected inversion (PEXSI) method [1-2]. The PEXSI method solves KSDFT without solving any eigenvalue and eigenvector, and directly evaluates physical quantities including electron density, energy, atomic force, density of states, and local density of states. The overall algorithm scales as at most O(N^2) for all materials including insulators, semiconductors and the difficult metallic systems. The PEXSI method can be efficiently parallelized over 10,000 - 100,000 processors on high performance machines. It has been integrated into standard electronic structure software packages such as SIESTA for ab initio materials simulation over 20,000 atoms [3]. Recently we have been able to use PEXSI to study electronic structure of large scale graphene nanoflakes [4] and phosphorene nanoribbons [5] to unprecedented scale (more than 10,000 atoms).[1] L. Lin, J. Lu, L. Ying, R. Car and W. E, Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems, Commun. Math. Sci. 7, 755, 2009

[2] L. Lin, M. Chen, C. Yang and L. He, Accelerating atomic orbital-based electronic structure calculation via pole Expansion and selected inversion, J. Phys. Condens. Matter 25, 295501, 2013

[3] L. Lin, A. Garcia, G. Huhs and C. Yang, SIESTA-PEXSI: Massively parallel method for efficient and accurate ab initio materials simulation without matrix diagonalization, J. Phys. Condens. Matter 26, 305503, 2014

[4] W. Hu, L. Lin, C. Yang and J. Yang, Electronic structure of large-scale graphene nanoflakes, J. Chem. Phys. 141, 214704, 2014

[5] W. Hu, L. Lin and C. Yang, Edge reconstruction in armchair phosphorene nanoribbons revealed by discontinuous Galerkin density functional theory, submitted