### Examples: Electronic Structure

#### PEXSI: POLE Expansion and Selected Inversion:

#### Pole expansion and selected inversion method (PEXSI) for accelerating the electronic structure calculation. The sequential version of PEXSI has been applied to Hamiltonian matrices discretized by atomic orbitals, and can perform first principle KSDFT calculation for a large carbon nanotube with more than 10,000 atoms on a single processor using single-zeta basis function. In order to fully reveal the capability of the PEXSI method and accelerate the electronic structure calculation for large scale systems in practice, the massively parallel PEXSI software is under active development and is being integrated into the electron structure software SIESTA. The massively parallel PEXSI code can efficiently use more than 10,000 cores simultaneously to calculate large scale electronic structure.

1. 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]

#### Adaptive Local Basis Representations:

In KSDFT, uniform discretization of the Kohn-Sham Hamiltonian generally results in a large number (500~5000 or more) of basis functions per atom in order to resolve the rapid oscillations of the Kohn-Sham orbitals around the nuclei. The adaptive local basis functions are constructed by solving Kohn-Sham problems locally in the real space. This approach automatically and systematically builds the rapid oscillations of the Kohn-Sham orbitals around the nuclei as well as environmental effects into the basis functions. The resulting basis functions are localized in the real space, and are discontinuous in the global domain. The continuous Kohn-Sham orbitals and the electron density are evaluated from the discontinuous basis functions using the discontinuous Galerkin (DG) framework. Numerical examples indicate that the adaptive local basis functions can reach very high accuracy (in the order of 1meV) with a very small number (4~40) of basis functions per atom for metallic and insulating systems. The construction of the localized basis set with a small number of basis functions per atom is a crucial step in ensuring that our new method has a small preconstant.

1. L. Lin, J. Lu, L. Ying and W. E, Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation, J. Comput. Phys. 231, 2140, 2012

2. L. Lin, L. Ying, Element orbitals for Kohn-Sham density functional theory, Phys. Rev. B 85, 235144, 2012

3. L. Lin, J. Lu, L. Ying and W. E, Optimized local basis function for Kohn-Sham density functional theory, J. Comput. Phys 231, 4515, 2012

#### Self-Consistent Field Iterations:

Another challenge for large scale electronic structure calculation for metallic systems is to devise self-consistent field iterations (SCF). The number of iterations can increase rapidly with the system size and this can make large scale electronic structure calculations prohibitively expensive. We developed an efficient "elliptic" preconditioner for accelerating the SCF iteration for solving the KSDFT for large scale inhomogeneous metallic systems. Our elliptic preconditioner solves an elliptic equation, which can be derived from a piecewise constant approximation to the polarizability matrix of the system. Such an elliptic equation can be efficiently solved with O(N) cost using techniques such as multigrid method or hierarchical matrix method. The elliptic preconditioner is effective for accelerating the SCF iterations for 1D systems with mixed metal and insulator component, as well as for 3D metallic systems with a large portion of vacuum area, for which standard SCF iteration techniques typically take a large number of steps to converge.

1. L. Lin and C. Yang, Elliptic preconditioner for accelerating the self consistent field iteration in Kohn-Sham density functional theory, SIAM J. Sci. Comput. 35, S277–S298, 2013 (Copper Mountain Special Issue)

#### Fast Evaluation Methods:

We present a numerical integration scheme for evaluating the convolution of a Green's function with a screened Coulomb potential on the real axis in the GW approximation of the self energy. Our scheme takes the zero broadening limit in Green's function first, replaces the numerator of the integrand with a piecewise polynomial approximation, and performs principal value integration on subintervals analytically. We give the error bound of our numerical integration scheme and show by numerical examples that it is more reliable and accurate than the standard quadrature rules such as the composite trapezoidal rule.

1. Fang Liu, Lin Lin, Derek Vigil-Fowler, Johannes Lischner, Alexander F. Kemper, Sahar Sharifzadeh, Felipe Homrich da Jornada, Jack Deslippe, Chao Yang, Jeffrey B. Neaton and Steven G. Louie, Numerical integration for ab initio many-electron self energy calculations within the GW approximation, submitted, http://arxiv.org/abs/1402.5433