Linear-scaling subspace-iteration algorithm with optimally localized nonorthogonal wave functions for Kohn-Sham density functional theory

We present a linear-scaling method for electronic structure computations in the context of Kohn-Sham density functional theory (DFT). The method is based on a subspace iteration, and takes advantage of the nonorthogonal formulation of the Kohn-Sham functional, and the improved localization properties of nonorthogonal wave functions. A one-dimensional linear problem is presented as a benchmark for the analysis of linear-scaling algorithms for Kohn-Sham DFT. Using this one-dimensional model, we study the convergence properties of the localized subspace-iteration algorithm presented. We demonstrate the efficiency of the algorithm for practical applications by performing fully three-dimensional computations of the electronic density of alkane chains.

Figure 1

(Color online) Chebyshev polynomial of degree 10, depicting its fast growth outside the interval $\left[-1,1\right]$.

Figure 2

(Color online) Band structure for different values of $\sigma$ and $a$. Choosing $a$ and $\sigma$ appropriately we can control the size of the energy-band gap. In figures (a) $a=1000;\sigma =0.15$, (b) $a=100;\sigma =0.3$, (c) $a=10;\sigma =0.3$, we show insulator systems with different size gaps. The gap in the system shown in figure (d) ($a=10;\sigma =0.45$) is almost zero, closer to what occurs in a metal.

Figure 3

(Color online) Band gap as function of $a$ and $\sigma$. (a) Fixed $\sigma$, $a$ varies from 0 to $10000$; the gap scales like $\sqrt{a}$. (b) Fixed $a$, $\sigma$ varies from 0.05 to 0.4; the gap scales like $1/\sigma$.

Figure 4

(Color online) Trajectory of the energy in the LSI iteration. (a) Logarithmic plot of the error in the energy for each step in the LSI iteration $\left(a=50,\sigma =0.3,R_c=2.0\right)$. (b) Logarithmic plot of the error in the energy in the LSI iteration $\left(a=50,\sigma =0.3,R_c=2.0\right)$; for each step, the errors before and after the truncation of the wave functions are both shown. (c) Remanent fluctuation in the energy $\left(a=50,\sigma =0.3,R_c=2.0\right)$. (d) Remanent fluctuation in the energy $\left(a=10,\sigma =0.4,R_c=2.0\right)$.

Figure 5

(Color online) The effect of the cutoff radius on the accuracy for the insulator case. (a) Logarithmic plot of the error in the energy for different cutoff radii. (b) The resulting wave functions for cutoff radii 0.2, 0.6, and 1.0.

Figure 6

(Color online) The effect of the cutoff radius on the accuracy for the metallic case. (a) Logarithmic plot of the error in the energy for different cutoff radii. (b) The resulting wave functions for cutoff radii 3.0, 6.0, and 9.0.

Figure 7

(Color online) Electron density of an alkane chain obtained with the LSI algorithm. (a) Atomic configuration, obtained using the package GAUSSIAN (Ref. 39). (b) Electron density.

Figure 8

(Color online) Timings obtained with the LSI code. Linear scaling is observed. (a) Localization, (b) Chebyshev filter; $n=10$, (c) Fermi energy, (d) inverse of the Gram matrix.