Linear algebraic calculation of the Green’s function for large-scale electronic structure theory

A linear algebraic method named the shifted conjugate-orthogonal conjugate-gradient method is introduced for large-scale electronic structure calculation. The method gives an iterative solver algorithm of the Green’s function and the density matrix without calculating eigenstates. The problem is reduced to independent linear equations at many energy points and the calculation is actually carried out only for a single energy point. The method is robust against the round-off error and the calculation can reach the machine accuracy. With the observation of residual vectors, the accuracy can be controlled, microscopically, independently for each element of the Green’s function, and dynamically, at each step in dynamical simulations. The method is applied to both a semiconductor and a metal.

Figure 1

Decay behavior of the ARN for a Si crystal with 512 atoms. A universal curve appears with four different reference energy points that are placed at the band bottom (B), in the valence band (V), in the gap (G), and in the conduction band (C). See Fig. 6a for the actual positions of these energy points.

Figure 2

Early stage in the decay behavior of the ARN for a Si crystal with 512 atoms and fcc Cu with 1568 atoms. In the Cu system, the two cases are plotted with the starting vectors $\mid j⟩$ of $s$ and $t_{2_g}$ bases on an atom.

Figure 3

Result of a slab for the Si(001) surface with asymmetric surface dimers. (a) Local density of states $[{\sum }_j-(1∕\pi )\mathrm{Im}{G}_{jj}]$ for surface and bulk atoms, calculated with the KS dimension of $n=30$. The surface atoms are classified into the upper and lower atoms of the asymmetric dimer, illustrated in the inset. The “bulk” atom is an atom on deeper layers of the present slab system. (b) Decay behavior of ARN for the atoms that appear in (a). The horizontal dotted line is a guide for the eye for a typical convergence criterion.

Figure 4

Result for fcc Cu with 1568 atoms: Partial density of states $-(1∕\pi )\mathrm{Im}{G}_{jj}$ for each orbital.

Figure 5

Result for fcc Cu with 1568 atoms: COHP and ICOHP for a first-nearest-neighbor atom pair (a) and PCOHP for a first- (b) or second- (c) nearest-neighbor atom pair.

Figure 6

(a) Well-converged Green’s function and (b) convergence behavior at different energy points. The calculation was carried out with 512 Si atoms and the top of the valence band is located at $E=0\mathrm{eV}$ (a) imaginary part of the well-converged Green’s function $[-(1∕\pi )\mathrm{Im}{G}_{jj}(E+i\gamma )]$ with $\gamma =0.002\mathrm{a.u.}$ (b) The iteration numbers to satisfy the converge criterion ${\parallel {\mathit{r}}_n^{\left(j\right)}\parallel }^2<{10}^{\epsilon }$ with $\epsilon =-4$ to $-16$. Note that four energy points are picked out, at the band bottom (B), in the valence band (V), in the gap (G), and in the conduction band (C). For discussion, see text.

Figure 7

Iteration dependence of (a) the residual norm (RN) in the present method and (b) the total residual norm (TRN). The symbols B, V, G, and C indicate the energy points that are shown by arrows in Fig. 6a.

Figure 8

Decay behavior of the ARN in the SDKS method for Si crystals with 512, 4096, and 32 768 atoms.

Figure 9

Energy dependence of the RN in the subspace diagonalization method for a Si crystal with 512 atoms. The KS dimension is $\upsilon =\left(\mathrm{a}\right)$ 30 and (b) 100.