Efficient and accurate linear algebraic methods for large-scale electronic structure calculations with nonorthogonal atomic orbitals

The need for large-scale electronic structure calculations arises recently in the field of material physics, and efficient and accurate algebraic methods for large simultaneous linear equations become greatly important. We investigate the generalized shifted conjugate orthogonal conjugate gradient method, the generalized Lanczos method, and the generalized Arnoldi method. They are the solver methods of large simultaneous linear equations of the one-electron Schrödinger equation and map the whole Hilbert space to a small subspace called the Krylov subspace. These methods are applied to systems of fcc Au with the NRL tight-binding Hamiltonian [F. Kirchhoff et al., Phys. Rev. B 63, 195101 (2001)]. We compare results by these methods and the exact calculation and show them to be equally accurate. The system size dependence of the CPU time is also discussed. The generalized Lanczos method and the generalized Arnoldi method are the most suitable for the large-scale molecular dynamics simulations from the viewpoint of CPU time and memory size.

Figure 1

Partial density of states for a system of Au 864 atoms by the NRL tight-binding Hamiltonian (Ref. 24) normalized to unity. (a) $s$ orbitals, (b) $p$ orbitals, and (c) $d$ orbitals. Comparison is for the GsCOCG and the exact calculation, which are almost identical to each other. Parameters in GsCOCG calculations are $\eta ={10}^{-3}$ Ry, $\tau =5\eta$. The energy interval of mesh points is ${10}^{-4}$ Ry. See Figs. 4, 5, and 7 for comparison.

Figure 2

Convergence behavior of residual norms for a system of Au 256 atoms by the NRL tight-binding Hamiltonian (Ref. 24). The spectrum extends between $-0.5$ and $1.5$ Ry. The inset in (b) shows the total density of states $D(\varepsilon )={\sum }_i{D}_{\mathit{ii}}(\varepsilon )$, where we use a finite imaginary number in the energy and the profile is of dense spiky peaks. (a) Residual norm $\parallel {\mathit{r}}_n^{(s,j)}\parallel$ at several energy points $\varepsilon =-0.5,0.0,0.5,\text{and}1.0$ Ry for the $s$ orbital. (b) Average residual norms $R_n^{(j)}$ with different three seed energies ($-0.5,0.0,\text{and}\hspace{0.5em}0.5$ Ry) for the $s$ orbital and they all overlap with each other.

Figure 3

Residual norm at the seed energies for a system of Au 864 atoms by the NRL tight-binding Hamiltonian (Ref. 24) in (a) $s$ orbitals and (b) $d$ orbitals. (a) Seed switches to $0.522$ Ry from $-0.5$ Ry at the 36th step and to $-0.299$ Ry at the 376th step in the $s$-orbital case. (b) Seed switches to $0.110$ Ry from $-0.5$ Ry at the 28th step and to $-0.034$ Ry at the 1002nd step in the $d$-orbital case. Once the calculation using one seed is converged and full convergence has not been achieved, one should choose the next seed and continue the calculation. The gray lines show the residual norm by energy shift ${\mathit{r}}_n^{\text{'}\sigma }$ before seed switching.

Figure 4

Partial density of states (pDOS), normalized to unity, and integrated density of states (IDOS) for a system of Au 864 atoms by the NRL tight-binding Hamiltonian (Ref. 24). Comparison is between those of the generalized Lanczos method (red solid lines) and those of the exact ones (blue solid lines). (a-1) and (a-2): pDOS and IDOS for $s$ orbitals. (b-1) and (b-2): pDOS and IDOS for $p$ orbitals. (c-1) and (c-2): pDOS and IDOS for $d$ orbitals. $\eta =5.0\times {10}^{-3}$ Ry.

Figure 5

Partial density of states, normalized to unity, and integrated density of states. Comparison is between those of the generalized Arnoldi method (red solid lines) and those of the exact ones (blue solid lines) for a system of Au 864 atoms by the NRL tight-binding Hamiltonian (Ref. 24). (a-1) and (a-2): pDOS and IDOS for $s$ orbitals. (b-1) and (b-2): pDOS and IDOS for $p$ orbitals. (c-1) and (c-2): pDOS and IDOS for $d$ orbitals. $\eta =5.0\times {10}^{-3}$ Ry.

Figure 6

Comparison of IDOS of $d$ orbitals for systems of Au 256 and 864 atoms by the NRL tight-binding Hamiltonian (Ref. 24). (a) The G-Lanczos (solid line) and exact calculation (chain line) for 864 atoms (red) and 256 (blue) atoms. (b) The G-Arnoldi (solid line) and exact calculation (chain line) for 864 atoms (red) and 256 (blue) atoms. The agreement between the results by the G-Lanczos or G-Arnoldi methods and those by the exact calculation is excellent.

Figure 7

Partial density of states and integrated ones without the modified Gram-Schmidt reorthogonalization of the generalized Lanczos method (red solid lines) and those by the exact ones (blue solid lines) for a system of Au 256 atoms by the NRL tight-binding Hamiltonian (Ref. 24). $N=50$. (a-1) and (a-2): pDOS and IDOS for $s$ orbitals. (b-1) and (b-2): pDOS and IDOS for $p$ orbitals. (c-1) and (c-2): pDOS and IDOS for $d$ orbitals. One should note that the normalization of the integrated density of states is broken without reorthogonalization procedure and that some “ghost” peaks appear due to incorrect mixing of states. $\eta =5.0\times {10}^{-3}$ Ry.