All-electron KKR Calculations for Metallic Systems with Thousands of Atoms Per Cell via Sparse Matrix Iterative Solvers

To perform electronic-structure calculations for inherently large systems, such as a quantum dots or interfaces like domain walls, we must perform the calculations over very large unit cells (10$^{4}$ to 10$^{8}$ atoms). For the inverse Green's function G$^{-1}$, KKR methods typically solve for G by direct inversion. Using a screened, k-space hybrid KKR, we solve Dyson's equation for the Green's function using a reference state via G = G$_{ref}$ [ I - (t - t$_{ref}$) G$_{ref}$]$^{-1}$, scattering matrices t and t$_{ref}$ are known and the non-Hermitian tensor G$_{ref}$ is chosen for convenience and sparsity [1]. The approach is O(N) for bandgap materials, whereas it is O(N$^2$) for metals but with a potentially large prefactor. Based upon Sparse Approximate Inverse (or SPAI) technique [2], we generalize the algorithm for complex, non-Hermitian matrices, then use the method as a preconditioner for the inversion to reduce the iteration counts (hence, reduce the prefactor) of the iterative Krylov-space inverses, such as TFQMR, to address large-scale metallic systems. Parallel iterative and energy contour solves are made also. We explore the numerical efficiency and scaling versus atoms per unit cells. \newline [1] Smirnov and Johnson, Comp. Phys. Comm. 148, 74-80 (2002). \newline [2] Grote and Huckle, SIAM J. Sci. Comput. 18, 8