The use of electronic structure calculations in understanding nano-systems is a natural area for high-quality computational modeling and simulation. This paper provides a fundamental study of how to select appropriate numerical methods for use in such applications. Vömel et al. consider current state-of-the-art eigensolvers that are used in electronic structure calculations and apply them to several test problems selected for modeling nano-systems.

The paper begins with a brief introduction to Kohn-Sham theory and self-consistent field iteration for the ground state. For nano-systems, the optical and electronic properties can be determined from interior eigenstates. The paper uses matrix-free computation, and considers valence band maximum (VBM) states that are highest-occupied states and conduction band minimum (CBM) states that are lowest-unoccupied states.

The paper provides an introduction to iterative Hermitian eigensolvers and considers five methods: preconditioned conjugate gradients (PCG), locally optimal block preconditioned conjugate gradients (LOBPCG), implicit restart Arnoldi/Lanczos (IRL), generalized Davidson + k (GD+k) (where k is the number of vectors used from the previous iteration), and the Jacobi-Davidson with quasi-minimal residual (QMR) method (JDQMR) as an inner solver.

These methods are explained in detail, and three test configurations for quantum dots and wires are described in Section 4. Detailed results and interpretation are given in Section 5, and a summary is provided in Section 6. These results lead to the recommendation of the PCG and the GD+k methods. However, GD+k is found to be two to three times faster than PCG. Additionally, the paper concludes with the encouraging comment that the authors have developed a bulk-based preconditioner for PCG for use in band edge calculations, and propose applying it to GD+k in future work. It will be interesting to see these results in the future.