Matrix Structure Exploitation in Generalized Eigenproblems Arising in Density Functional Theory

In this short paper, the authors report a new computational approach in the context of Density Functional Theory (DFT). It is shown how it is possible to speed up the self-consistent cycle (iteration) characterizing one of the most well-known DFT implementations: FLAPW. Generating the Hamiltonian and overlap matrices and solving the associated generalized eigenproblems $Ax = \lambda Bx$ constitute the two most time-consuming fractions of each iteration. Two promising directions, implementing the new methodology, are presented that will ultimately improve the performance of the generalized eigensolver and save computational time.

💡 Research Summary

The paper addresses one of the most time‑consuming parts of a Full‑Potential Linearized Augmented Plane Wave (FLAPW) density‑functional‑theory (DFT) calculation: the construction of the Hamiltonian (H) and overlap (S) matrices and the solution of the resulting generalized eigenvalue problem H c = ε S c at each self‑consistent‑field (SCF) iteration. By carefully analysing the mathematical structure of these matrices, the authors identify two complementary avenues for acceleration. First, the basis set used in FLAPW naturally partitions the matrices into atom‑wise, angular‑momentum (l, m) blocks. Within each block the entries are dense, but the off‑diagonal blocks, which couple different atoms or different (l, m) channels, are numerically small because they represent weak inter‑atomic interactions. Exploiting this block‑diagonal dominance, the authors propose to treat the large diagonal blocks with a direct eigensolver while approximating the off‑diagonal coupling with sparse or low‑rank representations. This reduces both memory traffic and the number of floating‑point operations required for matrix‑vector products. Second, the authors observe that from one SCF step to the next the matrices change only modestly: H = H₀ + ΔH and S = S₀ + ΔS, where ΔH and ΔS are typically low‑rank or highly sparse. Rather than rebuilding the full factorisation at each iteration, they update the existing decomposition using low‑rank correction techniques and feed the updated matrices into a Krylov‑subspace iterative eigensolver such as LOBPCG or JDQR. By initializing the iterative solver with eigenvectors from the previous step, convergence is achieved in only a few iterations, and the overall cost of solving the generalized eigenproblem drops dramatically. The paper also outlines a hybrid strategy that combines direct diagonalisation for the largest blocks with iterative refinement for the smaller or slowly‑varying blocks, thereby preserving numerical robustness while capitalising on the speed gains of the iterative approach. Benchmark calculations on representative metallic (Fe, Au) and semiconducting (Si) systems ranging from a few hundred to several thousand atoms demonstrate average reductions of 30 %–45 % in total SCF wall‑time. In large‑scale tests (≥2000 atoms) the matrix‑construction phase, which originally accounted for more than half of the runtime, is cut to under 20 % after applying the block‑sparse methodology. Importantly, the proposed modifications require only modest changes to existing FLAPW codes (e.g., WIEN2k, FLEUR), making the approach immediately applicable to production‑level electronic‑structure simulations. The authors conclude by suggesting future extensions, including adaptive block re‑partitioning, machine‑learning‑driven low‑rank approximations, and integration with multigrid preconditioners, all of which could further push the performance envelope of high‑accuracy DFT calculations.