Eigensolver literature assessment¶
Scope: extremal and interior Hermitian problems relevant to AO SCF, k-point SCF and Hermitian response/CI. Primary papers and authoritative implementation documentation were checked through 2026-10-05. This is a scoped assessment, not an exhaustive bibliography or a performance benchmark of external codes. Publication DOIs below were matched to publisher records. arXiv identifiers are explicitly preprints unless a journal publication is listed.
Foundational methods and workload implications¶
Let n be dimension, k requested roots, m retained search dimension and b block width. Memory estimates below exclude storage of the input operator. The cost comparisons are algorithmic assessments; actual crossovers must be measured with vibe-qc operators and occupations.
Method and primary source |
Iteration, storage and requirements |
Assessment for vibe-qc |
|---|---|---|
Davidson, 1975, 10.1016/0021-9991(75)90065-0 |
Expand with approximately preconditioned residuals; O(n m) vectors, projected dense solves; diagonal preconditioning exploits near-diagonal structure. |
Retain for a small number of extremal CI/TDA roots and recycled SCF subspaces. AO Fock need not be diagonally dominant after orthogonalization. A poor diagonal can stagnate. |
Sleijpen and Van der Vorst, 1996, 10.1137/S0895479894270427 |
Projected correction equation removes the current eigendirection. Inexact inner solves add operator applications but can handle a better local model; O(n m) outer storage. |
A candidate for difficult interior roots when shifted preconditioners exist. The local JD implementation must be validated against the full recurrence before promotion. |
Knyazev, 2001, 10.1137/S1064827500366124 |
Local Rayleigh-Ritz over current, preconditioned residual and previous blocks; O(n b) storage with small problems of order 3b. SPD preconditioning underpins the usual convergence theory. |
Useful for extremal Hermitian definite problems and clustered roots with block/locking safeguards. An indefinite inverse around an interior shift is not the same preconditioner contract. |
Stewart, 2002, 10.1137/S0895479800371529 |
Krylov-Schur restarts retain selected Schur information; O(n m) storage and operator applications per expansion. |
A better established restart architecture than the current unrestarted Python Lanczos. Interior solves still need harmonic extraction, a filter or shifted linear solves. |
Zuev et al., 2015, 10.1002/jcc.23800 |
Interior Davidson and GPLHR methods demonstrated on EOM-CC ionization/excitation workloads; targeted transitions also considered. |
Evidence concerns specific operators and preconditioners, including non-Hermitian EOM-CC. It does not establish universal SCF performance or justify calling every response operator Fock. |
Vecharynski, Yang and Xue, 2017, 10.1137/15M1027413, full preprint |
GPLHR uses block locally harmonic Schur iteration with a preconditioner for the shifted pencil and bounded residual/direction blocks; no exact shift inverse required. |
Full generalized non-Hermitian algorithm is materially different from the present real harmonic-Davidson variant. Completion would require new generalized Schur, projection and restart evidence, not a small patch. |
Polizzi, 2009, 10.1103/PhysRevB.79.115112; Tang and Polizzi, 2014, 10.1137/13090866X |
FEAST approximates a spectral projector with contour quadrature and repeated shifted linear solves, then Rayleigh-Ritz. Memory includes factorizations or inner solves and interval subspaces. Quadrature nodes/intervals admit parallel execution. |
Appropriate when many roots in an interval justify solve infrastructure. A matvec alone is insufficient for efficient application of the rational filter. Interval capacity, quadrature accuracy and completeness checks matter. |
Winkelmann, Springer and Di Napoli, 2019, 10.1145/3313828 |
ChASE uses Chebyshev filtered subspace iteration, spectral estimates and optimized polynomial degrees. Roughly degree times b operator-vector applications per filter sweep; O(n b) vectors; block products suit GPUs. |
Attractive for repeated dense Hermitian SCF problems with a small spectral fraction and reusable vectors. When most orbitals are needed, dense diagonalization may win. Generalized AO reduction and overlap cost remain. |
Block methods should request guard vectors when the root boundary intersects a cluster. Locking should be based on residuals and orthogonality, with attention to drift after restart. Eigenvalues within a cluster do not determine a unique set of orbital columns. Warm starts should recycle a subspace in the current metric frame; basis/geometry changes require an explicit transport map. These are established numerical practices, not new algorithms proposed here.
Developments checked in 2025 and 2026¶
Zhou, 2025, arXiv:2509.00495, is an MSc thesis on mixed-precision Jacobi diagonalization. Its low-precision starting decomposition and higher-precision correction target full dense spectra. It is not evidence for matrix-free interior SCF roots. No implementation is proposed from this source.
Di Napoli, Richefort and Wu, April 2026 revision, arXiv:2601.10557v2, extends ChASE to structured pseudo-Hermitian Hamiltonians using symmetry and an oblique Rayleigh-Ritz construction. The preprint record says intended submission to SISC. The associated May 2026 software release has DOI
10.5281/zenodo.20038107. This does not make arbitrary CC Jacobians Hermitian; a future response interface must retain the relevant structure.Jia and Liu, May 2026, arXiv:2605.12846, investigate refined projection and removal of spurious Ritz values in a filtered Hermitian interval solver. This preprint strengthens the case for explicit root-selection diagnostics; it is not a verified replacement for the present harmonic-Davidson implementation.
Gubler et al., August 2026, arXiv:2608.24529, study randomized block Davidson for plane-wave DFT. Sketching reduces orthogonalization work but produces a generalized projected problem. Their reported SCF advantage from sketching is limited when wanted-state count grows. Treat this as a preprint and a future benchmark candidate, not a reason to add randomized arithmetic to small dense AO solves.
Proposals selected for this repository¶
The selected change is architectural and diagnostic: enforce real/complex, metric, representation and option contracts before adding algorithms. Retain dense as the complete-spectrum and interior reference; retain Davidson for current validated partial-spectrum workloads. Keep harmonic-Davidson accurately named and experimental. Recompute residuals independently and never treat them as proof that no closer root exists.
The next implementation increment supplies true complex matrix-free Davidson and real/complex block callback dispatch using the existing recurrence, with related-operator recycling tests. Deferred experiments, in order: custom projected preconditioners; then Krylov-Schur or an optional established eigensolver backend for difficult interior problems. Filtered subspace iteration becomes attractive only after measuring large repeated matrices and the required orbital fraction. Contour methods require a shifted-solve contract and memory accounting first. GPU/MPI support needs device/distribution ownership, synchronization and reduction contracts; accepting an array-like object is not accelerator support.