Accelerating SCF Orbital Optimization with S-GEK/RVO: Efficient Subspace Compression and Robust Convergence

We present enhancements to the S-GEK/RVO method for self-consistent field (SCF) orbital optimization, aimed at improving computational efficiency and robustness. Building on a gradient-enhanced Kriging surrogate model and restricted-variance optimization, we introduce three key modifications: (i) a cost-effective subspace expansion using r-GDIIS or BFGS displacement predictions, (ii) a systematic undershoot mitigation strategy in flat energy regions, and (iii) rigorous coordinate and gradient transformations consistent with the exponential parametrization of orbital rotations. Benchmarking across an extensive set of molecular systems─including organic molecules, radicals, and transition-metal complexes─demonstrates that the new S-GEK/RVO variants consistently outperform the default (in OpenMolcas) r-GDIIS method in iteration count, convergence reliability, and wall time. These improvements make S-GEK/RVO a competitive alternative for SCF optimization and suggest broader applicability to other orbital optimization and localization problems.