FR-TO ΔSCF: A Robust and Systematic Framework for Core Excitations

We present a robust ΔSCF scheme for core-level excited states based on a freeze-and-release (FR) optimization strategy that stabilizes orbital relaxation. Unlike existing FR approaches that freeze specific canonical orbitals, we constrain transition orbitals (TOs) obtained from an inexpensive subspace projected Tamm-Dancoff approximation (TDA) calculation. These TOs provide a physically motivated representation of the target excitation and reduce the ambiguity inherent in cases where the transition cannot be associated with a single canonical orbital. In the "freeze stage", the TOs are held fixed while the remaining orbitals relax self-consistently to avoid variational collapse; the constraint is then lifted in the "release stage" to complete the ΔSCF optimization. Benchmarks on K- and L-edge excitations show excitation energies improved relative to iVI TDA (iterative vector interaction TDA) at computational cost comparable to ground-state SCF. The FR-TO procedure also converges more reliably than conventional MOM-type (maximum overlap method-type) schemes, particularly for systems with strong orbital mixing or ill-defined target-orbital character, providing a simple and systematic framework for core-level ΔSCF calculations.