TDΔSCF: Time-Dependent Density Functional Theory with a Non-Aufbau Reference for Near-Degenerate States

Near-degenerate electronic structures remain a major challenge for conventional single-reference density functional theory (DFT). To address this problem, we propose time-dependent ΔSCF (TDΔSCF), a novel linear-response scheme in which a non-Aufbau ΔSCF determinant serves as the reference for a subsequent TDDFT calculation. In contrast to collinear spin-flip (SF)-TDDFT, this formulation preserves the usual Coulomb and exchange-correlation response contributions while describing the target states from an electronically excited reference. We examine the performance of TDΔSCF for several prototypical problems involving near-degeneracy, including the torsional potential of ethylene, singlet-triplet gaps of representative diradicals, geometry optimizations of benzyne isomers, and bond-dissociation curves of hydrogen fluoride and F₂. Across these tests, TDΔSCF shows markedly weaker functional dependence than SF-TDDFT and often yields a more balanced description of challenging singlet states. In particular, it provides smooth torsional potentials, improved singlet-triplet gaps, a consistent monocyclic structure for singlet m-benzyne, and a more satisfactory description of bond dissociation without the spurious low-lying states found in SF-TDDFT. At the same time, the method exhibits a systematic tendency to overestimate singlet energies and can lose accuracy when the underlying ΔSCF reference is not well suited to the final state. We also identify a numerical instability that can arise in non-Aufbau calculations and trace its origin to the exchange-correlation potential near uncompensated nodal regions. These results highlight both the promise and the practical limitations of TDΔSCF as a low-cost method for singlet states with near-degenerate electronic structures.