Origin of anomalous functional dependence in TDΔSCF singlet–triplet gaps for O2 and benzynes

Determine the origin of the larger exchange–correlation functional dependence observed in time-dependent ΔSCF (TDΔSCF) predictions of singlet–triplet energy gaps for O2, o-benzyne, and p-benzyne, relative to other systems in the benchmark set. Specifically, ascertain whether inconsistencies in the underlying ΔSCF reference determinants across different functionals cause the observed variation by analyzing and comparing those ΔSCF reference states and quantifying their impact on the subsequent TDΔSCF results.

Background

The authors benchmark singlet–triplet gaps for atoms, small diatomics, carbenes, and benzyne isomers, comparing DFT, SF-TDDFT, and TDΔSCF. Overall, TDΔSCF shows markedly weaker functional dependence and lower mean absolute errors than SF-TDDFT across BLYP, B3LYP, and BHHLYP.

However, for three cases—O2, o-benzyne, and p-benzyne—the variation among functionals in TDΔSCF is substantially larger than for the rest of the set. The paper notes that the origin of this behavior is not fully understood and hypothesizes that functionals may converge to slightly different ΔSCF reference determinants, thereby affecting the TDΔSCF response step. A dedicated analysis of the ΔSCF references is suggested as necessary future work.

References

At present, the origin of this behavior is not entirely clear. One possible explanation is that the underlying $\Delta$SCF reference states are not fully consistent across the different functionals, so that the subsequent TD$\Delta$SCF calculations are effectively based on slightly different electronic references. A more detailed analysis of the corresponding $\Delta$SCF states would be needed to clarify this point, but we leave such an analysis for future work.

— TD$Δ$SCF: Time-Dependent Density Functional Theory with a Non-Aufbau Reference for near-degenerate states  (2603.29948 - Shibasaki et al., 31 Mar 2026) in Section 4.2 (Singlet–triplet energy gaps)