TD$Δ$SCF: Time-Dependent Density Functional Theory with a Non-Aufbau Reference for near-degenerate states

Abstract

Near-degenerate electronic structures remain a major challenge for conventional single-reference density functional theory (DFT). To address this problem, we propose time-dependent $Δ$SCF (TD$Δ$SCF), a novel linear-response scheme in which a non-Aufbau $Δ$SCF 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 TD$Δ$SCF 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 F$_2$. Across these tests, TD$Δ$SCF 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 $Δ$SCF 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 TD$Δ$SCF as a low-cost method for singlet states with near-degenerate electronic structures.

Figures (9)

• Figure 1: Illustration of SF-TDDFT and TD$\Delta$SCF. The two methods employ different reference states. (a) SF-TDDFT uses a high-spin configuration and accesses low-spin states via spin-flip excitations. (b) In TD$\Delta$SCF, the reference is an excited-state configuration and the target states are described by subsequent linear-response excitations.
• Figure 2: Torsional potential energy curves of ethylene computed with TD$\Delta$SCF (solid line), SF-TDDFT (dashed line) and MRCISD using the DZP basis set. A cusp is clearly observed near the barrier region for the pure functional (BLYP). Energies are referenced to the planar (0$^\circ$) geometry.
• Figure 3: Comparison of computed and experimental $\Delta E_{\rm ST}$ values (kcal/mol) obtained with different methods. For SF-TDDFT, the BLYP results are omitted; see the main text for details.
• Figure 4: Optimized singlet-state geometrical parameters of the benzyne systems obtained by SF-TDDFT and TD-$\Delta$SCF, together with the nuclear repulsion energies (bond lengths in Å, bond angles in degrees, and energies in hartree). BLYP is indicated in normal type, B3LYP in italics, and BHHLYP and 50-50 by underlining.
• Figure 5: Bond dissociation curves of hydrogen fluoride calculated with each method. Energies are plotted relative to the gray dashed line, which denotes the unrestricted dissociation limit.