Spin-adapted open-shell time-dependent density functional theory: towards a simple and accurate method for spin-flip-down excitations

TL;DR

Spin-flip TD-DFT commonly suffers from spin contamination due to an incomplete excitation manifold. The authors introduce XSF-TDA, a spin-adapted extension of the X-TD-DFT framework for spin-flip-down excitations, built on tensor configuration interaction singles (TCIS) with a tunable global hybridization factor $g_X$ to mitigate double counting. Benchmark results across spin-flip transitions, Be/Mg, HF, F2, and inverted singlet-triplet gaps show XSF-TDA achieves accuracy comparable to USF-TDA for clean states and substantially outperforms it for contaminated states, while outperforming MRSF-TDA in challenging cases of spin symmetry and state ordering. The method offers a practical, Hermitian approach to challenging open-shell excitations with potential impact on photochemistry and OLED-related systems, and lays groundwork for further refinements, including optimized $g_X$, multicollinear functionals, and spin-orbit/nonadiabatic extensions.

Abstract

A major challenge in using spin-flip time-dependent density functional theory (SF-TD-DFT) for spin-flip-down excitations is the presence of spin contamination. While several improved methods have been developed in the past, a simple and accurate method remains elusive. Here, based on our previous development on spin-adapted open-shell TD-DFT for spin-conserving excitations (X-TD-DFT) [Z. Li and W. Liu, J. Chem. Phys. 135, 194106 (2011)], we introduce a method termed as XSF-TDA for modeling spin-flip-down excitations, and provide an in-depth comparison of different methodologies for mitigating spin contamination in SF-TDA. Pilot applications to prototype systems demonstrate the promise of XSF-TDA over existing SF-TDA methods, including unrestricted SF-TDA (USF-TDA) and mixed-reference SF-TDA (MRSF-TDA), in describing bond breakings and inverted singlet-triplet gap systems.

Figures (5)

• Figure 1: Illustration of the excited-state spin contamination problem in SF-TD-DFT/SF-TDA due to the incompleteness of the excited state manifold generated by spin-flip-down single excitations. Configurations in red color represent the missing higher excitations with respect to the reference to achieve spin completeness/adaptation.
• Figure 2: Mean absolute errors (MAEs) in eV of vertical excitation energies computed with spin-flip TDA methods. D$\to$Q: spin-flip-up SF-TDA with UKS/ROKS reference; D$\leftarrow$Q and S$\leftarrow$T: spin-flip-down SF-TDA with UKS/ROKS reference. In spin-flip-down XSF-TDA method, Eq. \ref{['eq:XSFTDA']} is used, while in spin-flip-up XSF-TDA, the analog of Eq. \ref{['eq:SFTDA']} is used, as there is no spin contamination in such case.
• Figure 3: Results for the HF molecule using SF-TDA methods with the aug-cc-pVTZ basis set. (a) PECs obtained with USF-TDA (solid) and XSF-TDA (dashed); (b) PECs obtained with USF-TDA (solid) and MRSF-TDA (dashed); (c) $\Delta\langle \hat{S}^{2}\rangle$ obtained in USF-TDA; (d) Weights of the dominant two excitations obtained in USF-TDA/BHHLYP. The minimum of each curve in (a) and (b) has been shifted to zero for a better comparison. The cross marker in (a) and (b) represents the experimental bond dissociation energy 135.1 kcal/mol1970Bond.
• Figure 4: Results for the F2 molecule using SF-TDA methods with the aug-cc-pVTZ basis set. (a) PECs obtained with USF-TDA (solid) and XSF-TDA (dashed); (b) PECs obtained with USF-TDA (solid) and MRSF-TDA (dashed); (c) $\Delta \langle \hat{S}^2\rangle$ in USF-TDA. The minimum of each curve in (a) and (b) has been shifted to zero for a better comparison. The cross marker in (a) and (b) represents the experimental bond dissociation energy 37.0 kcal/mol2024Bond.
• Figure 5: Results for inverted single-triplet gap systems. (a) Schematic representation of heptazine (no. 1) and cyclazine (no. 2) and their derivatives considered in Ref. 2023heptazine as benchmark systems. (b) Singlet-triplet gaps $\Delta E_{\mathrm{ST}}$ (in eV) computed using TDA and SF-TDA methods with the 6-31G* (cross) and aug-cc-pVTZ (dot) basis sets. The theoretical best estimates (TBEs) are taken from Ref. 2023heptazine.