A Unified Formulation for ⟨Ŝ²⟩ in Two-Component TDDFT

Two-component linear-response time-dependent density functional theory (TDDFT) provides a unified framework that encompasses noncollinear excitations in noncollinear reference states, as well as both spin-conserving and spin-flip excitations in collinear reference states. In this work, we present a general formalism for evaluating the expectation value ⟨Ŝ²⟩ of electronically excited states obtained within two-component TDDFT. We then derive and analyze specialized forms of the resulting equations for collinear reference determinants, for which the two-component formalism decomposes into conventional spin-conserving and spin-flip TDDFT. The resulting working equations are systematically compared with previously proposed theoretical approaches. On the basis of our analysis, ⟨Ŝ²⟩ in the excited states is shown to arise from two distinct sources: (i) ⟨Ŝ²⟩₀ in the reference state and (ii) additional Δ⟨Ŝ²⟩ introduced by the excitation process itself. Finally, we evaluate the expectation value ⟨Ŝ²⟩ by performing two-component TDDFT calculations based on two-component DFT, unrestricted Kohn-Sham (UKS), and restricted open-shell Kohn-Sham (ROKS) reference states, respectively.