A Unified Formulation for $\langle \hat{S}^2 \rangle $ 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 $\langle \hat{S}^2 \rangle$ 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, $\langle \hat{S}^2 \rangle$ in the excited states is shown to arise from two distinct sources: (i) $\langle \hat{S}^2 \rangle_0$ in the reference state and (ii) additional $\Delta\langle \hat{S}^2 \rangle$ introduced by the excitation process itself. Finally, we evaluate the expectation value $\langle \hat{S}^2 \rangle$ 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.