The two components of this density matrix are subjected to separate spin-flipping operations, α → β for the M S = 1 component (as in the basic SF-TDDFT method discussed above) and β → α for the M S = -1 component. This is shown schematically in Fig. for the same (4e,4o) model that was used to introduce the basic method, using the same notation is used for the various subspaces of Slater determinants. Comparing Fig. to Fig. , it is evident that MRSF-TDDFT introduces many more electronic configurations as compared to SF-TDDFT. Operationally, this removes the majority of the spin contamination even thought the excitation space is not formally spin-complete because some of the necessary o-v determinants (shown in gray in Fig. ) cannot be generated by this procedure. However, these are higher-energy configurations and thus play little role in low-lying excited states, so that Ŝ2 is close to its spin-pure value in practice. Derivation of MRSF-TDDFT is based on the density matrix formulation of the time-dependent Kohn-Sham methodology, which means that the density matrix needs to be idempotent. This can be demonstrated for the density matrix in Eq. ( ) based on a complex rotation of the spin functions. Separation of the M S = 1 and M S = -1 triplet density matrices leads to two independent eigenvalue equations,