The formulation of SF-TDDFT that is outlined above is sometimes called the "collinear" approach. This was later generalized to an alternative "non-collinear" method, which does bring the semilocal part of the functional into the coupling matrix. (See Refs. 43 and 49 for additional discussion of the non-collinear approach. The name comes from generalized Hartree-Fock theory, where a non-collinear formalism is one that allows α and β spin to mix, as used for example in relativistic DFT. ) The non-collinear version of SF-TDDFT is found to improve the performance for some problematic diradicals, and is more accurate for excitation energies when functionals with a low percentage of HFX are used. A significant drawback, however, is that the non-collinear exchange-correlation kernel involves a numerically problematic factor of spin density (ρ α -ρ β ) in the denominator. Likely for this reason, the non-collinear formulation is less widely used and is not discussed further in this chapter.