In SA-SF-TDDFT, excitation operators in the TEOM formalism are truncated at the single excitation level. The single-excitation operators can be grouped into two categories: those with zero rank (singlet coupling) and those with rank one (triplet coupling). Because SF methods use a target state of lower spin angular momentum as compared to the reference state, only the triplet-coupled excitation operators are useful in this case. These can be represented in the MO basis as