A number of approximations can be introduced at this point. Just as in standard TDDFT, the Tamm-Dancoff approximation (TDA) can be applied by setting the B matrix to zero in Eq. ( ), as has been introduced in Ref. . More importantly, a hierarchy of approximations can be devised based on the subsystem structure of the matrices above, both in the full sTDDFT problem and in its TDA version. As a first step, the "uncoupled" (FDEu-TDDFT) approximation to the response problem above is obtained by considering only the subsystem-diagonal blocks (with elements A (ia) I ,(jb) I and B (ia) I ,(jb) I ) for a particular subsystem I, but neglecting the response of and response coupling to all other subsystems. Second, an approximate "coupled" (FDEc-TDDFT) solution can be obtained by generating a few of the lowest-lying FDEu-TDDFT excitations for several subsystems, and solving the subsystem TDDFT problem in the basis of those subsystem-localized excitations. This can be considered as a subspace projection of the full response problem in Eq. ( ) into a predefined space of subsystem-localized excitations, and is often useful when dealing with problems related to excitonic couplings in chromophore aggregates.