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\begin{align}  \left( A + B \right) \vec{x} = \omega \vec{x}  \end{align}  The Tamm-Dancoff approximation to these equations is identical to the ordinary TDDFT Tamm-Dancoff approximation. Note that all of the levels of theory we have discussed use the same Coulomb and $f_{\rm xc}$ matrix elements, so the code can calculate the results for multiple levels of theory with a small extra effort.  For a non-spin-polarized system, the excitations separate into a singlet and a triplet subspace,  which are superpositions of singlet and triplet KS transitions: