For historical reasons, the original eigenvalue problem in Eq. (2.2) is sometimes called the random phase approximation (RPA), in order to distinguish it from the simpler Hermitian eigenvalue problem in Eq. (2.6). That terminology is avoided here, however, so as not to confuse it with other methods known as RPA. Where we need to make a distinction, we refer to Eq. (2.2) as "full" TD-DFT and Eq. (2.6) as TD-DFT/ TDA. Use of the TDA is often essential for avoiding triplet instabilities and obtaining accurate triplet excitation energies. Triplet instabilities in the ground-state Kohn-Sham solution, indicating that an unrestricted wave function would lower the energy with respect to the closed-shell solution, manifest as negative excitation energies. This is common at bond-stretching geometries, where singlet and triplet states become quasidegenerate, but may also occur near the ground-state geometry, if the fraction of Hartree-Fock exchange is large, or for large values of the range separation parameter in LRC functionals. Beyond indicating an instability, solutions with negative excitation energies are not physically meaningful and can lead to convergence failure in solving Eq. (2.2), if the iterative algorithm is predicated on the excitation energies being positive. Invoking the TDA decouples the stability problem from the excitation energy problem and is used in most calculations that are described here.