Critical Assessment of TD-DFT for Excited States of Open-Shell Systems: I. Doublet-Doublet Transitions.

A benchmark set of 11 small radicals is set up to assess the performance of time-dependent density functional theory (TD-DFT) for the excited states of open-shell systems. Both the unrestricted (U-TD-DFT) and spin-adapted (X-TD-DFT) formulations of TD-DFT are considered. For comparison, the well-established EOM-CCSD (equation-of-motion coupled-cluster with singles and doubles) is also used. In total, 111 low-lying singly excited doublet states are accessed by all the three approaches. Taking the MRCISD+Q (multireference configuration interaction with singles and doubles plus the Davidson correction) results as the benchmark, it is found that both U-TD-DFT and EOM-CCSD perform well for those states dominated by singlet-coupled single excitations (SCSE) from closed-shell to open-shell, open-shell to vacant-shell, or closed-shell to vacant-shell orbitals. However, for those states dominated by triplet-coupled single excitations (TCSE) from closed-shell to vacant-shell orbitals, both U-TD-DFT and EOM-CCSD fail miserably due to severe spin contaminations. In contrast, X-TD-DFT provides balanced descriptions of both SCSE and TCSE. As far as the functional dependence is concerned, it is found that, when the Hartree-Fock ground state does not suffer from the instability problem, both global hybrid (GH) and range-separated hybrid (RSH) functionals perform grossly better than pure density functionals, especially for Rydberg and charge-transfer excitations. However, if the Hartree-Fock ground state is instable or nearly instable, GH and RSH tend to underestimate severely the excitation energies. The SAOP (statistically averaging of model orbital potentials) performs more uniformly than any other density functionals, although it generally overestimates the excitation energies of valence excitations. Not surprisingly, both EOM-CCSD and adiabatic TD-DFT are incapable of describing excited states with substantial double excitation characters.