Real-space vs interelectronic exchange mixing for the prediction of low-lying Rydberg excitations within time-dependent density functional theory

Low-lying Rydberg excitations are a challenge for linear-response time-dependent density functional theory because many functionals substantially underestimate their energies due to self-interaction error. The established remedy is range separation, which switches to full exact exchange (EXX) at large interelectronic distances. Local hybrid functionals instead control the EXX admixture through a real-space local mixing function (LMF), and range-separated local hybrids (RSLHs) combine both mixing schemes. Here, we analyze the two mixing coordinates by evaluating 87 Rydberg (3s and 3p) and 75 valence excitation energies from the QUEST database. A benchmark of 18 functionals shows that range-separated hybrids, some local hybrids, and RSLHs describe these states with comparable accuracy, with the best of each class reaching Rydberg mean absolute errors (MAEs) of 0.14-0.20 eV, but differences in their other ingredients prevent attributing this accuracy to either scheme. We therefore scan the LMF prefactor b and the range-separation parameter ω for a curated subset. A full-range EXX admixture, supplied through the LMF or a global fraction, describes these states more accurately than pure range separation (best Rydberg MAEs of 0.12 vs 0.20 eV). For RSLHs, the global Rydberg optimum lies at ω = 0, so range separation offers no further gain once real-space mixing is optimized. Real-space mixing incurs smaller barrier-height penalties when Rydberg-optimal parameters are transferred to ground-state properties, and a small ω can reconcile Rydberg and atomization-energy accuracy in LSDA-based RSLHs. For future RSLHs, we recommend encoding low-lying Rydberg states in the LMF, leaving the range-separation parameter for properties requiring long-range EXX, notably charge-transfer excitations.