Hybrid Diagonal Approximation in Time-Dependent Auxiliary Density Functional Theory

A hybrid diagonal approximation (HDA) for time-dependent auxiliary density functional theory (TD-ADFT) is presented. This newly implemented method allows the use of global and range-separated hybrid functionals in TD-ADFT for the calculation of vertical excitation energies and corresponding oscillator strengths. To preserve the exceptional computational efficiency and low-order scaling of TD-ADFT, only the diagonal elements of exact exchange are included in the TD-ADFT matrices. For singlet excitations, this approximation reaches accuracies comparable to four-center electron repulsion integral (ERI) implementations, albeit with a fraction of the computational cost. For triplet excitations, larger deviations are found with the HDA. Despite additional integral calculations, the low-order scaling of TD-ADFT is preserved with the HDA. We explain this by the intact index alignment between the ERIs and the excitation vectors, which remains unaltered in TD-ADFT with the HDA.