Although the theory discussed in Section 7.3 is known universally as “time-dependent” DFT (TDDFT), in truth it is the frequency-domain transformation of linear-response (LR) DFT,174, 342 and is sometimes given the additional designation of LR-TDDFT in order to distinguish it from the “real time” (RT) version of TDDFT that is described in this section. The phrase “real-time time-dependent DFT” (RT-TDDFT) is sufficiently awkward that the theory described here is also known as time-dependent Kohn-Sham (TDKS) theory,1292, Zhu:2021 terminology that is actually more consistent with the original language used by Gross and co-workers developing a theory based on the time-dependent Kohn-Sham equation.401, 403, 402 The TDKS approach is explicitly time-dependent, and amounts to propagation of time-dependent Kohn-Sham MOs following a perturbation of the ground-state density.
LR-TDDFT calculations are often the most efficient way to predict resonant electronic response frequencies and intensities when only a small number of low-lying excited states are desired. To obtain broadband spectra (in the x-ray regime, say), hundreds of excited states may be required, however. In such cases, the real-time approach may be preferable because it can be used to obtain the entire absorption spectrum (at all excitation energies) via Fourier transform of the time-dependent dipole moment function, without the need to compute the spectrum state-by-state. This is the theoretical basis of real-time electronic structure methods in general.926, 681 A perturbation creates a superposition of all (symmetry-allowed) excitations, and the Fourier components of the ensuing time evolution encode all of the excitation energies. This theory is described in somewhat more detail in the next section, following which the TDKS job control variables are described in Section 7.4.3. Calculation of broadband absorption spectra using the TDKS approach is discussed in Section 7.4.4. Users to the TDKS/RT-TDDFT code in Q-Chem are asked to cite Refs. 1292 and Zhu:2021, the latter of which also provides something of a tutorial for the calculation of broadband spectra using the TDKS approach.