Most applications of TDDFT today are in the linear response regime, to compute excitation spectra and response. The focus of this lecture is to derive the basic equations underlying these calculations. Two ingredients are needed in TDDFT linear response: the bare Kohn-Sham excitation energies and orbitals and the Hartree-exchange-correlation kernel which is the density-functional derivative of the Hartree-exchange-correlation potential. We discuss the common approximations made for the kernel, their general performance, and cases for which one must go beyond the usual approximations to get decent results. These include excitonic spectra in solids, double-excitations, long-range charge-transfer excitations.
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