  This chapter provides a basic introduction to excited-state extensions of
density functional theory (DFT), including time-dependent (TD-)DFT in both its
linear-response and its explicitly time-dependent formulations. As applied to
the Kohn-Sham DFT ground state, linear-response theory affords an
eigenvalue-type problem for the excitation energies in a basis of
singly-excited Slater determinants, and is widely known simply as "TDDFT"
despite its frequency-domain formulation. This form of TDDFT is the mostly
widely-used quantum-chemical method for excited states, due to a favorable
combination of low cost and reasonable accuracy. The chapter surveys the
accuracy of linear-response TDDFT, which is generally more sensitive to the
details of the exchange-correlation functional as compared to ground-state DFT,
and also describes some known systemic problems exhibited by this approach.
Some of those problems can be corrected on a case-by-case basis using
orbital-optimized, excited-state self-consistent field (SCF) calculations, in
what is known as excited-state Kohn-Sham theory or a "Delta-SCF" procedure, a
class of methods that includes restricted open-shell Kohn-Sham theory. Recent
successes of these approaches are highlighted, including double excitations and
core-level excitations. Finally, explicitly time-dependent (or "real-time")
TDDFT involves propagation of the molecular orbitals in time following an
external perturbation, according to the Kohn-Sham analogue of the
time-dependent Schroedinger equation. The time-dependent approach has been used
to model strong-field electron dynamics, and in the weak-field limit it
provides a route to broadband spectra based on the time evolution of the dipole
moment function. This is useful for describing high-energy excitations (as in
x-ray spectroscopy) and in systems where the density of states is high, as
demonstrated by a few examples.
