Kohn–Sham Time-Dependent Density Functional Theory with Applications to Linear and Nonlinear Properties

We review Kohn–Sham density-functional theory for time-dependent response functions up to and including cubic response. The working expressions are derived from an explicit exponential parametrization of the density operator and the Ehrenfest principle, alternatively the quasi-energy ansatz. While the theory retains the adiabatic approximation, implying that the time-dependency of the functional is obtained only implicitly—through the time-dependency of the density itself rather than through the form of the exchange-correlation functionals—our implementation generalizes previous time-dependent approaches in that arbitrary functionals can be chosen for the perturbed densities (energy derivatives or response functions). Thus, the response of the density can always be obtained using the stated density functional, or optionally different functionals can be applied for the unperturbed and perturbed densities, even different functionals for different response order. In particular, general density functionals beyond the local density approximation can be applied, such as hybrid functionals with exchange–correlation at the generalized gradient-approximation level and fractional exact Hartree–Fock exchange. We also review some recent progress in time-dependent density functional theory for open-shell systems, in particular spin-restricted and spin restricted-unrestricted formalisms for property calculations. We highlight a sample of applications of the theory.