TDDFT

Time-dependent density functional theory (TDDFT) is a quantum mechanical method used to calculate excited-state and time-dependent properties of many-electron systems. It extends density functional theory (DFT) to time-dependent phenomena, using the time-dependent electron density n(r,t) as the central variable. It is widely used for optical spectra, electron dynamics, and non-linear response.

The Runge-Gross theorem establishes a one-to-one mapping between time-dependent densities and external potentials for a given initial state, up to a time-dependent constant. The interacting many-electron problem is mapped to a non-interacting Kohn-Sham system with time-dependent orbitals psi_i(r,t) and a potential v_s = v_ext + v_H[n] + v_xc[n](r,t). The equations are i ∂/∂t psi_i = [ -1/2 ∇^2 + v_ext(r,t) + v_H[n](r,t) + v_xc[n](r,t) ] psi_i. The exchange-correlation potential depends on the density and its history (memory). In practice, approximations use the adiabatic approximation, evaluating ground-state v_xc at the instantaneous density.

In linear response TDDFT, an eigenvalue problem for excitation energies is formulated using the exchange-correlation kernel f_xc = δv_xc/δn, leading to Casida-type equations. Real-time TDDFT integrates the time-dependent Kohn-Sham equations directly to obtain spectra and dynamics without explicit eigenvalue solutions, which is advantageous for strong fields and nonlinear responses.

Applications include ultraviolet-visible spectra, excitations, charge-transfer states, plasmonics, and ultrafast processes. Limitations include dependence on approximate XC functionals, neglect of memory effects in common approximations, and challenges for double excitations and long-range charge-transfer. Computational cost is typically favorable compared with wavefunction methods, with common scaling around N^3 for linear response. Many codes implement both real-time and linear-response TDDFT.