The effect of such fields on molecules and solids can be studied with TDDFT to extract features like excitation energies, frequency-dependent response properties, and photoabsorption spectra.
TDDFT is an extension of density-functional theory (DFT), and the conceptual and computational foundations are analogous – to show that the (time-dependent) wave function is equivalent to the (time-dependent) electronic density, and then to derive the effective potential of a fictitious non-interacting system which returns the same density as any given interacting system.
The issue of constructing such a system is more complex for TDDFT, most notably because the time-dependent effective potential at any given instant depends on the value of the density at all previous times.
Consequently, the development of time-dependent approximations for the implementation of TDDFT is behind that of DFT, with applications routinely ignoring this memory requirement.
[3] The most popular application of TDDFT is in the calculation of the energies of excited states of isolated systems and, less commonly, solids.
[4][5] The approach of Runge and Gross considers a single-component system in the presence of a time-dependent scalar field for which the Hamiltonian takes the form where T is the kinetic energy operator, W the electron-electron interaction, and Vext(t) the external potential which along with the number of electrons defines the system.
Later it was observed that an approach based on the Dirac action yields paradoxical conclusions when considering the causality of the response functions it generates.
The response functions from the Dirac action however are symmetric in time so lack the required causal structure.
An approach which does not suffer from this issue was later introduced through an action based on the Keldysh formalism of complex-time path integration.
An alternative resolution of the causality paradox through a refinement of the action principle in real time has been recently proposed by Vignale.
[7] Linear-response TDDFT can be used if the external perturbation is small in the sense that it does not completely destroy the ground-state structure of the system.
This is a great advantage as, to first order, the variation of the system will depend only on the ground-state wave-function so that we can simply use all the properties of DFT.
Other linear-response approaches include the Casida formalism (an expansion in electron-hole pairs) and the Sternheimer equation (density-functional perturbation theory).