Conserving approximations in time-dependent density functional theory

In the present work, we propose a theory for obtaining successively better approximations to the linear response functions of time-dependent density or current-density functional theory. The new technique is based on the variational approach to many-body perturbation theory (MBPT) as developed during the sixties and later expanded by us in the mid-nineties. Due to this feature, the resulting response functions obey a large number of conservation laws such as particle and momentum conservation and sum rules. The quality of the obtained results is governed by the physical processes built in through MBPT but also by the choice of variational expressions. We here present several conserving response functions of different sophistication to be used in the calculation of the optical response of solids and nanoscale systems.

Figure 1

Exchange-correlation kernel in the EXX approximation.

Figure 2

Exchange-correlation kernel in the $GW$ approximation.

Figure 3

Exchange-correlation kernel calculated from the LW variational approach to TDDFT in the Hartree-Fock approximation. The thin lines represent the noninteracting Green’s function $G_s$, while the thick lines represent the Green’s function $\tilde{G}$.

Figure 4

Exchange-correlation kernel at the $GW$ level in the $\Psi$ scheme.