Time-dependent TDLDA theory

Using a method analogous to that for static case, the linear response theory can be developed within the LDA for the case when the external electric field, characterized by the potential Vext r o) = E r Yie is time-dependent. This leads to the time-dependent density functional theory (TDLDA) [55]. 

Now we have a picture about how demanding temperature-dependent calculations are we need at each T and colO Monte Carlo points and 10 frequencies in order to cover the experimental range. Furthermore we need the three different directions jc, y and z. So we had to solve three million times the TDLDA integral equation with full inclusion of the ionic structure (via pseudopotential perturbation theory). Needless to say it seems almost impossible to perform calculations of this type for transition metals with the additional complication of the d-electrons ... 

In the frame of the DF Theory, two approaches can be considered for photoionization the conventional Kohn-Sham (KS) method and the time dependent version of the theory (TD-DFT). Since usually the Local Density Approximation (LDA) is employed, they are often referred as the LDA and the TDLDA methods respectively. 

Figure 5.2 Absorption cross section of SissHse calculated using (1) tight-binding approach with local field effects (solid thick line), (2) the tight-binding energy levels with a classical model for the surface polarization contribution (dashed line) and (3) a time-dependent local density approximation (TDLDA) within density functional theory (solid thin line). TDLDA results from ref. 39.

Concurrent with these developments, the density functional formalism has emerged as an alternative to a Hartree Fock based description of the electronic structure of atoms, molecules and solids. In its most common form as a local density approximation (LDA) the question of atomic photoionization can again be addressed. Here too, one finds that the simplest approach falls to adequately charr-. acterize the experimental results in many cases. However, a recent straightforward generalization of density functional theory to time-dependent phenomena has been applied successfully to the problem of the optical response of atoms. In particular, highly accurate photoionization cross sections can be readily obtained. The purpose of the present article is to review this time-dependent local density approximation (TDLDA), illustrate its scope and limitations and compare it to the more familiar Hartree-Fock based methods. 

Finally, a necessary direction for future work is to seek examples where the TDLDA clearly fails. Such examples would provide the testing ground for similar schemes based on improved choices for Vjj Cx) as well as further impetus toward the development of a rigorous time-dependent density functional theory. 

---