TDLDA density approximation

The effective Time Dependent Kohn-Sham (TDKS) potential vks p (r>0 is decomposed into several pieces. The external source field vext(r,0 characterizes the excitation mechanism, namely the electromagentic pulse as delivered by a by passing ion or a laser pulse. The term vlon(r,/) accounts for the effect of ions on electrons (the time dependence reflects here the fact that ions are allowed to move). Finally, appear the Coulomb (direct part) potential of the total electron density p, and the exchange correlation potential vxc[p](r,/). The latter xc potential is expressed as a functional of the electronic density, which is at the heart of the DFT description. In practice, the functional form of the potential has to be approximated. The simplest choice consists in the Time Dependent Local Density Approximation (TDLDA). This latter approximation approximation to express vxc[p(r, /)]... 

In the following we describe the optical properties of metal clusters within the jellium model. We begin by introducing the TDLDA (time-dependent local density approximation)... 

The theoretical framework for all regimes is the (time-dependent) local-density approximation (TDLDA) which is much discussed also in Chapter 1 of this book. We thus will present here only a short discussion of the essential ingredients and compare it with the analog mean-field models in nuclear physics. This is done as a starter in the next section. 

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. 

It should be noted that the above TDLDA picture a priori involves two touchy approxmations. The first one consists in using the LDA which basically relies on the assumption of weakly varying (in space) electron density. This LDX approximation has been widely used in metal clusters arid does not raise problems with respect to the observables we arc interested in. The second approximation is to use in a dynamical context a functional which has been tuned to static problems. The extension of LDA to TDLDA is thus a further approximation which can he considered as adiabatic , in the sense that we are using, at each instant, the energy density as expressed... 

Table 12 Vertical excitation energies (in eV) for N2- TDLDA and TDGGA represent results from time-dependent density-functional calculations with either a local-density (LDA) or a generalized-gradient (GGA) approximation, whereas TDHF are similar results from time-dependent Hartree-Fock calculations. MRCCSD, SOPPA, MRTDHF are results from sophisticated configuration-interaction calculations, and CIS are less sophisticated configuration-interaction calculations. The final state and the electronic excitation are also shown and the results are compared with experimental values (Exp.). The results are from ref. 89...

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