TDLDA

PRACTICABLE FACTORIZED TDLDA FOR ARBITRARY DENSITY- AND CURRENT-DEPENDENT FUNCTIONALS... 

In fact, SRPA is the first TDLDA iteration with the initial wave function (7). A single iteration is generally not enough to get the complete convergence of TDLDA results. However, SRPA calculations demonstrate that high accuracy can be achieved even in this case if to ensure the optimal choice of the input operators Qk and Pk and keep sufficient amount of the separable terms (see discussion in Sec. 5). In this case, the first iteration already gives quite accurate results. 

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, /)]... 

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... 

As mentioned above, the electronic response of the cluster may be analyzed, at least during the early phases of the process, in terms of the collective Mie plasmon, ionization and energy deposit. These quantities can easily be accessed in the TDLDA. 

Such a non adiabatic coupled electrons+ions dynamics requires considerable numerical effort. Still our TDLDA scheme allows to perform such calculations, if not systematically, at least at a satisfactory level for the exploratory goal which we presently have. We have thus for example analysed this electronsions coupling in detail for a Nai2, with full ionic structure, irradiated by a gaussian laser pulse of 18 fs FHWM and at various frequencies... 

The TDLDA approach used here for describing electronic dynamics is basically an effective mean-field theory. In particular, the possibly crucial dynamical correlations are missing. While the latters are expected to play a minor role at low excitations, they may become dominant in far from... 

Figure J Absorption spectra of the S15H12 cluster calculated using different DFT based methods LDA-RPA black (black), GW-RPA red (dark gray), BSE violet (light gray), TDLDA blue (light black).

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]. 

Linear response theory (TDLDA) applied to the jellium model follows the Mie result, but only in a qualitative way the dipole absorption cross sections of spherical alkali clusters usually exhibit a dominant peak, which exausts some 75-90% of the dipole sum rule and is red-shifted by 10-20% with respect to the Mie formula (see Fig. 7). The centroid of the strength distribution tends towards the Mie resonance in the limit of a macroscopic metal sphere. Its red-shift in finite clusters is a quantum mechanical finite-size effect, which is closely related to the spill-out of the electrons beyond the jellium edge. Some 10-25% of the... 

The calculated photoabsorption spectrum of this cluster shows a collective excitation with a peak at 2.12 eV. The tail of the resonance extends up to 3 eV, and concentrates a sizable amount of strength, due to particle-hole transitions that interact with the collective excitation and lead to its broadening. One of the most important particle-hole transitions is that from the HOMO level to the continuum the energy of this ionization threshold is indicated by the arrow at 2.6 eV. Similar TDLDA calculations have been performed for pure Na [104] and pure K [127] clusters. Comparing the positions of the collective resonances, it can be concluded that the position of the resonance in K2oNa2o is closer to that in pure K clusters thus, the surface, made of K atoms, controls the frequency of the collective resonance. 

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...

State Excitation TDLDA TDGGA MRCCSD SOPPA MRTDHF TDHF CIS Exp. 

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)... 

In this equation x is the exact dynamical density-density correlation function calculated with the inclusion of all many-body effects. As one can show [5], the exact x is determined by solving the TDLDA integral equation... 

Figure 1.17 Dipole-allowed absorption in jellium Na2o and its interpretation. The continuous line gives the result from the TDLDA. The nature of the double structure between 0.5 and 1.0 can be understood in two steps. First, the TDLDA is compared with LDA, the independent-particle response (dashed line). Each peak corresponds to one arrow in the upper part of the figure. After turning on the interaction among excited pairs, bare pairs are transformed into dressed pairs. Note that there is a one-to-one correspondence between the spikes in the two curves. As explained in the text there is another effect of this interaction, namely the formation of a collective surface mode at about 0.9. This feature has no counterpart in the dashed curve. Furthermore there is one more collective effect at about 1.2. For more explanation see text. Reproduced with permission from Ekardt, Pacheco and Schone, Comments on Atomic and Molecular Physics, 31, 291 (1995). Copyright by OPA (Overseas Publishers Association) B.V...

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