Full Solution of the Kohn-Sham Equations

One of the most important uses of TDDFT is the calculation of photoabsorption spectra. This problem can be solved in TDDFT either by propagating the time-dependent Kohn-Sham equations [39] or by using linear-response theory. In this section we will be concerned by the former, relegating the latter to the next section. 

Let be the ground-state Kohn-Sham wave-functions for the system under study. We prepare the initial state for the time propagation by exciting the electrons with the electric field v r,t) = —kox S t), where = x,y,z. The amplitude ko must be small in order to keep the response of the system linear and dipolar. Through this prescription all frequencies of the system are excited with equal weight. At t = 0+ the initial state for the time evolution reads 

The Kohn-Sham orbitals are then further propagated during a finite time. The dynamical polarizability can be obtained from 

In the last expression 5n r,cu) stands for the Fourier transform of n r,t) — n r), where n r) is the ground-state density of the system. The quantity that is usually measured in experiments, the photo-absorption cross-section, is essentially proportional to the imaginary part of the djmamical polarizability averaged over the three spatial directions 

To illustrate the method, we present, in Fig. 4.2, the excitation spectrum of benzene calculated within the LDA/ALDA. The agreement with experiment is quite remarkable, especially when looking at the tt —tt resonance at around 7 eV. The spurious peaks that appear in the calculation at higher energies are artifacts caused by an insufficient treatment of the unbound states. We furthermore observe that such good results are routinely obtained when applying the LDA/ALDA to several finite systems, from small molecules to metallic clusters and biological systems. 

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