When Does It Not Work

In the previous sections we showed the results of several TDDFT calculations, most of them agreeing quite well with experiment. Clearly no physical theory works for all systems and situations, and TDDFT is not an exception. It is the purpose of this section to show some examples where the theory does not work. However, before proceeding with our task, we should specify what we mean by failures of TDDFT . TDDFT is an exact reformulation of the time-dependent many-body Schrodinger equation - it can only fail in situations where quantum-mechanics also fails. The key approximation made in practical applications is the approximation for the xc potential. Errors in the calculations should therefore be imputed to the functional used. As a large majority of TDDFT calculations use the ALDA or the adiabatic GGA, we will be mainly interested in the errors caused by these approximate functionals. Furthermore, and as we already mentioned in the previous section, there 

Our first example is the calculation of optical properties of long conjugated molecular chains [54]. For these systems, the local or gradient-corrected approximations can give overestimations of several orders of magnitude. The problem is related to a non-local dependence of the xc potential In a system with an applied electric field, the exact xc potential develops a linear part that counteracts the applied field [54,55]. This term is completely absent in both the LDA and the GGA, but is present in more non-local functionals like the EXX. 

A related problem occurs in solids [56]. In fact, the ALDA does not work properly for the calculation of excitations of non-metallic solids, especially in systems like wide-band gap semiconductors. For infinite systems, the Coulomb potential is (in momentum space) 47r/q. It is then clear from the response (see 4.69) that if /xc is to correct the non-interacting response for 7 0 it will have to contain a term that behaves asymptotically as 

1/g when q 0. This is not the case for the local or gradient-corrected approximations. Several attempts have been made to correct this problem from which we mention [57,58,59,60]. 

Another problematic system for the ALDA is the streched H2 molecule [61,62]. Prom a comparison with exact results it was found that the ALDA fails to reproduce even qualitatively the shape of the potential curves for the and states. A detailed analysis of the problem shows that the failure is related to the breakdown of the simple local approximation to the kernel. 

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The series of standard solutions method works satisfactorily most of the time. When does it not work well ... 

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