0 Calculations of single particle spectra in density functional theory

In the context of the density functional theory we consider the single particle ex-citation spectra of electron systems. As a result, we have related the single particle excitations with the eigenvalues of the corresponding Kohn-Sham equations. In a case when the exchange correlation functional is approximated by the exchange functional the coupling equations are very simple, while the single particle spectra do not coincide neither with the eigenvalues of the Kohn-Sham equations nor with the ones of the Hartree-Fock equations. The density functional theory (DFT), that originated from the pioneer work of Hohen-berg and Kohn [1], has been extremely effective in describing the ground state of finite many-electron systems. Such a success gave birth to many papers concerned with generalizations of DFT, which would permit to describe the excitation spectra also (see e.g. [2-5]). The generalizations, on the theoretical grounds, originated mainly from the Runge-Gross theorem , which helped to transform DFT into the time-dependent density functional theory TDDFT [6]. Both, DFT and TDDFT, are based on the one-to-one correspondence between particle densities of the considered systems and external potentials acting upon these particles. For the sake of simplicity and definiteness, while dealing with TDDFT, we assume that the time dependent part of the considered electronic system's density ρ(r, t), which is created under the action of an external time-dependent field λv ext (r, t), is developing from the system's ground state. Unfortunately, the one-to-one correspondence establishes only the existence of the functionals in principle, leaving aside a very important question on how one can construct them in reality. This is why the successes of DFT and TDDFT strongly depend upon the availability of good approximations for the functionals. This shortcoming was resolved to large extent in [2, 7, 8] where an exact equations connecting the action