  A fully analytical approximation for the observable characteristics of
many-electron atoms is developed via a complete and orthonormal hydrogen-like
basis with a single-effective charge parameter for all electrons of a given
atom. The basis completeness allows us to employ the secondary-quantized
representation for the construction of regular perturbation theory, which
includes in a natural way correlation effects, converges fast and enables an
effective calculation of the subsequent corrections. The hydrogen-like basis
set provides a possibility to perform all summations over intermediate states
in closed form, including both the discrete and continuous spectra. This is
achieved with the help of the decomposition of the multi-particle Green
function in a convolution of single-electronic Coulomb Green functions. We
demonstrate that our fully analytical zeroth-order approximation describes the
whole spectrum of the system, provides accuracy, which is independent of the
number of electrons and is important for applications where the Thomas-Fermi
model is still utilized. In addition already in second-order perturbation
theory our results become comparable with those via a multi-configuration
Hartree-Fock approach.
