Fundamentals of gauge-invariant relativistic many-body perturbation theory (PT) with optimized ab initio zeroth approximation in theory of relativistic multi-electron systems are presented. The problem of construction of optimal one-electron representation is directly linked with a problem of the correct accounting for multielectron exchange-correlation effects and gauge-invariance principle fulfilling in atomic calculations. New approach to construction of optimal PT zeroth approximation is based on accurate treating the lowest order multielectron effects, in particular, the gauge-dependent radiative contribution for the certain class of photon propagator (for instance, the Coulomb, Feynman, Babushkin ones) gauge. This value is considered to be a typical representative of important multielectron exchange-correlation effects, whose minimization is a reasonable criteria in the searching for optimal PT one-electron orbital basis. This procedure derives an undoubted profit in the routine many-body calculations as it provides the way of refinement of the atomic characteristics calculations, based on the "first principles". The relativistic density-functional approximation is taken as the zeroth one. There have taken into account all exchangecorrelation corrections of the second order and dominated classes of the higher orders diagrams (polarization interaction, quasiparticles screening, etc.). New form of multi-electron polarization functional is used. As illustration, the results of computing energies, transition probabilities for some heavy ions are presented.

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An investigation of spectra, radiative and autoionization characteristics for the rare-earth elements is of a great interest as for development atomic spectroscopy as different applications in plasma chemistry, astrophysics, laser physics, quantum electronics etc. We present and review the results of studying spectra and autoionization resonance characteristics of a few lanthanide elements (ytterbium and thulium). Computing the spectra and autoionization resonance parameters is carried out within the relativistic many-body perturbation theory (RMBPT) and generalized relativistic energy approach. The accurate results on the autoionization resonance energies and widths are presented with correct accounting for the exchange-correlation and relativistic corrections and compared with other available theoretical and experimental data. In this chapter, we present a brief review of the theoretical and experimental works on spectroscopy of some lanthanide atoms. Spectroscopy of the Rydberg autoionization resonances in rare-earth atoms in an external electromagnetic field is expected to be very complex and unusual.

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