  In this chapter we focus first on the theoretical methods and relevant
computational approaches to calculate the electronic structure of atoms,
molecules, and clusters containing heavy elements for which relativistic
effects become significant. In particular, we discuss the mean-field
approximation of the Dirac equation for many-electron systems, and its
self-consistent numerical solution by using either radial mesh or Gaussian
basis sets. The former technique is appropriate for spherical symmetric
problems, such as atoms, while the latter approach is better suited to study
non-spherical non-periodic polycentric systems, such as molecules and clusters.
We also outline the pseudopotential approximation in relativistic context to
deal with the electron-ion interaction in extended systems, where the
unfavourable computational scaling with system size makes it necessary. As test
cases we apply our theoretical and numerical schemes to the calculation of the
electronic structure i) of the gold atom, and ii) of the superatom W@Au, where
the inclusion of spin-orbit effects is crucial to the accurate understanding of
the electronic properties. Furthermore, we describe the extension of our
relativistic approach to deal with nuclear reactions driven by the weak force,
such as the electron capture and $\beta$-decay, also at finite temperature in
astrophysical scenarios, using the Fermi-Dirac statistics. The latter processes
are indeed major drivers of the nucleosynthesis of the elements in stars and,
thus, their understanding is crucial to model the chemical evolution of the
Universe. Finally, we show the application of our relativistic quantum
mechanical framework to the assessment of the elastic differential scattering
cross section of electrons impinging on molecular targets, notably liquid
water.
