In this chapter, we apply the formalism of hydrogenic and multi-electron atoms and build the periodic table of ground-state elements. Examination of the table shows that all elements in a given column share the same Russell-Saunders state symbol; they have identical orbital and total angular momentum states and valence electron multiplicities. These columns are formally grouped, and we show how each group shares the same spectral characteristics (the transition energies differ, but the relationships between transitions are identical from one element to another in a group). We then introduce the idea of iso-electronic sequences, which neatly explain the many lithium-like and sodium-like ions (CIV, NV, OVI, NeVIII, MgII, etc.) that have hydrogenic-like spectral series, including zero-volt resonant fine-structure doublets. We then provide accurate tables of ionization potentials and describe the physical reasons for the ion-to-ion trends in these potentials. We conclude the chapter with a complete suite of Grotrian diagrams (visual representations of the energy states and allowed electron transitions) for ions commonly studied using quasar absorption lines.

The energy structures and transition energies of single-electron atoms and ions are presented. Five Nobel Prizes in Physics were awarded for the theories discussed in this chapter. We first review the Bohr model, which was based on quantized angular momentum and classical circular orbits. The wave model of Schrödinger followed, in which spherical boundary conditions quantized polar and azimuthal standing waves. The energies were identical to Bohr’s, but transition selection rules dictated the change in angular momentum of the system during absorption and emission. Dirac incorporated electron spin and relativistic energies, resulting in energy shifts and fine structure splitting of the energy levels for non-zero angular momentum states. Feynman and Swinger incorporated quantization of the electric vector potential. This physics broke energy degeneracies in the Dirac model and correctly predicted the famous Lamb shift. In this chapter, each of these models are described in detail. The final full characterization of the energy levels and transitions are presented. The chapter ends with a discussion on isotope shifting and transitions to the continuum (ionization/recombination).