The “many electrons problem” for determining atomic energy levels cannot be solved analytically. It must be solved numerically using approximation techniques applied to each ion for each element. The industry standard approach is called the Hartree-Fock method, which incorporates a three-tiered Hamiltonian approximation. In this chapter we describe how these approximations yield the Russell-Saunders vector model, for which we describe quantized vector addition. We then summarize the Russell-Saunders term and state symbols so commonly used to precisely notate atomic transitions. It is through this formalism that we come to understand that a given energy structure/transition does not describe a single active or optical electron but applies to the full bound multi-electron ionic system. We also describe intermediate coupling schemes and the j-j coupling scheme for heavier nuclei. We then derive the line strengths and oscillator strengths for both term-averaged and fine-structure transitions. Line emission power and line absorption cross sections are derived and the dipole selection rules for multi-electron ions are presented.

For a proper quantum mechanical description of multiple-particle systems, we must account for the indistinguishability of fundamental particles. The symmetrization postulate requires that the quantum state vector of a system of identical particles be either symmetric or antisymmetric with respect to exchange of any pair of identical particles within the system. Nature dictates that integer spin particles – bosons – have symmetric states, while half-integer spin particles – fermions – have antisymmetric states. The best-known manifestation of this is the Pauli exclusion principle, which limits the number of electrons in given atomic levels and leads to the structure of the periodic table.

Our modern understanding of atoms, molecules, solids, atomic nuclei, and elementary particles is largely based on quantum mechanics. Quantum mechanics grew in the mid-1920s out of two independent developments: the matrix mechanics of Werner and the wave mechanics of Erwin Schrödinger. For the most part this chapter follows the path of wave mechanics, which is more convenient for all but the simplest calculations. The general principles of the wave mechanical formulation of quantum mechanics are laid out and provide a basis for the discussion of spin, identical particles. and scattering processes. The general principles are supplemented with the canonical formalism to work out the Schrödinger equation for charged particles in a general electromagnetic field. The chapter ends with the unification of the approaches of wave and matrix mechanics by Paul Dirac, and a modern approach, known as Hilbert space, is briefly described.