  This Ph.D. tutorial discusses ultra-long-range Rydberg molecules, the exotic
bound states of a Rydberg atom and one or more ground state atoms immersed in
the Rydberg electron's wave function. This novel chemical bond is distinct from
an ionic or covalent bond, and is accomplished by a very different mechanism:
the Rydberg electron, elastically scattering off of the ground state atoms,
exerts a weak attractive force sufficient to form the molecule in long-range
oscillatory potential wells. In the last decade this topic has burgeoned into a
vibrant and mature subfield of atomic and molecular physics following the
rapidly developing capability of experiment to observe and manipulate these
molecules. This tutorial focuses on three areas where this experimental
progress has demanded more sophisticated theoretical descriptions: the
structure of polyatomic molecules, the influence of electronic and nuclear
spin, and the behavior of these molecules in external fields. The main results
are a collection of potential energy curves and electronic wave functions which
together describe the physics of Rydberg molecules. Additionally, to facilitate
future progress in this field, this tutorial provides a general overview of the
current state of experiment and theory.
