  The strong interaction between individual Rydberg atoms provides a powerful
tool exploited in an ever-growing range of applications in quantum information
science, quantum simulation, and ultracold chemistry. One hallmark of the
Rydberg interaction is that both its strength and angular dependence can be
fine-tuned with great flexibility by choosing appropriate Rydberg states and
applying external electric and magnetic fields. More and more experiments are
probing this interaction at short atomic distances or with such high precision
that perturbative calculations as well as restrictions to the leading
dipole-dipole interaction term are no longer sufficient. In this tutorial, we
review all relevant aspects of the full calculation of Rydberg interaction
potentials. We discuss the derivation of the interaction Hamiltonian from the
electrostatic multipole expansion, numerical and analytical methods for
calculating the required electric multipole moments, and the inclusion of
electromagnetic fields with arbitrary direction. We focus specifically on
symmetry arguments and selection rules, which greatly reduce the size of the
Hamiltonian matrix, enabling the direct diagonalization of the Hamiltonian up
to higher multipole orders on a desktop computer. Finally, we present example
calculations showing the relevance of the full interaction calculation to
current experiments. Our software for calculating Rydberg potentials including
all features discussed in this tutorial is available as open source.
