  Eliashberg theory provides a theoretical framework for understanding the
phenomenon of superconductivity when pairing between two electrons is mediated
by phonons, and retardation effects are fully accounted for. However, when a
direct Coulomb interaction between two electrons is also present, this
interaction is only partially accounted for. In this work we use a well-defined
Hamiltonian, the Hubbard-Holstein model, to examine this competition more
rigorously, using exact diagonalization on a two-site system. We find that the
direct electron-electron repulsion between two electrons has a significantly
more harmful effect on pairing than suggested through the standard treatment of
this interaction.
