In the analysis of molecular systems, one is led to the study of a quantum mechanical Hamiltonian for the electrons that is a function of $n$ parameters that describe the positions of the nuclei. As the parameters are varied, the spectrum of the electron Hamiltonian can change. The way in which the graphs of the discrete eigenvalues cross one another depends on the symmetry group of the Hamiltonian function. We ify generic crossings of minimal multiplicity eigenvalues under all possible symmetry circumstances. For each of the eleven types of crossings, we derive a normal form for the Hamiltonian function near the crossing.