Classification and normal forms for avoided crossings of quantum-mechanical energy levels

When using the Born-Oppenheimer approximation for molecular systems, one encounters a quantum mechanical Hamiltonian for the electrons that depends on several parameters that describe the positions of the nuclei. As these parameters are varied, the spectrum of the electron Hamiltonian may vary. In particular, discrete eigenvalues may approach very close to one another at `avoided crossings' of the electronic energy levels. We give a definition of an avoided crossing and classify generic avoided crossings of minimal multiplicity eigenvalues. There are six distinct types that depend on the dimension of the nuclear-configuration space and on the symmetries of the electron Hamiltonian function.