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This largely self-contained book provides a thorough discussion of the mathematics of molecular dynamics and fills a gap in the theoretical knowledge of the subject. Most knowledge of molecular propagation is derived from the time-dependent Born-Oppenheimer approximation. This approximation breaks down at electron energy level crossings, which are prevalent in molecular systems. Hagedorn classifies generic, minimal multiplicity level crossings and derives a normal form for the electron Hamiltonian near each type of crossing. He then extends the time-dependent Born-Oppenheimer approximation to accommodate the propagation of nuclei through each type of electron energy level crossing.