The Born-Oppenheimer approximation separates electronic and nuclear motion. At leading order, fixed nuclear positions and electric charges specify an electronic potential energy curve, on which the atomic nuclei move with their own masses.
A basis state chosen to retain a useful electronic character, such as a localized charge arrangement, as the nuclei move. A diabatic state need not diagonalize the electronic Hamiltonian; off-diagonal coupling between such states produces the adiabatic states.
An electronic energy eigenstate at a fixed nuclear geometry. Its wavefunction and energy change with nuclear positions. Two same-symmetry states can exchange their dominant electronic character through an avoided crossing.
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The Born-Oppenheimer approximation is a fundamental concept in molecular quantum mechanics that simplifies the study of molecular systems by decoupling the motion of nuclei and electrons. The core idea is based on the significant difference in mass between nuclei (which are heavy) and electrons (which are much lighter). This mass difference leads to different time scales for their motions.
