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The Born-Oppenheimer approximation is a way to simplify the complicated Schrödinger equation for a molecule. The nucleus and electrons are attracted to each other with the same magnitude of electric charge, thus they exert the same force and momentum. While exerting the same kind of momentum, the nucleus, with a much larger mass in comparison to electron’s mass, will have a very small velocity that is almost negligible. Born-Oppenheimer takes advantage of this phenomenon and makes the assumption that since the nucleus is way heavier in mass compared to the electron, its motion can be ignored while solving the electronic Schrödinger equation; that is, the nucleus is assumed to be stationary while electrons move around it. The motion of the nuclei and the electrons can be separated and the electronic and nuclear problems can be solved with independent wavefunctions.
The wavefunction for the molecule thus becomes:
Ψmolecule= Ψelectronx Ψnuclei
The principle of Born-Oppenheimer can be applied to calculate the bond length energy between molecules. By focusing on the specific separation between nucleus and electron, their wavefunction can be calculated. Thus, a molecule’s energy in relationship with its bond length can be examined.