Born-Oppenheimer energy surface from molecular vibrations

The Born-Oppenheimer (B.O.) approximation is the cornerstone of most theories dealing with the effect of isotopic substitution on molecular properties. Within the framework of this approximation, the potential energy surface for the vibrational-rotational motions of a molecular system depends on the nuclear charges of the substituent atoms and on the number of electrons in the system but is independent of the masses of the nuclei. Thus isotope effects arise from the fact that nuclei of different mass move differently on the same potential surface. 

Fig. 3. Vibrational population distributions of N2 formed in associative desorption of N-atoms from ruthenium, (a) Predictions of a classical trajectory based theory adhering to the Born-Oppenheimer approximation, (b) Predictions of a molecular dynamics with electron friction theory taking into account interactions of the reacting molecule with the electron bath, (c) Born—Oppenheimer potential energy surface, (d) Experimentally-observed distribution. The qualitative failure of the electronically adiabatic approach provides some of the best available evidence that chemical reactions at metal surfaces are subject to strong electronically nonadiabatic influences. (See Refs. 44 and 45.)...

The equilibrium structure of a molecule is conceived as the hypothetical vibrationless state, i.e. the state in which all intramolecular modes of vibration are imagined as frozen at the minima of their potential energy curves. This concept, and indeed the entire concept of a potential energy surface which arises from the electronic structure of a molecule, depends on the Born-Oppenheimer approximation which is virtually always made in studies of molecular structure. 

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