Explain the origin of the molecular potential within the Born-Oppenheimer approximation

The Born-Oppenheimer approximation is a cornerstone of quantum chemistry that simplifies the calculation of molecular energies. By assuming that electrons move so quickly compared to the motion of the nuclei that their motion can be treated separately from the nuclei, it reduces a multi-dimensional problem down to two separate problems: one for each nuclear and electronic subsystem. The result is an approximate separation of energy into components due to each subsystem. This approximation was developed by Max Born and Robert Oppenheimer in 1927, paving the way for much deeper understanding of molecular structures and interactions.


Explain the origin of the molecular potential within the Born-Oppenheimer approximation


The underlying theory behind this approximation is based on wave mechanics, which describes physical systems in terms of wave functions rather than particles moving through space like classical mechanics does. According to wave mechanics, electron wave functions have an associated probability density that describes where they are most likely located at any given time. In molecules with multiple atoms bound together, these electron wave functions extend across all atoms; however, under certain conditions it can be assumed that these electron wave functions primarily depend upon only one nucleus at a time—a situation known as partial localization or diabatic states (where electrons remain localized about particular atomic centers). This allows us to decompose the single molecule problem into many smaller problems involving individual atom–electron pairs or “localized” states within a single molecule structure.

In effect then, we assume there exists some potential defined over nuclear coordinates such that when integrated over electronic variables yields total energy values identical to what we would expect from fully calculating both sets of variables together—this is known as the molecular potential energy surface (PES). We call this reduced form of PES created via Born-Oppenheimer approximation—the adiabatic PES since it assumes every nucleus remains in its own local minimum throughout all calculations. Subsequently then, an exact solution for this PES can be obtained by solving a set of Schrödinger equations independently for each atom–electron pair in order to produce diabatic states which represent different configurations corresponding to various positions/orientations of nuclei within same molecule; this step completes derivation process leading us back full circle—from original formulation down reducing number dimensions required describing system from three coupled differential equations solved simultaneously (full Hamiltonian) down just two uncoupled ones (adiabatic Hamiltonians), yielding final product: easy calculable yet accurate enough representation our system’s behavior with regard total energy value per molar unit volume being function alone internuclear distance(s) between atoms constituting molecular aggregate!

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