The Potential Energy Surface in Internal Coordinates¶
Within the Born–Oppenheimer approximation, the electronic energy is a function of the nuclear positions alone. Rather than working directly with \(3N\) Cartesian coordinates, it is far more natural to describe a molecule with internal coordinates: bond lengths \(r\), bond angles \(\theta\), and dihedral angles \(\phi\), which directly reflect chemical bonding and are invariant to overall translation and rotation of the molecule.
The Potential Energy Surface is then a function
of all of these internal coordinates together. A geometry optimization finds a stationary point of this function over all coordinates simultaneously. A coordinate scan instead studies how \(E\) behaves along just one of these coordinates, with the others allowed to adjust (relaxed scan) or held fixed (rigid scan).