to map the potential energy of the system as a function of geometric parameters , , and so on. The potential energy at given values of the geometric parameters is represented as a hyper-surface (when ) or a surface (when ). Mathematically, it can be written as For the quantum mechanical interpretation, a PES is typically defined within the Born–Oppenheimer approximation (in order to distinguish between nuclear and electronic motion and energy) which states that the nuclei are stationary relative to the electrons. In other words, the approximation allows the kinetic energy of the nuclei (or movement of the nuclei) to be neglected and therefore the nuclei repulsion is a constant value (as static point charges) and is only considered when calculating the total energy of the system. The electronic energy is then taken to depend parametrically on the nuclear coordinates, meaning a new electronic energy () must be calculated for each corresponding atomic configuration. PES is an important concept in computational chemistry and greatly aids in geometry and transition state optimization. Degrees of freedom An -atom system is defined by coordinates: for each atom. These degrees of freedom can be broken down to include 3 overall translational and 3 (or 2) overall rotational degrees of freedom for a non-linear system (for a linear system). However, overall translational or rotational degrees do not affect the potential energy of the system, which only depends on its internal coordinates. Thus an -atom system will be defined by (non-linear) or (linear) coordinates. These internal coordinates may be represented by simple stretch, bend, torsion coordinates, or symmetry-adapted linear combinations, or redundant coordinates, or normal modes coordinates, etc. For a system described by -internal coordinates a separate potential energy function can be written with respect to each of these coordinates by holding the other parameters at a constant value allowing the potential energy contribution from a particular molecular motion (or interaction) to be monitored while the other parameters are defined. Consider a diatomic molecule AB which can macroscopically visualized as two balls (which depict the two atoms A and B) connected through a spring which depicts the bond. As this spring (or bond) is stretched or compressed, the potential energy of the ball-spring system (AB molecule) changes and this can be mapped on a 2-dimensional plot as a function of distance between A and B, i.e. bond length. The concept can be expanded to a