A potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems. The "(hyper)surface" name comes from the fact that the total energy of an atom arrangement can be represented as a curve or (multidimensional) surface, with atomic positions as variables. The best visualization for a layman would be to think of a landscape, where going North-South and East-West are two independent variables (the equivalent of two geometrical parameters of the molecule), and the height of the land we are on would be the energy associated with a given value of such variables.
There is a natural correspondence between potential energy surfaces as they exist (as polynomial surfaces) and their application in potential theory, which associates and studies harmonic functions in relation to these surfaces.
For example, the Morse potential and the simple harmonic potential well are common one-dimensional potential energy surfaces (potential energy curves) in applications of quantum chemistry and physics.
These simple potential energy surfaces (which can be obtained analytically), however, only provide an adequate description of the very simplest chemical systems. To model an actual chemical reaction, a potential energy surface must be created to take into account every possible orientation of the reactant and product molecules and the electronic energy of each of these orientations.
Typically, the electronic energy is obtained for each of tens of thousands of possible orientations, and these energy values are then fitted numerically to a multidimensional function. The accuracy of these points depends upon the level of theory used to calculate them. For particularly simple surfaces (such as H + H2), the analytically derived LEPS (London-Eyring-Polanyi-Sato) potential surface may be sufficient. Other methods of obtaining such a fit include cubic splines, Shepard interpolation, and other types of multidimensional fitting functions.
Once the potential energy surface has been obtained, several points of interest must be determined. Perhaps the most important is the global minimum for the energy value. This global minimum, which can be found numerically, corresponds to the most stable nuclear configuration. Other interesting features are the reaction coordinate (the path along the potential energy surface that the atoms "travel" during the chemical reaction), saddle points or local maxima along this coordinate (which correspond to transition states), and local minima along this coordinate (which correspond to reactive intermediates).
Single-point energy is the potential energy of a molecule for a given arrangement of the atoms in the molecule. The single point energy is simply a numeric value of the potential energy surface.
Outside of physics and chemistry, "potential energy" surfaces may be associated with a cost function, which may be explored in order to minimize the function.
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