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 adiabatic theorem is an important concept in Quantum mechanics. In Atomic molecular and optical physics as well as in Quantum chemistry, molecular Hamiltonian is the name given to the Hamiltonian representing the Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics A chemical reaction is a process that always results in the interconversion of Chemical substances The substance or substances initially involved in a chemical reaction are called 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. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. Potential theory may be defined as the study of Harmonic functions Definition and comments The term "potential theory" arises from the fact that In Mathematics, Mathematical physics and the theory of Stochastic processes a harmonic function is a twice continuously differentiable function
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. The Morse potential, named after physicist Philip M Morse, is a convenient model for the Potential energy of a Diatomic molecule. This article is about the harmonic oscillator in classical mechanics Quantum chemistry is a branch of Theoretical chemistry, which applies Quantum mechanics and Quantum field theory to address issues and problems in Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons
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. In the mathematical field of Numerical analysis, a spline is a special function defined Piecewise by Polynomials In interpolating
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). In Chemistry, a reaction coordinate is an abstract one-dimensional Coordinate which represents progress along a Reaction pathway. The transition state of a Chemical reaction is a particular configuration along the Reaction coordinate. A reaction intermediate or an intermediate is a Molecular entity that is formed from the reactants (or preceding intermediates and reacts further to give the directly
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. In Mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function