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 to a given value of such variables. The Born-Oppenheimer approximation, also known as the adiabatic approximation, is a technique used in quantum chemistry and condensed matter physics in order to de-couple the motion of nuclei and electrons. ... The Molecular Hamiltonian or Electronic Hamiltonian is an operator in quantum mechanics (and in particular quantum chemistry) which describes the motions of electrons and nuclei in a polyatomic molecule. ... Fig. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... A chemical reaction occurs when vapours of hydrogen chloride and ammonia meet to form a cloud of a new substance, ammonium chloride Chemical reaction is a process that results in the interconversion of chemical substances [1]. The substance or substances initially involved in a chemical reaction are called reactants. ...

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 in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... An open surface with X-, Y-, and Z-contours shown. ... Potential theory may be defined as the study of harmonic functions. ... In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ...

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. ... A harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement : where is a positive constant. ... Linus Pauling, as a pioneer of the valence bond theory, is one of the first quantum chemists. ... Fig. ...