Potential energy surface
Encyclopedia
A potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics
and statistical mechanics
to model chemical reaction
s 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
surface
s) and their application in potential theory
, which associates and studies harmonic function
s 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 state
s), and local minima along this coordinate (which correspond to reactive intermediate
s).
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 the 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.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
to model chemical reaction
Chemical reaction
A chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Chemical reactions can be either spontaneous, requiring no input of energy, or non-spontaneous, typically following the input of some type of energy, such as heat, light or electricity...
s 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
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s) and their application in potential theory
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
, which associates and studies harmonic function
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
s in relation to these surfaces.
For example, the Morse potential
Morse potential
The Morse potential, named after physicist Philip M. Morse, is a convenient model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond...
and the simple harmonic potential well
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
are common one-dimensional potential energy surfaces (potential energy curves) in applications of quantum chemistry
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
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
Reaction coordinate
In chemistry, a reaction coordinate is an abstract one-dimensional coordinate which represents progress along a reaction pathway. It is usually a geometric parameter that changes during the conversion of one or more molecular entities....
(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 state
Transition state
The transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest energy along this reaction coordinate. At this point, assuming a perfectly irreversible reaction, colliding reactant molecules will always...
s), and local minima along this coordinate (which correspond to reactive intermediate
Reactive intermediate
In chemistry a reactive intermediate is a short-lived, high energy, highly reactive molecule. When generated in a chemical reaction it will quickly convert into a more stable molecule. Only in exceptional cases can these compounds be isolated and stored, e.g. low temperatures, matrix isolation...
s).
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 the numeric value of the potential energy surface .
Outside of physics and chemistry, "potential energy" surfaces may be associated with a cost function
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
, which may be explored in order to minimize the function.