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Quantum Mechanical Potentials
This is a list of potential energy functions that are frequently used in quantum mechanics and have any meaning. One-dimensional potentials * Rectangular potential barrier * Delta potential (aka "contact potential") * Delta potential#Double delta potential, Double delta potential * Step potential * Periodic potential * Barrier potential * Gaussian potential * Eckart potential Wells * Quantum well * Potential well * Finite potential well * Infinite potential well * Double-well potential * Semicircular potential well * Circular potential well * Spherical potential well * Triangular potential well Interatomic potentials * Interatomic potential * Bond order potential * EAM potential * Coulomb potential * Buckingham potential * Lennard-Jones potential * Morse potential * Morse/Long-range potential * Pöschl–Teller_potential#Rosen–Morse_potential, Rosen–Morse potential * Trigonometric Rosen–Morse potential * Stockmayer potential * Pöschl–Teller potential * Axilrod–Tell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potential Energy
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity or those in a spring. The term ''potential energy'' was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to the ancient Greek philosopher Aristotle's concept of Potentiality and Actuality, ''potentiality''. Common types of potential energy include gravitational potential energy, the elastic potential energy of a deformed spring, and the electric potential energy of an electric charge and an electric field. The unit for energy in the International System of Units (SI) is the joule (symbol J). Potential energy is associated with forces that act on a body in a way that the total Work (physics), work done by these forces on the body depends only on the initial and final positions of the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Potential Well
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated withi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axilrod–Teller Potential
In molecular physics, the Axilrod–Teller potential (also known as the Axilrod–Teller–Muto or ATM potential) describes the interaction potential (energy) between three atoms, capturing effects beyond simple pairwise attractions. It builds on the London dispersion forces, using perturbation theory to correct for the influence of a third atom. Its strength depends on properties of the atoms and their arrangement in space. This potential is often used in models of rare gases and molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the Motion (physics), physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamics ( ... simulations to study complex attraction effects. Formula The ATM potential is given by: : V_= E_ \left \frac \right where r_ is the distance between atoms i and j, and \gamma_ is the angle between the vectors \mathbf_ and \mathbf_. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pöschl–Teller Potential
In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl (credited as G. Pöschl) and Edward Teller, is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by .... Definition In its symmetric form is explicitly given by : V(x) =-\frac\mathrm^2(x) and the solutions of the time-independent Schrödinger equation : -\frac\psi''(x)+ V(x)\psi(x)=E\psi(x) with this potential can be found by virtue of the substitution u=\mathrm, which yields : \left 1-u^2)\psi'(u)\right+\lambda(\lambda+1)\psi(u)+\frac\psi(u)=0 . Thus the solutions \psi(u) are just the Legendre functions P_\lambda^\mu(\tanh(x)) with E=-\frac, and \lambda=1, 2, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stockmayer Potential
The Stockmayer potential is a mathematical model for representing the interactions between pairs of atoms or molecules. It is defined as a Lennard-Jones potential with a point electric dipole moment The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system: that is, a measure of the system's overall Chemical polarity, polarity. The International System of Units, SI unit for electric .... A Stockmayer liquid consists of a collection of spheres with point dipoles embedded at the centre of each. These spheres interact both by Lennard-Jones and dipolar interactions. In the absence of the point dipoles, the spheres face no rotational friction and the translational dynamics of such LJ spheres have been studied in detail. This system, therefore, provides a simple model where the only source of rotational friction is dipolar interactions. The interaction potential may be written as V(r) = 4 \varepsilon_\left left(\frac\right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Rosen–Morse Potential
The trigonometric Rosen–Morse potential, named after the physicists Nathan Rosen and Philip M. Morse, is among the exactly solvable quantum mechanical potentials. Definition In dimensionless units and modulo additive constants, it is defined as where r is a relative distance, \lambda is an angle rescaling parameter, and R is so far a matching length parameter. Another parametrization of same potential is which is the trigonometric version of a one-dimensional hyperbolic potential introduced in molecular physics by Nathan Rosen and Philip M. Morse and given by, a parallelism that explains the potential's name. The most prominent application concerns the V_^(\chi) parametrization, with \ell non-negative integer, and is due to Schrödinger who intended to formulate the hydrogen atom problem on Albert Einstein's closed universe, R^1\otimes S^3, the direct product of a time line with a three-dimensional closed space of positive constant curvature, the hypersphere S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pöschl–Teller Potential
In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl (credited as G. Pöschl) and Edward Teller, is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by .... Definition In its symmetric form is explicitly given by : V(x) =-\frac\mathrm^2(x) and the solutions of the time-independent Schrödinger equation : -\frac\psi''(x)+ V(x)\psi(x)=E\psi(x) with this potential can be found by virtue of the substitution u=\mathrm, which yields : \left 1-u^2)\psi'(u)\right+\lambda(\lambda+1)\psi(u)+\frac\psi(u)=0 . Thus the solutions \psi(u) are just the Legendre functions P_\lambda^\mu(\tanh(x)) with E=-\frac, and \lambda=1, 2, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse/Long-range Potential
The Morse/Long-range potential (MLR potential) is an interatomic interaction model for the potential energy of a diatomic molecule. Due to the simplicity of the regular Morse potential (it only has three adjustable parameters), it is very limited in its applicability in modern spectroscopy. The MLR potential is a modern version of the Morse potential which has the correct theoretical long-range form of the potential naturally built into it. It has been an important tool for spectroscopists to represent experimental data, verify measurements, and make predictions. It is useful for its extrapolation capability when data for certain regions of the potential are missing, its ability to predict energies with accuracy often better than the most sophisticated ''ab initio'' techniques, and its ability to determine precise empirical values for physical parameters such as the dissociation energy, equilibrium bond length, and long-range constants. Cases of particular note include: # the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse Potential
The Morse potential, named after physicist Philip M. Morse, is a convenient Interatomic potential, interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the oscillation, vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity of real bonds and the non-zero transition probability for overtone band, overtone and Hot band#Combination bands, combination bands. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. Due to its simplicity (only three fitting parameters), it is not used in modern spectroscopy. However, its mathematical form inspired the MLR (Morse/Long-range potential, Morse/Long-range) potential, which is the most popular potential energy function used for fitting spectroscopic data. Potential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lennard-Jones Potential
In computational chemistry, molecular physics, and physical chemistry, the Lennard-Jones potential (also termed the LJ potential or 12-6 potential; named for John Lennard-Jones) is an intermolecular pair potential. Out of all the intermolecular potentials, the Lennard-Jones potential is probably the one that has been the most extensively studied. It is considered an archetype model for simple yet realistic intermolecular interactions. The Lennard-Jones potential is often used as a building block in molecular models (a.k.a. force fields) for more complex substances. Many studies of the idealized "Lennard-Jones substance" use the potential to understand the physical nature of matter. Overview The Lennard-Jones potential is a simple model that still manages to describe the essential features of interactions between simple atoms and molecules: Two interacting particles repel each other at very close distance, attract each other at moderate distance, and eventually stop intera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buckingham Potential
In theoretical chemistry, the Buckingham potential is a formula proposed by Richard Buckingham which describes the Pauli exclusion principle and van der Waals energy \Phi_(r) for the interaction of two atoms that are not directly bonded as a function of the interatomic distance r. It is a variety of interatomic potentials. :\Phi_(r) = A \exp \left(-Br\right) - \frac Here, A, B and C are constants. The two terms on the right-hand side constitute a repulsion and an attraction, because their first derivatives with respect to r are negative and positive, respectively. Buckingham proposed this as a simplification of the Lennard-Jones potential, in a theoretical study of the equation of state for gaseous helium, neon and argon. As explained in Buckingham's original paper and, e.g., in section 2.2.5 of Jensen's text,F. Jensen, ''Introduction to Computational Chemistry'', 2nd ed., Wiley, 2007, the repulsion is due to the interpenetration of the closed electron shells. "There is there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coulomb Potential
Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physics), work needed to move a test charge from a reference point to a specific point in a static electric field. The test charge used is small enough that disturbance to the field is unnoticeable, and its motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is Earth (electricity), earth or a point at infinity, although any point can be used. In classical electrostatics, the electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is a scalar (physics), scalar quantity denoted by or occasi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
