In
theoretical chemistry
Theoretical chemistry is the branch of chemistry which develops theoretical generalizations that are part of the theoretical arsenal of modern chemistry: for example, the concepts of chemical bonding, chemical reaction, valence, the surface o ...
, the Buckingham potential is a
formula proposed by
Richard Buckingham which describes the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulat ...
and
van der Waals energy for the interaction of two atoms that are not directly bonded as a function of the
interatomic distance . It is a variety of
interatomic potential
Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space.M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, Oxford, England, 198 ...
s.
:
Here,
,
and
are constants. The two terms on the right-hand side constitute a repulsion and an attraction, because their first
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s with respect to
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
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or intern ...
for
gas
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
eous
helium
Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
,
neon and
argon
Argon is a chemical element with the symbol Ar and atomic number 18. It is in group 18 of the periodic table and is a noble gas. Argon is the third-most abundant gas in Earth's atmosphere, at 0.934% (9340 ppmv). It is more than twice as ...
.
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 shell
In chemistry and atomic physics, an electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called the "K shell"), followed by the "2 shell" (or ...
s. "There is therefore some justification for choosing the repulsive part (of the potential) as an
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
". The Buckingham potential has been used extensively in simulations of
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the 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 dynamic "evolution" of t ...
.
Because the exponential term converges to a constant as
→
, while the
term diverges, the Buckingham potential becomes attractive as
becomes small. This may be problematic when dealing with a structure with very short interatomic distances, as any nuclei that cross a certain threshold will become strongly (and unphysically) bound to one another at a distance of zero.
[
]
Modified Buckingham (Exp-Six) potential
The modified Buckingham potential, also called the "exp-six" potential, is used to calculate the interatomic forces for gases based on Chapman and Cowling collision theory. The potential has the form
where is the interatomic potential between atom i and atom j, is the minimum potential energy, is the measurement of the repulsive energy steepness which is the ratio , is the value of where is zero, and is the value of which can achieve the minimum interatomic potential . This potential function is only valid when , as the potential will decay towards as . This is corrected by identifying , which is the value of at which the potential is maximized; when , the potential is set to infinity.
Coulomb–Buckingham potential
The Coulomb–Buckingham potential is an extension of the Buckingham potential for application to ionic systems (e.g. ceramic materials). The formula for the interaction is
:
where ''A'', ''B'', and ''C'' are suitable constants and the additional term is the electrostatic potential energy.
The above equation may be written in its alternate form as
:
where is the minimum energy distance, is a free dimensionless parameter and is the depth of the minimum energy.
Beest Kramer van Santen (BKS) potential
The BKS potential is a force field that may be used to simulate the interatomic potential
Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space.M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, Oxford, England, 198 ...
between Silica glass atoms. Rather than relying only on experimental data, the BKS potential is derived by combining ''ab initio'' quantum chemistry methods on small silica clusters to describe accurate interaction between nearest-neighbors, which is the function of accurate force field. The experimental data is applied to fit larger scale force information beyond nearest neighbors. By combining the microscopic
The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded as the scale be ...
and macroscopic
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic.
Overview
When applied to physical phenomena a ...
information, the applicability of the BKS potential has been extended to both the silica polymorphs and other tetrahedral network oxides systems systems that have same cluster structure, such as aluminophosphates, carbon
Carbon () is a chemical element with the symbol C and atomic number 6. It is nonmetallic and tetravalent—its atom making four electrons available to form covalent chemical bonds. It belongs to group 14 of the periodic table. Carbon mak ...
and silicon
Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic ta ...
.
The form of this interatomic potential is the usual Buckingham form, with the addition of a Coulomb force term. The formula for the BKS potential is expressed as
:
where is the interatomic potential between atom i and atom j, and are the charges magnitudes, is the distance between atoms, and , and are constant parameters based on the type of atoms.
The BKS potential parameters for common atoms are shown below:
An updated version of the BKS potential introduced a new repulsive term to prevent atom overlapping. The modified potential is taken as
where the constant parameters were chosen to have the following values for Silica glass:
References
External links
Buckingham potential
o
SklogWiki
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Theoretical chemistry
Computational chemistry
Thermodynamics
Chemical bonding
Intermolecular forces
Quantum mechanical potentials