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 ${\displaystyle \Phi _{12}(r)}$ for the interaction of two atoms that are not directly bonded as a function of the interatomic distance ${\displaystyle r}$. It is a variety of interatomic potentials.

${\displaystyle \Phi _{12}(r)=A\exp \left(-Br\right)-{\frac {C}{r^{6}}}}$

Here, ${\displaystyle A}$, ${\displaystyle B}$ and ${\displaystyle C}$ are constants. The two terms on the right-hand side constitute a repulsion and an attraction, because their first derivatives with respect to ${\displaystyle r}$ 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.^[1]

As explained in Buckingham's original paper and, e.g., in section 2.2.5 of Jensen's text,^[2] the repulsion is due to the interpenetration of the closed electron shells. "There is therefore some justification for choosing the repulsive part (of the potential) as an exponential function". The Buckingham potential has been used extensively in simulations of molecular dynamics.

Because the exponential term converges to a constant as ${\displaystyle r}$→${\displaystyle 0}$, while the ${\displaystyle r^{-6}}$ term diverges, the Buckingham potential becomes attractive as ${\displaystyle r}$ 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.^[2]

Modified Buckingham (Exp-Six) potential[]

The modified Buckingham potential, also called exp-six potential, is proposed to calculate the interatomic forces for gases based on Chapman and Cowling collision theory.^[3] The potential has form

${\displaystyle \Phi _{12}(r)={\frac {\epsilon }{1-6/\alpha }}\left[{\frac {6}{\alpha }}\exp \left[\alpha \left(1-{\frac {r}{r_{min}}}\right)\right]-\left({\frac {r_{min}}{r}}\right)^{6}\right]}$

where ${\displaystyle \Phi _{12}(r)}$ is the interatomic potential between atom i and atom j, ${\displaystyle \epsilon }$ is the minimum potential energy, ${\displaystyle \alpha }$ is the measurement of the repulsive energy steepness which is the ratio ${\displaystyle \sigma /r_{min}}$, ${\displaystyle \sigma }$ is the value of ${\displaystyle r}$where potential ${\displaystyle \Phi _{12}(r)}$ is zero, and ${\displaystyle r_{min}}$ is the value of ${\displaystyle r}$ which can achieve minimum interatomic potential ${\displaystyle \epsilon }$. This potential function can only be used when ${\displaystyle r>r_{max}}$to calculate a valid value. The ${\displaystyle r>r_{max}}$ is the value of ${\displaystyle r}$ to achieve maximum potential ${\displaystyle \Phi _{12}(r)}$.When ${\displaystyle r\leq {r_{max}}}$, the potential is set to infinity.

Coulomb–Buckingham potential[]

Example Coulomb–Buckingham potential curve.

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

${\displaystyle \Phi _{12}(r)=A\exp \left(-Br\right)-{\frac {C}{r^{6}}}+{\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}r}}}$

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

${\displaystyle \Phi (r)=\varepsilon \left\{{\frac {6}{\alpha -6}}\exp \left(\alpha \left[1-{\frac {r}{r_{0}}}\right]\right)-{\frac {\alpha }{\alpha -6}}\left({\frac {r_{0}}{r}}\right)^{6}\right\}+{\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}r}}}$

where ${\displaystyle r_{0}}$ is the minimum energy distance, ${\displaystyle \alpha }$ is a free dimensionless parameter and ${\displaystyle \varepsilon }$ is the depth of the minimum energy.

Beest Kramer van Santen (BKS) potential[]

The BKS potential is one of most popular force field used to solve the problem of simulate the interatomic potentialbetween Silica glass atoms.^[4] 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 and macroscopic 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 and silicon.

The form of this interatomic potential is usual Buckingham form which contains Coulomb force term and covalent contribution. The formula for the BKS potential is expressed as

${\displaystyle \Phi _{12}(r)=\left[A_{12}\exp \left(-B_{12}r_{12}\right)-{\frac {C_{12}}{r_{12}^{6}}}\right]+{\frac {q_{1}q_{2}}{r_{12}}}}$

where ${\displaystyle \Phi _{12}(r)}$ is the interatomic potential between atom i and atom j, ${\displaystyle q_{1}}$and ${\displaystyle q_{2}}$ are the charges magnitudes, ${\displaystyle r_{12}}$ is the distance between atoms, ${\displaystyle A_{ij}}$,${\displaystyle B_{ij}}$ and ${\displaystyle C_{ij}}$ are constant parameters based on the type of atoms.

The short-range contribution is represented by the first term of the BKS potential formula, which includes both covalent contribution and repulsion contribution inside the small cluster, while the longe-range contribution is calculated through the second Coulomb force term, which shows the electrostatic interaction.^[5] The decisive factor for the accuracy of BKS potential energy is the accuracy of short-range interaction constant parameters, which can be computed through the comparison with ab initio potential surface.

The BKS potential parameters for common atoms are shown below:^[5]

The updated version of BKS potential introduce a new repulsive term to prevent atom overlapping induced by van den Waals force.^[6]

${\displaystyle \Phi _{12}(r)=\left[A_{12}\exp \left(-B_{12}r_{12}\right)-{\frac {C_{12}}{r_{12}^{6}}}\right]+{\frac {q_{1}q_{2}}{r_{12}}}+{\frac {D_{12}}{r_{12}^{24}}}}$

where the constant parameter D has settled value for Silica glass:

References[]

External links[]

• Buckingham potential on SklogWiki