User:Ruoxia/Buckingham potential

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 potential 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 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


 * $$\Phi_{12}(r) = \left[ A_{12} \exp \left(-B_{12}r_{12}\right) - \frac{C_{12}}{r_{12}^6}\right] + \frac{q_1q_2}{r_{12}} $$

where $$\Phi_{12}(r) $$ is the interatomic potential between atom i and atom j, $$q_1 $$and $$q_2 $$ are the charges magnitudes, $$r_{12} $$ is the distance between atoms, $$A_{ij} $$,$$B_{ij} $$ and $$C_{ij} $$ 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. 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 : The updated version of BKS potential introduce a new repulsive term to prevent atom overlapping induced by van den Waals force.

$$\Phi_{12}(r) = \left[ A_{12} \exp \left(-B_{12}r_{12}\right) - \frac{C_{12}}{r_{12}^6}\right] + \frac{q_1q_2}{r_{12}} + \frac{D_{12}}{r_{12}^{24}} $$

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

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. The potential has form

$$\Phi_{12}(r) = \frac{\epsilon}{1-6/\alpha}\left[ \frac6\alpha \exp \left[\alpha\left(1-\frac{r}{r_{min}}\right)\right] - \left(\frac{r_{min}}{r}\right)^6\right] $$

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