Struct Buckingham 

pub struct Buckingham;

Buckingham (Exponential-6) potential for Van der Waals interactions.

Physics

Models short-range repulsion exponentially and long-range attraction with an $r^{-6}$ term, providing a more physically accurate repulsion wall than Lennard-Jones.

• Formula: $$ E = D_0 \left[ \frac{6}{\zeta-6} \exp\left(\zeta(1 - \frac{r}{R_0})\right) - \frac{\zeta}{\zeta-6} \left(\frac{R_0}{r}\right)^6 \right] $$
• Derivative Factor (diff): $$ D = -\frac{1}{r} \frac{dE}{dr} = \frac{6\zeta D_0}{r(\zeta-6)R_0} \left[ \exp\left(\zeta(1 - \frac{r}{R_0})\right) - \left(\frac{R_0}{r}\right)^7 \right] $$

Parameters

For computational efficiency, the physical parameters ($D_0, R_0, \zeta$) are pre-computed into the standard Buckingham form ($A, B, C$):

• a: Repulsion pre-factor $A = \frac{6 D_0}{\zeta-6} e^{\zeta}$.
• b: Repulsion decay constant $B = \zeta / R_0$.
• c: Attraction pre-factor $C = \frac{\zeta D_0 R_0^6}{\zeta-6}$.
• r_max_sq: Squared distance of the local energy maximum $r_{\text{max}}^2$.
• two_e_max: Twice the energy at the local maximum, $2 E(r_{\text{max}})$.

Pre-computation

Use Buckingham::precompute to convert physical constants into optimized parameters: $(D_0, R_0, \zeta) \to (A, B, C, r_{max}^2, 2E_{max})$. This involves computing the $A, B, C$ form and finding the reflection point via Newton’s method.

Inputs

• r_sq: Squared distance $r^2$ between two atoms.

Implementation Notes

• The kernel operates on the computationally efficient $A, B, C$ form.
• For $r < r_{\text{max}}$, the energy is reflected about the local maximum: $E_{\text{ref}}(r) = 2 E_{\text{max}} - E(r)$. This produces a repulsive wall that diverges to $+\infty$ as $r \to 0$ via the $C/r^6$ attraction term, while maintaining $C^1$ continuity at $r_{\text{max}}$ (where $E’(r_{\text{max}}) = 0$).
• A branchless sign-flip replaces the traditional constant penalty, providing physically correct short-range behavior at zero additional runtime cost.
• Requires one sqrt and one exp call, making it computationally more demanding than LJ.
• Power chain inv_r2 -> inv_r6 -> inv_r8 is used for the attractive term.

Implementations

impl Buckingham

pub fn precompute<T: Real>(d0: T, r0: T, zeta: T) -> (T, T, T, T, T)

Pre-computes optimized kernel parameters from physical constants.

Input

• d0: Energy well depth $D_0$.
• r0: Equilibrium distance $R_0$.
• zeta: Steepness parameter $\zeta$ (must be $> 6$).

Output

Returns (a, b, c, r_max_sq, two_e_max):

• a: Repulsion pre-factor $A = \frac{6 D_0}{\zeta-6} e^{\zeta}$.
• b: Repulsion decay constant $B = \zeta / R_0$.
• c: Attraction pre-factor $C = \frac{\zeta D_0 R_0^6}{\zeta-6}$.
• r_max_sq: Squared distance of the local energy maximum.
• two_e_max: Twice the energy at the local maximum.

Computation

$$ A = \frac{6 D_0}{\zeta - 6} e^{\zeta}, \quad B = \frac{\zeta}{R_0}, \quad C = \frac{\zeta D_0 R_0^6}{\zeta - 6} $$

The reflection point $r_{max}$ is found by solving $dE/dr = 0$ via Newton’s method.

Trait Implementations

impl Clone for Buckingham

fn clone(&self) -> Buckingham

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for Buckingham

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Default for Buckingham

fn default() -> Buckingham

Returns the “default value” for a type.

impl<T: Real> PairKernel<T> for Buckingham

fn energy(r_sq: T, (a, b, c, r_max_sq, two_e_max): Self::Params) -> T

Computes only the potential energy.

Formula

$$ E(r) = \begin{cases} A e^{-Br} - C/r^6 & r \ge r_{\text{max}} \ 2 E_{\text{max}} - (A e^{-Br} - C/r^6) & r < r_{\text{max}} \end{cases} $$

fn diff(r_sq: T, (a, b, c, r_max_sq, _two_e_max): Self::Params) -> T

Computes only the force pre-factor $D$.

Formula

$$ D(r) = \text{sign}(r) \left( \frac{A B e^{-Br}}{r} - \frac{6C}{r^8} \right) $$

where $\text{sign}(r) = +1$ for $r \ge r_{\text{max}}$ and $-1$ otherwise. At the maximum, $D(r_{\text{max}}) = 0$ from both sides, ensuring $C^1$ continuity.

This factor is defined such that the force vector can be computed by a single vector multiplication: $\vec{F} = -D \cdot \vec{r}$.

fn compute( r_sq: T, (a, b, c, r_max_sq, two_e_max): Self::Params, ) -> EnergyDiff<T>

Computes both energy and force pre-factor efficiently.

This method reuses intermediate calculations to minimize operations.

type Params = (T, T, T, T, T)

Associated constants/parameters required by the potential (e.g., $k, r_0$).

impl Copy for Buckingham