Struct SplinedBuckingham 

pub struct SplinedBuckingham;

Splined Buckingham (Exp-6) potential with $C^2$ continuous short-range regularization.

Physics

This kernel enhances the standard Buckingham potential by replacing the problematic short-range region ($r < r_{spline}$) with a quintic (5th degree) polynomial, $P_5(r)$. This guarantees that the energy, force ($C^1$), and force derivative ($C^2$) are continuous everywhere, which is critical for stable molecular dynamics simulations.

• Formula: $$ E(r) = \begin{cases} D_0 \left[ \frac{6}{\zeta-6} \exp\left(\zeta(1 - \frac{r}{R_0})\right) - \frac{\zeta}{\zeta-6} \left(\frac{R_0}{r}\right)^6 \right] & r \ge r_{spline} \ P_5(r) = \sum_{i=0}^{5} p_i r^i & r < r_{spline} \end{cases} $$
• Derivative Factor (diff): $$ D = -\frac{1}{r} \frac{dE}{dr} = \begin{cases} \frac{6\zeta D_0}{r(\zeta-6)R_0} \left[ \exp\left(\zeta(1 - \frac{r}{R_0})\right) - \left(\frac{R_0}{r}\right)^7 \right] & r \ge r_{spline} \ -\frac{1}{r} \sum_{i=1}^{5} i \cdot p_i r^{i-1} & r < r_{spline} \end{cases} $$

Parameters

For computational efficiency, the physical parameters ($D_0, R_0, \zeta$) are pre-computed into two sets for the long-range and short-range parts:

• Long-Range Part ($r \ge r_{spline}$):

  • a: The repulsion pre-factor $A = \frac{6 D_0}{\zeta-6} e^{\zeta}$.
  • b: The repulsion decay constant $B = \zeta / R_0$.
  • c: The attraction pre-factor $C = \frac{\zeta D_0 R_0^6}{\zeta-6}$.

• Short-Range Part ($r < r_{spline}$):

  • r_spline_sq: The squared distance threshold for switching to the polynomial.
  • p0..p5: The six coefficients of the quintic polynomial $P_5(r)$, pre-computed to satisfy $C^2$ continuity at $r_{spline}$ and boundary conditions at $r=0$.

Inputs

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

Implementation Notes

• A branchless selection mechanism is used to switch between the Buckingham and polynomial forms, making it SIMD-friendly.
• The polynomial is evaluated using Horner’s method for improved numerical stability and reduced floating-point operations.
• The entire computation, including both paths, is executed to avoid pipeline stalls, making the runtime performance constant and predictable.

Trait Implementations

impl Clone for SplinedBuckingham

fn clone(&self) -> SplinedBuckingham

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for SplinedBuckingham

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

Formats the value using the given formatter.

impl Default for SplinedBuckingham

fn default() -> SplinedBuckingham

Returns the “default value” for a type.

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

fn energy(r_sq: T, params: Self::Params) -> T

Computes only the potential energy, selecting between Exp-6 and polynomial forms.

Formula

$$ E(r) = \begin{cases} A e^{-Br} - C/r^6 & r \ge r_{spline} \ P_5(r) & r < r_{spline} \end{cases} $$

fn diff(r_sq: T, params: Self::Params) -> T

Computes only the force pre-factor $D$, selecting between Exp-6 and polynomial forms.

Formula

$$ D(r) = \begin{cases} \frac{A B e^{-B r}}{r} - \frac{6 C}{r^8} & r \ge r_{spline} \ -\frac{P’5(r)}{r} & r < r{spline} \end{cases} $$

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, params: 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, T, T, T, T, T)

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

impl Copy for SplinedBuckingham