Struct UmbrellaInversion 

pub struct UmbrellaInversion;

Umbrella inversion potential for sp³ centers.

Physics

Models a specific pyramidal geometry for sp³ hybridized atoms (e.g., amines, phosphines). The central atom $I$ is bonded to three neighbors $J, K, L$, and the potential penalizes deviations from the target out-of-plane angle $\psi_0$.

• Formula: $$ E = \frac{1}{2} C (\cos\psi - \cos\psi_0)^2 $$
• Derivative Factor (diff): $$ \Gamma = \frac{dE}{d(\cos\psi)} = C (\cos\psi - \cos\psi_0) $$

Parameters

• c_half: Half force constant $C_{half} = C/2$.
• cos_psi0: Cosine of equilibrium angle $\cos_{\psi,0} \neq 0$.

Pre-computation

Use UmbrellaInversion::precompute to convert physical constants into optimized parameters: $(C, \psi_0°) \to (C/2, \cos_{\psi,0})$.

Inputs

• cos_psi: Cosine of the out-of-plane angle $\psi$.

Implementation Notes

• Pure polynomial evaluation; no trigonometric functions required.
• All intermediate calculations are shared between energy and force computations.
• Branchless and panic-free.
• DREIDING convention: force constant $C = K_{inv}/3$ (split among three permutations).

Implementations

impl UmbrellaInversion

pub fn precompute<T: Real>(c: T, psi0_deg: T) -> (T, T)

Pre-computes optimized kernel parameters from physical constants.

Input

• c: Force constant $C$.
• psi0_deg: Equilibrium out-of-plane angle $\psi_0$ in degrees.

Output

Returns (c_half, cos_psi0):

• c_half: Half force constant $C/2$.
• cos_psi0: Cosine of equilibrium angle $\cos_{\psi,0}$.

Computation

$$ C_{half} = C / 2, \quad \cos_{\psi,0} = \cos(\psi_0 \cdot \pi / 180) $$

Trait Implementations

impl<T: Real> AngleKernel<T> for UmbrellaInversion

fn energy(cos_psi: T, (c_half, cos_psi0): Self::Params) -> T

Computes only the potential energy.

Formula

$$ E = C_{half} (\cos\psi - \cos_{\psi,0})^2 $$

fn diff(cos_psi: T, (c_half, cos_psi0): Self::Params) -> T

Computes only the derivative factor $\Gamma$.

Formula

$$ \Gamma = 2 C_{half} (\cos\psi - \cos_{\psi,0}) $$

This factor allows computing forces via the chain rule: $$ \vec{F} = -\Gamma \cdot \nabla (\cos\psi) $$

fn compute(cos_psi: T, (c_half, cos_psi0): Self::Params) -> EnergyDiff<T>

Computes both energy and derivative factor efficiently.

This method reuses intermediate calculations to minimize operations.

type Params = (T, T)

Associated constants/parameters required by the potential (e.g., $C, \theta_0$).

impl Clone for UmbrellaInversion

fn clone(&self) -> UmbrellaInversion

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for UmbrellaInversion

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

Formats the value using the given formatter.

impl Default for UmbrellaInversion

fn default() -> UmbrellaInversion

Returns the “default value” for a type.

impl Copy for UmbrellaInversion