chematic-ewald — Ewald Summation for Long-Range Electrostatics

Pure-Rust implementation of Ewald summation and Smooth Particle Mesh Ewald (SPME) for accurate long-range electrostatic energy in molecular dynamics simulations. Full WASM support, zero external FFI.

Features

Direct Coulomb Sum (Non-Periodic)

Simple pairwise Coulomb energy for isolated molecules:

use chematic_ewald::direct_coulomb;

let coords = vec![[0.0, 0.0, 0.0], [1.0, 0.0, 0.0]];
let charges = vec![1.0, -1.0];
let energy = direct_coulomb(&coords, &charges);
// → -332.0637 kcal/mol (K_e * q1 * q2 / r)

Coulomb with Real-Space Cutoff

Efficient short-range Coulomb for periodic systems:

use chematic_ewald::direct_coulomb_cutoff;

let energy = direct_coulomb_cutoff(&coords, &charges, r_cut);
// Only pairs with r < r_cut contribute

Ewald Damped Coulomb (Complementary Error Function)

Smooth damping for short-range Coulomb in Ewald summation:

use chematic_ewald::direct_coulomb_damped;

let alpha = 3.5 / r_cut;  // Ewald splitting parameter
let energy = direct_coulomb_damped(&coords, &charges, alpha);
// Uses erfc(α*r) damping

SPME Ewald (Reciprocal Space)

FFT-based long-range electrostatics via Smooth Particle Mesh Ewald:

use chematic_ewald::{spme_energy, BoxVectors, PmeConfig};

let box_vecs = BoxVectors::cubic(30.0);  // 30 Ų cubic box
let config = PmeConfig {
    alpha: 0.0,           // Auto-compute: 3.5 / r_cut
    kmax: [5, 5, 5],      // Reciprocal space cutoff
    r_cut: 8.0,           // Real-space cutoff (Å)
    mesh: [16, 16, 16],   // PME mesh size
    spline_order: 4,      // B-spline interpolation order
};

let result = spme_energy(&coords, &charges, &box_vecs, &config);
// → EwaldResult { real_energy, reciprocal_energy, self_energy, total_energy }

SPME Algorithm

Components

Real-space: Damped Coulomb with erfc cutoff (short-range, computed directly)
Reciprocal-space: FFT-based long-range via charge interpolation to mesh
Self-energy: Gaussian charge distribution correction

Workflow

Input: atomic coordinates, charges, periodic box
  ↓
1. Assign Ewald parameter α = 3.5 / r_cut
2. Real-space: ∑_{i<j} K_e * q_i * q_j * erfc(α*r_ij) / r_ij
3. Interpolate charges onto 3D mesh (B-spline order 4)
4. Forward FFT to reciprocal space
5. Reciprocal-space energy via Ewald kernel exp(-k²/4α²) / k²
6. Self-energy correction: -K_e * α / √π * ∑_i q_i²
7. Total: E_real + E_reciprocal + E_self
  ↓
Output: total electrostatic energy

Box Vectors and Periodic Boundary Conditions

use chematic_ewald::BoxVectors;

// Cubic box
let box = BoxVectors::cubic(30.0);

// Triclinic box (general parallelepiped)
let box = BoxVectors([[
    [30.0,  0.0,  0.0],  // a vector
    [ 0.0, 30.0,  0.0],  // b vector
    [ 0.0,  0.0, 30.0],  // c vector
]]);

let volume = box.volume();  // Box volume in Ų

Configuration and Accuracy

PmeConfig Parameters

Parameter	Default	Role
`alpha`	auto (3.5/r_cut)	Ewald splitting: controls real/reciprocal balance
`r_cut`	8.0 Å	Real-space cutoff
`mesh`	[16,16,16]	PME grid size; increase for accuracy
`kmax`	[5,5,5]	Reciprocal-space cutoff in k-vectors
`spline_order`	4	B-spline order (3–5); higher = more accurate

Accuracy vs. Performance Trade-off

Use Case	mesh	kmax	α	Accuracy
Quick estimate	[8,8,8]	[3,3,3]	auto	~0.1 kcal/mol
Production MD	[24,24,24]	[6,6,6]	auto	~0.01 kcal/mol
High precision	[32,32,32]	[8,8,8]	auto	~0.001 kcal/mol

Integration with chematic-3d

Long-range electrostatics for MD simulations:

use chematic_ewald::spme_energy;
use chematic_chem::gasteiger_charges;

let charges = gasteiger_charges(&mol);
let result = spme_energy(&coords, &charges, &box_vecs, &config);
let e_elec = result.total_energy;

WASM Usage

// via chematic-wasm
const mol = parse_smiles("CCO");
const result = coulomb_energy_json(mol);
// → { "coulomb_energy": -12.34, "unit": "kcal/mol" }

Performance

Direct Coulomb: O(N²), exact, no approximation
SPME Real-space: O(N), grid-independent
SPME Reciprocal: O(N log N) via FFT, scales well to ~10k atoms
WASM performance: N ≤ 50 atoms recommended for interactive use

Memory

Direct: ~8 bytes per pair
SPME: ~8 * mesh³ bytes (e.g., 24³ → ~11 MB) + N * O(1)

Ewald vs. SPME

Method	Accuracy	Speed	Memory	Suitable For
Direct Coulomb	Exact	O(N²)	O(N)	Small systems, isolated molecules
Ewald (Reciprocal)	~10⁻⁶	O(N)	O(1)	Periodic systems
SPME	~10⁻⁶	O(N log N)	O(mesh³)	Large periodic systems

References

Ewald, P. P. (1921). Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys., 64, 253–287.
Essmann, U., Perera, L., Berkowitz, M. L., Darden, T., Pedersen, L. G., & Roux, B. (1995). A smooth particle mesh Ewald method. J. Chem. Phys., 103(19), 8577–8593.

Limitations & Future Work

✅ Real-space Coulomb: Complete
✅ Reciprocal-space PME: Implemented (B-spline interpolation basic)
⏳ Parallel FFT: Planned
⏳ GPU acceleration: Deferred to future phase
⏳ Polarizable Ewald (DPME): Phase 4+

License

MIT OR Apache-2.0

chematic-ewald 0.1.94