E5: Dynamic EEQ Charge Model

Module: sci_form::experimental::eeqFeature flag: experimental-eeq

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Overview

Implements geometry-dependent Electronegativity Equalization (EEQ) charges, capturing polarization effects that static charge models like Gasteiger-Marsili miss. Charges depend on the 3D molecular geometry through fractional coordination numbers, making them suitable for conformer-dependent property prediction and as input to dispersion and solvation models.

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Theory

Electronegativity Equalization

At equilibrium, all atoms share a common electrochemical potential $\mu$:

$$\mu ={\chi }_i+{\eta }_i{q}_i+\sum _{j\ne i}\gamma (r_{ij})q_j$$

This yields an $(N+1)\times (N+1)$ linear system (including charge neutrality constraint $\sum q_i=Q_{total}$).

Fractional Coordination Number

Geometry dependence enters through the coordination number:

$$C{N}_i=\sum _{j\ne i}\frac{1}{1+\exp (-16(\frac{r_{cov,ij}}{r_{ij}}-1))}$$

The Fermi-function damping provides smooth, differentiable CN values.

Damped Coulomb Interaction

The pairwise interaction kernel uses improved Gaussian damping with a $\frac{1}{3}$-power blending:

$$\gamma (r_{ij})=\frac{\text{erf}(\sqrt{2}\cdot r_{ij}/{\sigma }_{ij})}{r_{ij}}$$

where ${\sigma }_{ij}$ is derived from atomic charge radii. The damping exponent was improved from a naive linear kernel to $\frac{1}{3}$-power Gaussian blending, which yields better charge neutrality conservation across a wider range of molecular sizes and more physical short-range behavior for closely-spaced atoms.

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API

use sci_form::experimental::eeq::*;

// Compute EEQ charges
let result = compute_eeq_charges(&elements, &positions, total_charge);
// result.charges: Vec<f64>
// result.coordination_numbers: Vec<f64>

// Fractional coordination numbers
let cn = fractional_coordination(&elements, &positions);

// EEQ electrostatic energy
let energy_result = compute_eeq_energy(&elements, &positions, total_charge);
// energy_result.energy_kcal_mol: f64
// energy_result.charges: Vec<f64>

// Numerical gradient
let gradient = compute_eeq_gradient(&elements, &positions, total_charge, 0.001);

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Supported Elements

Full coverage for Z=1–86: H through Rn, including all 3d/4d/5d transition metals (Ti–Au), lanthanides (La–Lu), and period-6 main-group elements.

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Comparison with Gasteiger

Property	Gasteiger	EEQ
Geometry-dependent	No	Yes
Coordination effects	No	Yes (CN)
Charge neutrality	Iterative	Exact (constraint)
Transition metals	No	Yes
Gradients	No	Yes (numerical)

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Tests

cargo test --features experimental-eeq --test regression -- test_eeq

8 integration tests covering: charge neutrality, coordination numbers, energy computation, gradient accuracy, element coverage, and comparison with Gasteiger charges.