E7: DFT-D4 Dispersion Correction

Module: sci_form::experimental::d4Feature flag: experimental-d4

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Overview

Implements the DFT-D4 dispersion correction with geometry-dependent $C_6$/$C_8$ coefficients and Becke-Johnson (BJ) damping. Augments EHT, UFF, and MMFF94 with quantum-precision van der Waals interactions. Includes optional three-body Axilrod-Teller-Muto (ATM) terms.

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Theory

Dynamic $C_6$ Coefficients

Unlike fixed-parameter methods (D3), D4 uses coordination-number-dependent dispersion coefficients:

$$C_6^{AB}=\sum _{a,b}{W}_a{W}_b{C}_6^{AB,{\text{ref}}_{ab}}$$

where weights $W_a$ are Gaussian functions of the D4 coordination number deviation from reference values.

D4 Coordination Number

$$C{N}_i^{D4}=\sum _{j\ne i}\frac{1}{1+\exp (-16(\frac{r_{cov,ij}}{r_{ij}}-1))}$$

Two-Body Dispersion

$$E_{\text{disp}}^{(2)}=-\sum _{A<B}[s_6\frac{C_6^{AB}}{r^6}{f}_{\text{damp}}^{(6)}+s_8\frac{C_8^{AB}}{r^8}{f}_{\text{damp}}^{(8)}]$$

with Becke-Johnson damping:

$$f_{\text{damp}}^{(n)}(r)=\frac{r^n}{r^n+(a_1\sqrt{C_8/C_6}+a_2{)}^n}$$

Three-Body ATM Term

$$E_{\text{disp}}^{(3)}=s_9\sum _{A<B<C}{C}_9^{ABC}\frac{3\cos {\theta }_A\cos {\theta }_B\cos {\theta }_C+1}{(r_{AB}{r}_{BC}{r}_{CA}{)}^3}$$

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API

use sci_form::experimental::d4::*;

// D4 coordination numbers
let cn = d4_coordination_number(&elements, &positions);

// Reference C6 coefficients
let c6 = get_c6_reference(6, 8); // C-O pair

// Full D4 energy
let config = D4Config {
    s6: 1.0, s8: 0.95,
    a1: 0.45, a2: 4.0,
    three_body: false,
    ..Default::default()
};
let result = compute_d4_energy(&elements, &positions, &config);
// result.two_body_energy: f64  (Hartree)
// result.three_body_energy: f64
// result.total_energy: f64

// Numerical gradient
let gradient = compute_d4_gradient(&elements, &positions, &config);

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Default Parameters

Parameter	Value	Description
s6	1.0	$C_6$ scaling
s8	0.95	$C_8$ scaling
a1	0.45	BJ damping parameter
a2	4.0	BJ damping radius (Bohr)
s9	1.0	Three-body scaling
three_body	false	Enable ATM three-body term

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Element Coverage

H, B, C, N, O, F, Si, P, S, Cl, Br, I plus xTB transition metals: Ti, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ru, Pd, Ag, Pt, Au.

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Tests

cargo test --features experimental-d4 --test regression -- test_d4

8 integration tests covering: coordination numbers, C6 reference values, C6 symmetry, BJ damping behavior, two-body energy negativity, three-body terms, gradient accuracy, and multi-element systems.