E10: Constant Potential Method (CPM)

Module: sci_form::experimental::cpmFeature flag: experimental-cpm

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Overview

Extends charge and electronic property calculations by coupling the molecule to a "virtual electrode" at electrochemical potential $\mu$. Charges fluctuate to equilibrate with the electrode, enabling simulation of quantum capacitance, charge response to potential, and electrochemical energy surfaces.

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Theory

Grand Potential

The grand potential functional couples electronic energy with the electrode potential:

$$\mathrm{\Omega }(\{q_i\},\mu )=E_{\text{elec}}(\{q_i\})-\mu \sum _i{q}_i$$

Minimizing with respect to charges $\{q_i\}$ yields the equilibrium charge distribution at potential $\mu$.

Charge Equilibration at Fixed $\mu$

The SCF iteration solves:

$$q_i^{(n+1)}=\frac{\mu -{\chi }_i-\sum _{j\ne i}{J}_{ij}{q}_j^{(n)}}{{\eta }_i}$$

with damped update (50/50 mixing) and Coulomb interaction:

$$J_{ij}=\frac{14.3996}{\epsilon \cdot r_{ij}}\text{(kcal/mol per e²)}$$

Electrochemical Energy Surface

Scanning $\mu$ produces the charge response curve $Q(\mu )$ and capacitance:

$$C(\mu )=\frac{\partial Q}{\partial \mu }$$

computed via finite differences over the $\mu$ scan.

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API

use sci_form::experimental::cpm::*;

// CPM charges at a given potential
let config = CpmConfig {
    mu_ev: -4.44,           // SHE reference (eV)
    dielectric: 78.5,       // water
    max_iter: 100,
    charge_tol: 1e-6,
};
let result = compute_cpm_charges(&elements, &positions, &config);
// result.charges: Vec<f64>
// result.total_charge: f64
// result.grand_potential: f64
// result.converged: bool
// result.iterations: usize

// Electrochemical surface scan
let surface = compute_cpm_surface(&elements, &positions, mu_min, mu_max, n_points, dielectric);
// surface.mu_values: Vec<f64>
// surface.total_charge: Vec<f64>
// surface.free_energy: Vec<f64>
// surface.capacitance: Vec<f64>

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Potential Scale

Reference	$\mu$ (eV)
SHE (Standard Hydrogen Electrode)	$-4.44$
+1 V vs SHE	$-3.44$
-1 V vs SHE	$-5.44$
Typical scan range	$[-5.5,-3.5]$

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Applications

• Quantum capacitance of molecular electrodes
• Redox potential estimation: $E^0=-\mathrm{\Delta }\mathrm{\Omega }/nF$
• Conformer ranking under electrochemical bias
• Charge response analysis for molecular electronics

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Tests

cargo test --features experimental-cpm --test regression -- test_cpm

8 integration tests covering: charge equilibration convergence, total charge response to potential, monotonic charge-potential relationship, grand potential computation, capacitance positivity, electrochemical surface scanning, solvent dielectric effects, and multi-element systems.