Extended Hückel Theory (EHT)

sci-form implements a full Extended Hückel Theory (EHT) pipeline for semiempirical electronic-structure calculations. EHT calculates molecular orbital (MO) energies, coefficients, and volumetric orbital densities from a 3D geometry without self-consistent field iteration.

Overview

EHT was introduced by Hoffmann (1963) as an extension of Hückel theory to all valence electrons (not just π electrons). It requires only:

1. The molecular geometry (Cartesian coordinates)
2. A set of Valence State Ionization Potentials (VSIP) per element
3. Slater-type orbital exponents per element

The pipeline has five phases:

Phase	Description
B1	Build atomic orbital basis from element parameters
B2	Compute overlap matrix S and Hamiltonian H
B3	Solve generalized eigenproblem HC = SCE
B4	Evaluate MO densities on 3D volumetric grid
B5	Extract isosurfaces via Marching Cubes

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Phase B1: Atomic Basis Set

Each atom contributes one or two valence orbitals (s and p) described as Slater-type orbitals (STOs) expanded in a minimal STO-3G Gaussian basis.

Slater-Type Orbitals

A Slater orbital of principal quantum number $n$ and exponent $\zeta$ (bohr⁻¹):

$${\chi }_{nlm}(r,\theta ,\varphi )=N\cdot r^{n-1}{e}^{-\zeta r}\cdot Y_l^m(\theta ,\varphi )$$

where $N$ is a normalization constant and $Y_l^m$ are spherical harmonics.

STO-3G Expansion

Each Slater orbital is approximated by 3 Gaussian primitives fitted to reproduce the STO shape:

$$\chi (\mathbf{r})=\sum _{k=1}^3{d}_k\cdot G_k({\alpha }_k,\mathbf{r}-{\mathbf{R}}_A)$$

where $G_k({\alpha }_k,\mathbf{r})=N_k\exp (-{\alpha }_k|\mathbf{r}{|}^2)$ is a normalized Gaussian primitive.

VSIP Parameter Table

The following Hoffmann parameters are used as diagonal Hamiltonian elements:

Element	Orbital	VSIP (eV)	Slater $\zeta$ (bohr⁻¹)
H	1s	−13.6	1.300
C	2s	−21.4	1.625
C	2p	−11.4	1.625
N	2s	−26.0	1.950
N	2p	−13.4	1.950
O	2s	−32.3	2.275
O	2p	−14.8	2.275
F	2s	−40.0	2.425
F	2p	−18.1	2.425
Cl	3s	−26.3	2.183
Cl	3p	−14.2	1.733
Br	4s	−22.07	2.588
Br	4p	−13.1	2.131
I	5s	−18.0	2.679
I	5p	−12.7	2.322

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Phase B2: Overlap Matrix and Hamiltonian

Overlap Matrix $S$

$$S_{ij}=\int {\chi }_i(\mathbf{r}){\chi }_j(\mathbf{r})d\mathbf{r}$$

Computed analytically using the Gaussian product theorem: the product of two Gaussians centered at $\mathbf{A}$ and $\mathbf{B}$ is a single Gaussian centered at the weighted midpoint $\mathbf{P}$:

$$G_a(\alpha ,\mathbf{r}-\mathbf{A})\cdot G_b(\beta ,\mathbf{r}-\mathbf{B})=K_{AB}\cdot G_{a+b}(\gamma ,\mathbf{r}-\mathbf{P})$$

where $\gamma =\alpha +\beta$, $\mathbf{P}=\frac{\alpha \mathbf{A}+\beta \mathbf{B}}{\gamma }$, and $K_{AB}=\exp (-\frac{\alpha \beta }{\gamma }|\mathbf{A}-\mathbf{B}{|}^2)$.

This allows exact analytical evaluation of all $⟨s|s⟩$, $⟨s|p⟩$, and $⟨p|p⟩$ integrals without numerical quadrature.

Hamiltonian Matrix $H$

Diagonal elements (Hückel approximation):

$$H_{ii}={\text{VSIP}}_i$$

Off-diagonal elements (Wolfsberg-Helmholtz approximation):

$$H_{ij}=\frac{1}{2}K\cdot S_{ij}\cdot (H_{ii}+H_{jj})$$

where $K=1.75$ is the Wolfsberg-Helmholtz constant. This is the key approximation of EHT: the resonance integral between two orbitals is proportional to their overlap and the average of their ionization potentials.

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Phase B3: Generalized Eigenproblem

The EHT secular equation is the generalized eigenproblem:

$$\mathbf{H}\mathbf{C}=\mathbf{S}\mathbf{C}\mathbf{E}$$

where $\mathbf{E}$ is a diagonal matrix of orbital energies and the columns of $\mathbf{C}$ are MO coefficient vectors. sci-form solves this using Löwdin symmetric orthogonalization:

Step 1: Diagonalize S

$$\mathbf{S}=\mathbf{U}\mathbf{\Lambda }{\mathbf{U}}^T$$

Step 2: Compute $S^{-1/2}$

$${\mathbf{S}}^{-1/2}=\mathbf{U}\text{diag}\left({\lambda }_i^{-1/2}\right){\mathbf{U}}^T$$

Eigenvalues smaller than ${10}^{-10}$ are discarded to avoid numerical instability.

Step 3: Transform to Standard Form

$${\mathbf{H}}^{\prime }={\mathbf{S}}^{-1/2}\mathbf{H}{\mathbf{S}}^{-1/2}$$

Step 4: Diagonalize $H^{\prime }$

$${\mathbf{H}}^{\prime }{\mathbf{C}}^{\prime }={\mathbf{C}}^{\prime }\mathbf{E}$$

Step 5: Back-Transform Coefficients

$$\mathbf{C}={\mathbf{S}}^{-1/2}{\mathbf{C}}^{\prime }$$

Eigenvalues are sorted in ascending order. The HOMO and LUMO indices are identified from the number of valence electrons (filled two electrons per orbital).

Output: EhtResult

pub struct EhtResult {
    pub energies: Vec<f64>,      // MO energies (eV), ascending
    pub coefficients: Vec<Vec<f64>>, // C matrix: coefficients[ao][mo]
    pub n_electrons: usize,       // total valence electrons
    pub homo_index: usize,        // HOMO (0-based)
    pub lumo_index: usize,        // LUMO (0-based)
    pub homo_energy: f64,         // eV
    pub lumo_energy: f64,         // eV
    pub gap: f64,                 // HOMO-LUMO gap (eV)
}

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Phase B4: Orbital Grid Evaluation

Once the MO coefficients are known, any orbital ${\psi }_m$ can be evaluated on a 3D volumetric grid:

$${\psi }_m(\mathbf{r})=\sum _{\mu =1}^{N_{\text{AO}}}{C}_{\mu m}{\chi }_{\mu }(\mathbf{r})$$

The orbital density (for visualization) is:

$${\rho }_m(\mathbf{r})=|{\psi }_m(\mathbf{r}){|}^2$$

The grid is parallelized over chunks using Rayon, evaluating all AO basis functions at each grid point.

Grid Parameters

pub struct VolumetricGrid {
    pub origin: [f64; 3],    // lower-left corner (Å)
    pub spacing: f64,         // grid spacing (Å), default 0.15
    pub dims: [usize; 3],    // (Nx, Ny, Nz)
    pub values: Vec<f32>,    // scalar field (f32 for compact transfer)
}

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Phase B5: Marching Cubes Isosurface

Isosurfaces of MO density are extracted using the Marching Cubes algorithm (Lorensen & Cline 1987):

1. For each voxel cube in the grid, determine which vertices are inside/outside the isovalue threshold
2. Use the 256-case lookup table to identify the edge intersections
3. Interpolate vertex positions on crossed edges
4. Return a triangle mesh

The isosurface mesh is returned as:

pub struct IsosurfaceMesh {
    pub vertices: Vec<[f32; 3]>,    // xyz positions
    pub normals: Vec<[f32; 3]>,     // per-vertex normals
    pub indices: Vec<u32>,           // triangle indices
}

---

API

Rust

use sci_form::{eht_calculate, eht_orbital_mesh};

// Run EHT calculation
let result = eht_calculate("C6H6", None)?;  // benzene
println!("HOMO: {:.2} eV", result.homo_energy);
println!("LUMO: {:.2} eV", result.lumo_energy);
println!("Gap:  {:.2} eV", result.gap);

// Get orbital isosurface as JSON mesh
let mesh_json = eht_orbital_mesh("C6H6", result.homo_index, 0.05, 0.15, None)?;

Python

import sci_form

result = sci_form.eht_calculate("c1ccccc1")  # benzene (aromatic SMILES)
print(f"HOMO = {result.homo_energy:.2f} eV")
print(f"LUMO = {result.lumo_energy:.2f} eV")
print(f"Gap  = {result.gap:.2f} eV")
print(f"Coefficients shape: {len(result.coefficients)} x {len(result.coefficients[0])}")

# Get orbital grid for visualization
grid = sci_form.eht_orbital_mesh("c1ccccc1", result.homo_index, 0.05, 0.15)

TypeScript / WASM

import init, { eht_calculate, eht_orbital_grid_typed } from 'sci-form-wasm';

await init();

const result = JSON.parse(eht_calculate("c1ccccc1"));
console.log(`HOMO = ${result.homo_energy.toFixed(2)} eV`);
console.log(`Gap  = ${result.gap.toFixed(2)} eV`);

// Get orbital volume as a flat Float32Array (Nx × Ny × Nz)
const gridInfo = JSON.parse(compute_esp_grid_info("c1ccccc1", ...));
const orbital = eht_orbital_grid_typed("c1ccccc1", result.homo_index, 0.05, 0.15);
// orbital: Float32Array — ready for Three.js Data3DTexture

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Capabilities and Limitations

Property	Value
Supported elements	H, C, N, O, F, Si, P, S, Cl, Br, I
Basis set	STO-3G (minimal, 1–2 orbitals per atom)
Correlation	None (single-determinant)
Relativistic effects	No
Self-consistency	No (one-shot diagonalization)
Cost	$O(N^3)$ diagonalization + $O(N^2)$ overlap

EHT is appropriate for qualitative visualization of frontier orbitals (HOMO/LUMO), rough HOMO-LUMO gap estimates, and as a fast electronic descriptor. For quantitative accuracy, DFT or post-Hartree-Fock methods are required.

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References

1. Hoffmann, R. An Extended Hückel Theory. I. Hydrocarbons. J. Chem. Phys. 1963, 39, 1397–1412.
2. Wolfsberg, M.; Helmholtz, L. The Spectra and Electronic Structure of the Tetrahedral Ions MnO₄⁻, CrO₄²⁻, and ClO₄⁻. J. Chem. Phys. 1952, 20, 837–843.
3. Löwdin, P.-O. On the Non-Orthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals. J. Chem. Phys. 1950, 18, 365–375.
4. Lorensen, W. E.; Cline, H. E. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. SIGGRAPH 1987, 21, 163–169.