Algorithm Overview

sci-form is a computational chemistry library with two main areas: 3D conformer generation and quantum-chemistry-inspired property computation. This page describes both pipelines at a high level.

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Part 1: ETKDGv2 Conformer Pipeline

sci-form implements ETKDGv2 (Experimental Torsion Knowledge Distance Geometry v2) to generate 3D molecular conformers from SMILES strings.

The 9-Step Pipeline

Phase 1: Topology → Bounds (Steps 1–3)

Build a molecular graph and derive distance constraints between all atom pairs. Constraints form a bounds matrix $B$ where $l_{ij}\le d_{ij}\le u_{ij}$.

• 1-2 bounds: Bond lengths from UFF parameters
• 1-3 bounds: From bond lengths + equilibrium angles (law of cosines)
• 1-4 bounds: From torsion angle cis/trans extremes
• VDW bounds: Lower bounds from van der Waals radii
• Smoothing: Floyd-Warshall to enforce the triangle inequality

→ Details: Bounds Matrix, SMILES Parsing

Phase 2: Embedding → Optimization (Steps 4–6)

Generate 3D (or 4D) coordinates from the distance constraints using distance geometry:

1. Pick random distances from the smoothed bounds (MinstdRand RNG)
2. Convert distances to a metric matrix via the Cayley-Menger transform
3. Extract coordinates via eigendecomposition (power iteration solver)
4. Minimize distance violations using BFGS with a bounds violation force field

$$T_{ij}=\frac{1}{2}(D_{0i}+D_{0j}-d_{ij}^2)$$

where $D_{0i}=\frac{1}{N}\sum _k{d}_{ik}^2-\frac{1}{N^2}\sum _{k<l}{d}_{kl}^2$

→ Details: Distance Geometry, Embedding

Phase 3: Refinement → Output (Steps 7–9)

After obtaining valid 3D coordinates, refine using the ETKDG force field:

• CSD torsion preferences: 846 SMARTS patterns with Fourier coefficients
• UFF inversions: Out-of-plane energy for SP2 centers
• Distance constraints: Maintain bond lengths and angles
• Validation: Reject conformers failing tetrahedral/planarity/stereo checks

→ Details: ETKDG Refinement, Force Fields, Validation

The Retry Loop

The pipeline uses a retry loop with up to $10N$ iterations:

1. Metric matrix has zero or negative eigenvalues → retry
2. Energy/atom after bounds minimization exceeds 0.05 → retry
3. Tetrahedral centers fail volume test → retry
4. Chiral volume signs don't match → retry
5. Planarity check fails → retry
6. Double-bond geometry is wrong → retry

After $N/4$ consecutive failures, fall back to random box placement (uniform from $[-5,5{]}^3$).

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Part 2: Property Computation Pipeline

Once a conformer is generated, sci-form can compute a range of molecular properties.

Gasteiger Charges (no QM)

Fast empirical electronegativity equalization — requires only topology + atomic numbers, runs in microseconds.

→ Details: Population Analysis

Extended Hückel Theory (EHT)

EHT is the gateway to most quantum properties:

1. Build STO-3G basis — contracted Gaussian orbitals for each atom
2. Overlap matrix S — $S_{\mu \nu }=⟨{\varphi }_{\mu }|{\varphi }_{\nu }⟩$
3. Wolfsberg-Helmholtz H — $H_{\mu \nu }=\frac{K}{2}{S}_{\mu \nu }(H_{\mu \mu }+H_{\nu \nu })$
4. Löwdin ortho — $\tilde{H}=S^{-1/2}H{S}^{-1/2}$, diagonalize → orbital energies ${\epsilon }_i$ and MO coefficients
5. Fill $N_e$ electrons from lowest orbital up → HOMO/LUMO

From EHT:

Property	Method	Key Output
Population analysis	Mulliken/Löwdin	Per-atom charges
Dipole moment	Bond + lone-pair	Vector + Debye magnitude
Orbital grids	STO-3G on 3D grid	Float32 volumetric array
Isosurface mesh	Marching cubes	Vertices, normals, triangles
DOS/PDOS	Gaussian smearing	Total DOS, per-atom DOS

→ Details: Density of States, Population Analysis, Dipole Moments

Electrostatic Potential

Coulomb ESP on a 3D grid from Mulliken charges:

$$V(\mathbf{r})=\sum _i\frac{q_i^{\text{Mulliken}}}{|\mathbf{r}-{\mathbf{r}}_i|}$$

Color-mapped (red = negative, white = zero, blue = positive). Gaussian Cube export.

→ Details: Electrostatic Potential

Force Fields

• UFF — Universal Force Field, 50+ element types (including transition metals)
• MMFF94 — Merck Molecular Force Field: quartic stretch, cubic bend, 3-term Fourier torsion, Halgren 14-7 vdW

→ Details: Force Fields, Strain Energy

Molecular Alignment

Two algorithms:

• Kabsch SVD — optimal rotation, $O(N\cdot min(N,3{)}^2)$
• Quaternion alignment — Coutsias 2004 4×4 eigenproblem, faster for large $N$

→ Details: Molecular Alignment

Materials Assembly

Node/linker SBUs + topology → periodic crystal structure (MOF-type).

→ Details: Materials Assembly

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References

• Riniker & Landrum, J. Chem. Inf. Model. 2015, 55, 2562 (ETKDGv2)
• Wang et al., J. Chem. Inf. Model. 2020, 60, 2044 (ETKDGv3)
• Blaney & Dixon, Rev. Comput. Chem. 1994, 5, 299 (Distance geometry)
• Wolfsberg & Helmholtz, J. Chem. Phys. 1952, 20, 837 (EHT)
• Mulliken, J. Chem. Phys. 1955, 23, 1833 (Population analysis)
• Gasteiger & Marsili, Tetrahedron 1980, 36, 3219 (Charge equalization)
• Coutsias et al., J. Comput. Chem. 2004, 25, 1849 (Quaternion alignment)
• Halgren, J. Comput. Chem. 1996, 17, 490 (MMFF94)