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Q-Shape is a web-based tool for analyzing the geometry of coordination complexes using Continuous Shape Measures (CShM). This documentation provides in-depth scientific and algorithmic details for researchers and developers.
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- Mathematical definition and derivation
- Normalization and scale invariance
- Interpretation scale
- Historical context and key references
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- Optimal rotation problem
- Jacobi SVD implementation
- Reflection handling
- Numerical considerations
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- Assignment problem formulation
- Munkres algorithm details
- Complexity analysis
- Exhaustive search comparison
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Reference Geometries
- Complete catalog of 92 polyhedra (CN 2-12+)
- SHAPE 2.1 and CoSyMlib sources
- Coordinate definitions
- Symmetry classifications
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Quality Metrics
- Bond length statistics
- Angular distortion index
- Bond length uniformity
- Overall quality scores
- RMSD approximation
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Ring Detection & Hapticity
- π-coordination handling
- Aromatic ring detection
- Centroid representation
- Sandwich complex analysis
| CShM Value | Classification | Confidence |
|---|---|---|
| < 0.1 | Perfect | 100% |
| 0.1 - 0.5 | Excellent | 95% |
| 0.5 - 1.5 | Very Good | 90% |
| 1.5 - 3.0 | Good | 80% |
| 3.0 - 7.5 | Moderate | 60% |
| 7.5 - 15.0 | Poor | 30% |
| > 15.0 | No Match | 10% |
Continuous Shape Measure: $$S(Q, P) = 100 \times \min_{{R, \pi}} \frac{\sum_{i=1}^{N} |\mathbf{q}i - R \cdot \mathbf{p}{\pi(i)}|^2}{\sum_{i=1}^{N} |\mathbf{q}_i|^2}$$
Kabsch Rotation:
Approximate RMSD:
┌─────────────────────────────────────────────────────────────────┐
│ INPUT STRUCTURE │
│ (XYZ coordinates) │
└───────────────────────────┬─────────────────────────────────────┘
│
▼
┌─────────────────────────────────────────────────────────────────┐
│ 1. COORDINATION SPHERE DETECTION │
│ - Identify metal center │
│ - Find coordinating atoms within radius │
│ - Detect rings and hapticity │
└───────────────────────────┬─────────────────────────────────────┘
│
▼
┌─────────────────────────────────────────────────────────────────┐
│ 2. COORDINATE PREPROCESSING │
│ - Center at metal position │
│ - Normalize to unit sphere │
│ - Extract ligand vectors │
└───────────────────────────┬─────────────────────────────────────┘
│
▼
┌─────────────────────────────────────────────────────────────────┐
│ 3. CShM CALCULATION (for each reference geometry) │
│ │
│ ┌─────────────────────────────────────────────────┐ │
│ │ a. HUNGARIAN ALGORITHM │ │
│ │ Find optimal atom-to-vertex assignment │ │
│ └─────────────────────┬───────────────────────────┘ │
│ │ │
│ ▼ │
│ ┌─────────────────────────────────────────────────┐ │
│ │ b. KABSCH ALIGNMENT │ │
│ │ Find optimal rotation (SVD) │ │
│ └─────────────────────┬───────────────────────────┘ │
│ │ │
│ ▼ │
│ ┌─────────────────────────────────────────────────┐ │
│ │ c. COMPUTE CShM │ │
│ │ Calculate mean squared deviation │ │
│ └─────────────────────────────────────────────────┘ │
│ │
└───────────────────────────┬─────────────────────────────────────┘
│
▼
┌─────────────────────────────────────────────────────────────────┐
│ 4. RESULTS RANKING │
│ - Sort geometries by CShM (lowest = best) │
│ - Calculate quality metrics │
│ - Generate interpretation │
└───────────────────────────┬─────────────────────────────────────┘
│
▼
┌─────────────────────────────────────────────────────────────────┐
│ OUTPUT REPORT │
│ - Best matching geometry │
│ - CShM values for all tested geometries │
│ - Quality metrics (ADI, BLUI, OQS) │
│ - Visualization of aligned structures │
└─────────────────────────────────────────────────────────────────┘
| CN | Count | Examples |
|---|---|---|
| 2 | 3 | Linear, V-shape, L-shape |
| 3 | 4 | Trigonal planar, T-shaped, Pyramidal |
| 4 | 4 | Tetrahedral, Square planar, See-saw |
| 5 | 5 | TBP, Square pyramid, Pentagon |
| 6 | 5 | Octahedral, Trigonal prism, Hexagon |
| 7 | 7 | Pentagonal bipyramid, Capped octahedron |
| 8 | 13 | Square antiprism, Cube, Dodecahedron |
| 9 | 9 | Tricapped trigonal prism, Muffin |
| 10 | 9 | Bicapped square antiprism |
| 11 | 7 | Capped pentagonal antiprism |
| 12 | 5 | Icosahedron, Cuboctahedron |
| 20+ | 4 | Dodecahedron, Truncated structures |
Total: 92 reference geometries
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Zabrodsky, H.; Peleg, S.; Avnir, D. "Continuous Symmetry Measures." J. Am. Chem. Soc. 1992, 114, 7843-7851.
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Pinsky, M.; Avnir, D. "Continuous Symmetry Measures. 5. The Classical Polyhedra." Inorg. Chem. 1998, 37, 5575-5582.
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Casanova, D.; Cirera, J.; Llunell, M.; Alemany, P.; Avnir, D.; Alvarez, S. J. Am. Chem. Soc. 2004, 126, 1755-1763.
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Alvarez, S.; Alemany, P.; Casanova, D.; Cirera, J.; Llunell, M.; Avnir, D. "Shape maps and polyhedral interconversion paths in transition metal chemistry." Coord. Chem. Rev. 2005, 249, 1693-1708.
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Kabsch, W. "A solution for the best rotation to relate two sets of vectors." Acta Crystallogr. A 1976, 32, 922-923.
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Kuhn, H. W. "The Hungarian Method for the Assignment Problem." Naval Res. Logist. Quarterly 1955, 2, 83-97.
- Llunell, M.; Casanova, D.; Cirera, J.; Bofill, J. M.; Alemany, P.; Alvarez, S. SHAPE 2.1; Universitat de Barcelona: Barcelona, 2013.
- Frontend: React.js, Three.js (3D visualization)
- Algorithms: Pure JavaScript (no external math libraries)
- SVD: Custom Jacobi SVD implementation
- Workers: Web Workers for background calculations
| Metric | Value |
|---|---|
| CShM per geometry | ~10 ms |
| Full CN=6 analysis | ~50 ms |
| Full CN=12 analysis | ~150 ms |
| Memory footprint | < 50 MB |
- Chrome 88+
- Firefox 85+
- Safari 14+
- Edge 88+
Q-Shape is designed for publication in peer-reviewed scientific journals. All algorithms and implementations follow best practices for:
- Numerical accuracy: Double precision throughout
- Reproducibility: Deterministic algorithms (no random initialization)
- Traceability: Full provenance of reference geometries
- Documentation: Complete scientific documentation
Q-Shape is open source software for academic and research use.
Last updated: December 2025