Series Overview
This three-part series explores one of the most elegant questions in computational chemistry: can mathematical signatures capture molecular shape? Inspired by Mark Kac’s famous question “Can One Hear the Shape of a Drum?”, we investigate whether Coulomb matrix eigenvalues can distinguish between constitutional isomers—molecules with identical formulas but different structures.
What You’ll Learn
- Mathematical foundations: How eigenvalues encode structural information
- Computational validation: Building robust molecular simulation pipelines
- Unsupervised analysis: Testing natural clustering in eigenvalue space
- Supervised learning: Extracting hidden patterns with labeled data
- Practical limitations: Understanding when elegant math meets real-world constraints
The Journey
Part 1 establishes the computational foundation, validating our pipeline against literature results and exploring the eigenvalue space through principal component analysis.
Part 2 applies unsupervised clustering techniques (Dunn Index, silhouette analysis) to test whether eigenvalues naturally separate different molecular shapes.
Part 3 employs supervised learning (k-NN, logistic regression) to determine if labels can unlock patterns that unsupervised methods missed.
Key Insights
This series reveals some lessons about molecular representations:
- Mathematical appeal doesn’t always translate to practical utility
- Local structure can exist where global clustering fails
- Molecular complexity challenges even sophisticated descriptors
- The choice between supervised and unsupervised approaches can matter significantly
Perfect for readers interested in computational chemistry, molecular machine learning, or the intersection of mathematical beauty and practical limitations.