Can You Hear the Shape of a Molecule? An Introduction to Spectral Identifiability
In 1966, mathematician Mark Kac asked a famous question: “Can one hear the shape of a drum?” — that is, can you reconstruct a drum’s geometry just from its resonant frequencies? The answer turned out to be no in general (Gordon, Webb, and Wolpert found counterexamples in 1992), but the question opened up deep connections between geometry and spectral theory.
I’ve been asking an analogous question for chemistry: Can one determine a molecule’s structure from its vibrational spectrum?
The Forward Problem
Every molecule vibrates. When you shine infrared light on a molecule or scatter laser light off it (Raman spectroscopy), you excite specific vibrational modes. The frequencies and intensities of these modes depend on the molecule’s geometry and bonding — its force constants.
The forward map is well-understood: given a molecular structure, we can compute the IR and Raman spectrum using the Wilson GF method. But the inverse problem — going from spectrum back to structure — is much harder.
Why Symmetry Matters
Molecular symmetry creates fundamental blind spots. If a molecule has a center of inversion (like CO2 or benzene), the mutual exclusion principle says that IR-active modes are Raman-inactive and vice versa. No single technique sees everything.
We quantify this with the Information Completeness Ratio R(G,N), which measures what fraction of a molecule’s vibrational degrees of freedom are observable via combined IR + Raman. For most organic molecules (low symmetry), R = 1.0 and everything is observable. But for highly symmetric molecules like cubane (Oh symmetry), R drops to 0.45 — over half the vibrational information is hidden.
What the Math Says
Our analysis of 130,831 molecules from the QM9 dataset shows that 99.9% have all modes observable. The “hard” molecules — those with silent modes — are rare but include important cases like benzene and methane.
More excitingly, Jacobian rank analysis on 999 real molecular geometries shows that the combined IR + Raman forward map has full rank at every tested point, with a 4:1 overdetermination ratio. This is strong numerical evidence that the inverse problem generically has a unique solution (up to symmetry equivalence).
Implications for ML
This theory has direct implications for machine learning models in spectroscopy:
- Models should perform worse on high-symmetry molecules (lower R)
- Combining IR and Raman inputs should help most for centrosymmetric molecules
- The 4:1 overdetermination suggests the learning problem is well-conditioned
We’re testing these predictions with a symmetry-stratified evaluation framework. Early results are encouraging — stay tuned for the full paper.