QM9 HOMO-LUMO Gap Prediction

What this research found

The HOMO–LUMO gap — the energy difference between a molecule's highest occupied and lowest unoccupied orbital — sets its optical and reactivity behaviour, and predicting it from geometry alone is far cheaper than computing it. A gradient-boosted tree ensemble was trained on 130,831 curated QM9 molecules encoded as sorted Coulomb matrices, reaching a held-out mean absolute error of 0.3126 eV with an R² of 0.8785. Stratifying the error by gap size then showed that the narrow-gap molecules of greatest practical interest are predicted 1.77 times less accurately, and systematically too high.

• The selected gradient-boosting model reached a test mean absolute error of 0.3126 eV (312.6 meV), a root-mean-square error of 0.4479 eV and R² = 0.8785 across 10,831 held-out molecules, against 1.0627 eV for a constant-mean baseline. Test error reproduced validation error to within 1 meV.
• Errors are right-skewed: the median absolute error is 0.225 eV, with 53.8% of molecules predicted within 0.25 eV of the reference calculation, 81.5% within 0.5 eV and 96.3% within 1.0 eV.
• Narrow-gap molecules — the lowest decile of true gaps, at or below 5.2763 eV, 1084 molecules — were predicted with a mean absolute error of 553.9 meV and a root-mean-square error of 0.7931 eV, roughly 1.77 times worse than the test set as a whole.
• That degradation is bias, not noise. The model is essentially unbiased overall (mean signed error +0.015 eV) but over-predicts narrow gaps by +0.50 eV on average, with 84.8% of narrow-gap errors positive. The highest decile shows the mirror image at −0.31 eV, the signature of regression toward the population mean at both edges.
• Non-linearity matters more than network depth here. Ridge regression on the same features reached only 0.7128 eV validation error (R² = 0.504), while a multilayer perceptron reached 0.4054 eV and showed clear overfitting, with training error of 0.1512 eV against validation error of 0.4054 eV.
• The practical implication is that a model reported only by aggregate error would overstate its usefulness for finding semiconductors, dyes or photocatalysts by nearly a factor of two, and would systematically miss genuinely narrow-gap candidates by predicting their gaps as larger than they are.

How it was done

Starting from the 133,885 QM9 molecules — up to nine heavy atoms of carbon, nitrogen, oxygen and fluorine with properties computed at the B3LYP/6-31G(2df,p) level — the canonical geometric-consistency filter removed the 3054 structures whose relaxed geometry fails the round-trip structure test, leaving 130,831. Each molecule was encoded as a sorted, zero-padded Coulomb matrix flattened to 435 dimensions, a descriptor invariant to translation, rotation and atom ordering. A single fixed permutation split the data into 110,000 training, 10,000 validation and 10,831 test molecules, with feature scaling fit on the training partition only to avoid leakage. Ridge regression, histogram-based gradient boosting and a multilayer perceptron were tuned and compared on validation error; the gradient-boosting model was selected and evaluated on the test set exactly once. Test error was then stratified by gap decile, and per-molecule predictions were released with their original dataset indices.

Data sources

• QM9 (Ramakrishnan et al., Scientific Data 1:140022, 2014) — 133,885 molecules with B3LYP/6-31G(2df,p) properties, 130,831 retained after filtering
• QM9 published list of 3054 geometry-inconsistent molecules
• Coulomb matrix descriptor of Rupp et al., Phys. Rev. Lett. 108:058301 (2012)
• scikit-learn and NumPy