QM9: A Quantum Chemistry Benchmark Dataset

Updated 7 October 2025

QM9 dataset is a benchmark collection featuring approximately 134,000 small organic molecules with optimized 3D geometries and DFT-calculated quantum-chemical properties.
It underpins advances in quantum chemistry machine learning by enabling evaluations of methods like GNNs, MPNNs, and kernel-based regression for property prediction.
The dataset catalyzes molecular discovery by standardizing tasks in property prediction, generative modeling, and methodological extensions in computational chemistry.

The QM9 dataset is a foundational resource in quantum chemistry and molecular machine learning, comprising geometrically optimized structures and computed quantum-chemical properties for approximately 134,000 small organic molecules. Molecules in QM9 are restricted to H, C, N, O, and F atoms with up to nine heavy atoms, and were initially characterized at the B3LYP/6-31G(2df,p) density functional theory (DFT) level. This dataset serves as one of the most important benchmarks for the development, evaluation, and widespread comparison of machine learning models for property prediction, generative modeling, and quantum chemical analysis. It has also given rise to several major datasets that extend its representations or calculated properties, which are used broadly to accelerate quantum chemistry and molecular discovery.

1. Composition, Properties, and Calculations

QM9 contains roughly 130,000–134,000 neutral, stable organic molecules. Each molecule is provided as a 3D geometry with atomic coordinates and associated chemical graph representation, and was obtained via systematic enumeration and subsequent geometric optimization. The dataset includes 13 quantum-chemical properties per molecule, computed using DFT at the B3LYP/6-31G(2df,p) level:

Atomization energies (at 0 K, room temperature, enthalpy, free energy)
Electronic properties: HOMO, LUMO, and energy gap
Vibrational properties: zero point vibrational energy, highest vibrational frequency
Electron distribution: dipole moment, polarizability, spatial extent
Thermochemical data: heat capacity

All calculations were performed after full geometry optimization, and explicit hydrogen atoms are included for each molecule (Gilmer et al., 2017). Derivatives and extensions of QM9 compute further properties, such as GW-level frontier orbital energies (Fediai et al., 2023), NMR shielding constants (Gupta et al., 2020), quantum Hamiltonian matrices (Yu et al., 2023), numerical Hessians in solvents (Williams et al., 15 Aug 2024), and molecular descriptors for ML (Khan et al., 2023). The dataset is available in machine-accessible formats, supporting direct use in data-driven modeling.

2. Role in Machine Learning for Quantum Chemistry

QM9 is the principal benchmark for evaluating quantum chemistry–oriented machine learning models, particularly graph neural networks (GNNs), message passing neural networks (MPNNs), kernel-based regression, molecular generative models, and structure–property prediction frameworks.

MPNNs and GNNs: The dataset enabled the systematic evaluation of MPNNs with message functions (e.g., edge network, GG-NN) and update/readout strategies (e.g., GRU, set2set) in predicting energies and electronic properties with accuracy surpassing older hand-crafted descriptors like Coulomb matrices or bag-of-bonds (Gilmer et al., 2017).
Kernel methods: Compact many-body distribution functionals (MBDFs) and local descriptors (FCHL, SOAP, CM) have shown exceptional performance in kernel ridge regression and Gaussian process frameworks for rapid property prediction (Gupta et al., 2020, Khan et al., 2023).
Mutual information maximization: Incorporating variational information constraints on edge features leads to significant improvements in regression accuracy and generalization (Chen et al., 2019).
Uncertainty quantification: Bayesian extensions of MPNN/GNNs were shown to provide enhanced calibration and out-of-distribution generalization, crucial for new scaffold discovery (Lamb et al., 2020).
LLMs: Recent studies demonstrated that LLMs like LLaMA 3, when fine-tuned on QM9 SMILES strings, can perform regression with errors only 5–10× higher than deep graph-based models, outperforming baseline random forests for several properties (Jacobs et al., 9 Sep 2024).

For property prediction, models are typically benchmarked against mean absolute error (MAE) respective to chemical accuracy, geometric and energetic similarity, and in generative tasks, metrics such as validity, uniqueness, and Fréchet distances.

3. Dataset Extensions: New Properties and Benchmarks

The QM9 dataset has been structurally and functionally extended to address limitations and to provide new machine learning benchmarking tasks:

Dataset/Extension	Properties/Novelty	Reference
QM9-NMR	13C NMR shieldings for 134k molecules (vacuum + solvents), 0.8M+ C atoms	(Gupta et al., 2020)
Hessian QM9	Complete Hessian (second-derivative) matrices, vacuum and solvents	(Williams et al., 15 Aug 2024)
GW-QM9	GW-level HOMO/LUMO energies (>130k molecules); enables delta- and transfer-learning	(Fediai et al., 2023)
QH9	Full Hamiltonian matrices, MD trajectories, >130k structures	(Yu et al., 2023)

These resources provide reference data for vibrational frequencies, spectroscopic prediction, dynamic properties, and the acceleration of quantum electronic structure calculations.

4. Structural, Unsupervised, and Manifold Analysis

Unsupervised learning applied to QM9 reveals that intrinsic data dimensionality is much lower than the number of descriptive features (approx. 5 of 19). Studies using UMAP and hierarchical clustering elucidate a two-level structure: an outer region of outlier molecules and an inner core of inliners (well-clustered), with a molecule's atomic count strongly correlated with its outlier/inliner status. Most of the predictive power when inferring atomic composition is retained even when reducing to a two-dimensional latent space (Valdés et al., 2023).

Key implications include:

Highly redundant molecular property representations, promoting low-dimensional modeling.
Clear segmentation of the chemical space, aiding targeted inverse design and model robustness.

5. Advanced Machine Learning Techniques: Methodological Insights

State-of-the-art modeling on QM9 has involved a wide range of learning architectures and methodologies:

Weighted skip-connections: Enhanced interpretability by allowing the model to learn the importance of different representation layers; in QM9, atom-type embeddings dominate due to chemical composition’s role in energy variation (Nicoli et al., 2018).
Mutual information maximization: Directly constraining edge-feature transformations increases regression accuracy for quantum properties and molecular bioactivity by preserving relational chemical information (Chen et al., 2019).
Compact representations: MBDF descriptors enable linear scaling and rapid kernel evaluation, attaining accuracy competitive with high-dimensional molecular representations (Khan et al., 2023).
Generative modeling: Diffusion and transformer models trained on QM9 are capable of both inverse molecular design and property-specific molecule generation, showing strong generalization to new tasks such as deep eutectic solvent design (Luu et al., 2023, Huang et al., 2023).
Pretraining and equivariant architectures: Equivariant pretraining with physics-based losses on 3D molecular graphs (e.g., node-level force prediction) leads to improved downstream property prediction (Jiao et al., 2022).
Minimal multilevel machine learning (M3L): Learning corrections across multiple quantum chemistry levels reduces high-level (e.g., CCSD(T)) data needs by orders of magnitude, attaining chemical accuracy at dramatically reduced cost (Heinen et al., 2023).
Hamiltonian prediction: SE(3)-equivariant neural architectures (e.g., QHNet) can directly predict full Hamiltonians, enabling rapid surrogate modeling for electronic structure tasks (Yu et al., 2023).

6. Impact, Limitations, and Prospects

The introduction of QM9 established standardized benchmarks and protocol for molecular ML. Its influence has led to reproducible property prediction comparisons and the advancement of methods that can generalize across chemical space, handle diverse molecular sizes, and assimilate gradually more complex physical constraints.

Nevertheless, QM9’s restriction to small, neutral, closed-shell molecules with up to nine heavy atoms, limited element types, and ground-state, vacuum computations presents limitations for broader chemical applicability. Efforts are ongoing to extend the number of elements, increase system size, improve electronic structure accuracy (e.g., GW, CCSD(T)), and incorporate properties measured or calculated in solution or under experimental conditions (Williams et al., 15 Aug 2024, Fediai et al., 2023).

The dataset and its derivatives are frequently cited as the preferred testbed for model development, but reaching chemical accuracy for larger or more chemically diverse molecules is an open challenge. Future directions include extending QM9-level curation to more elements and reactions, linking experiment with computation, and designing more efficient transfer learning and uncertainty quantification frameworks that facilitate robust, physically-informed generalization beyond the original QM9 chemical space.