OMol_CSH_58k: Hamiltonian Matrix Dataset

• The paper introduces a Hamiltonian matrix dataset that integrates full electronic structure data with extensive orbital interactions for 56,657 molecules, enabling enhanced pretraining strategies.
• OMol_CSH_58k is characterized by a comprehensive sampling of atomic elements, orbital interactions, and interatomic distances, with a 12 Å cutoff to balance sparsity and accuracy.
• The dataset underpins HELM’s equivariant neural network architecture, demonstrating up to 2× lower MAE in energy predictions through effective Hamiltonian pretraining.

OMol_CSH_58k is a curated Hamiltonian matrix dataset of molecules released alongside HELM, “Hamiltonian-trained Electronic-structure Learning for Molecules,” for integrating orbital interaction data within the Hamiltonian matrix $\mathbf{H}$ into training pipelines for atomic-level properties. It is described as a dataset with unprecedented elemental diversity, molecular size, and basis set, and is intended to support machine learning on electronic structure across much of the periodic table rather than only narrow organic subdomains (Kaniselvan et al., 30 Sep 2025).

1. Definition and scope

OMol_CSH_58k contains 56,657 molecules and 58 atomic elements. The included elements are all first 83 atomic elements except first-row transition metals (Sc–Zn) and lanthanides (Ce–Lu); these are omitted to avoid impractical memory scaling from high angular momentum orbitals (“g” functions). Molecular sizes range from 10 to 150 atoms, with mean $\approx 59$, and the dataset samples 4,258 edges per molecule on average. The max interatomic distance sampled is up to 15 Å, with cutoff/clip at 12 Å for matrix sparsity and practicality (Kaniselvan et al., 30 Sep 2025).

The dataset is explicitly organized around three sampled degrees of freedom: atomic elements, orbital interactions, and inter-atomic distances. The stated sampling strategy is to ensure wide elemental diversity, including $\sim 1000$ occurrences of each heavy element, while also covering all orbital/block types for included atoms and a broad range of distances, which is emphasized as especially important for long-range and heavy-atom interactions (Kaniselvan et al., 30 Sep 2025).

A common point of confusion is the dataset name. The label “CSH” does not denote a narrowly defined carbon–sulfur–hydrogen chemistry dataset. Instead, the defining object is the Kohn–Sham Fock/Hamiltonian matrix for a broad molecular collection. This suggests that the name should be interpreted operationally, through the electronic-structure content and benchmark role, rather than through a compositional restriction.

2. Provenance, selection, and dataset splits

OMol_CSH_58k is a subset of the Open Molecules 2025 (OMol25) dataset (>110M molecules with DFT-level data). The downsampling is described as being curated specifically for maximal diversity in electronic structure data, with two stated goals: maximizes element coverage per structure and avoids redundancy in orbital interactions and long-range sparsity (Kaniselvan et al., 30 Sep 2025).

The released splits distinguish the main training corpus from two evaluation sets that probe different generalization regimes. The “all elements” test set evaluates performance across the full elemental span, whereas the “common organic elements” test set restricts chemistry to a smaller subset.

Split	Molecules	Elements
Train	56,657	all 58
Test 1 (“OMol_common_1k”)	1,006	H, C, N, O, F, S, Cl, Br
Test 2 (“OMol_all_5k”)	4,937	all 58

Additional structural statistics are reported for these partitions. The training set has Avg #Atoms = 59, Avg #Edges = 4,258, #H_ii (intra) = 3,362,755, and #H_ij (inter) = 241,227,574. “OMol_common_1k” has Avg #Atoms = 98, Avg #Edges = 9,086, #H_ii = 98,399, and #H_ij = 9,140,166. “OMol_all_5k” has Avg #Atoms = 75, Avg #Edges = 7,396, #H_ii = 370,924, and #H_ij = 36,514,364. All three splits are marked as containing energies (Kaniselvan et al., 30 Sep 2025).

These split definitions matter methodologically because the dataset is not only large; it is also stratified to test whether learned representations survive both a restricted organic regime and a chemically broad regime. A plausible implication is that benchmark performance on OMol_common_1k and OMol_all_5k probes different failure modes: compositional extrapolation in the latter and representation quality within a more standard molecular subset in the former.

3. Electronic-structure content and representation

For each structure, OMol_CSH_58k stores the full Kohn–Sham Fock/Hamiltonian matrix together with total energies and atomic forces. The basis set is def2-TZVPD (with diffuse functions), described as triple-zeta valence, polarization, and diffuse basis, and it supports high angular momentum orbitals (up to f-functions; l_{max}=6) (Kaniselvan et al., 30 Sep 2025).

The Hamiltonian is framed through the standard Kohn–Sham expansion

$$\psi_i(\mathbf{r}) = \sum_\mu C_{\mu i} \chi_\mu(\mathbf{r}),$$

with the matrix equation

$$\sum_\nu F_{\mu\nu} C_{\nu i} = \epsilon_i \sum_\nu S_{\mu\nu} C_{\nu i},$$

where $F_{\mu\nu}$ is the Fock/Hamiltonian matrix, $S_{\mu\nu} = \langle \chi_\mu | \chi_\nu \rangle$ is the overlap, $\epsilon_i$ are orbital energies, and $C_{\mu i}$ are molecular orbital coefficients (Kaniselvan et al., 30 Sep 2025).

The total DFT energy is written in terms of the density matrix $P$,

$\approx 59$0

and

$\approx 59$1

with

$\approx 59$2

This formulation is central to the dataset’s rationale: the Hamiltonian is treated not as an auxiliary artifact of a quantum-chemistry calculation, but as a rich supervisory target that contains structured electronic information underlying energies and forces (Kaniselvan et al., 30 Sep 2025).

The dataset further resolves orbital interactions into angular-momentum channels. Each block of $\approx 59$3 for orbital interaction $\approx 59$4 is decomposed as

$\approx 59$5

The paper states that this irreducible-representation (irrep) decomposition allows the ML model to process the data with proper rotation/equivariant symmetry. A representative Clebsch–Gordan form is also given: $\approx 59$6 For OMol_CSH_58k, the reported features per node/edge are $\approx 59$7, corresponding to 4096 elements total per node/edge for targets (Kaniselvan et al., 30 Sep 2025).

4. Construction details and normalization choices

The quantum-chemical calculations are described as Kohn–Sham DFT, with functional details inherited from OMol25; the paper mentions $\approx 59$8B97M-V and $\approx 59$9B97X-D for various benchmarks. Within OMol_CSH_58k itself, the defining stored object is the def2-TZVPD Hamiltonian matrix rather than only scalar observables (Kaniselvan et al., 30 Sep 2025).

A key implementation choice is the interatomic distance cutoff: 12 Å for matrix labels. The stated justification is that above this distance, matrix elements are negligible, specifically $\sim 1000$0, which balances accuracy and data size. This is not a generic graph cutoff applied only to neighborhood construction; it is part of the label-generation protocol for the Hamiltonian targets (Kaniselvan et al., 30 Sep 2025).

The dataset also uses special normalization for the scalar ($\sim 1000$1) blocks, because heavy elements have much larger trace contributions. This normalization detail is important for cross-element learning, since the dataset’s elemental span includes heavy atoms and aims to avoid a regime in which target magnitudes are dominated by a small subset of basis blocks.

Per-molecule data consist of the Hamiltonian matrix $\sim 1000$2, the total electronic energy $\sim 1000$3, and atomic forces. Per-atom and per-edge data contain all orbital interactions, including both intra- and inter-atomic cases, in the decomposed representation above (Kaniselvan et al., 30 Sep 2025). This suggests a supervisory density much larger than conventional energy-only datasets, because every structure contributes a large number of local and pairwise electronic labels.

5. HELM and Hamiltonian pretraining

OMol_CSH_58k underpins HELM, a scalable neural network based on equivariant message passing with two heads: a Hamiltonian prediction head that predicts $\sim 1000$4 blocks for all intra- and interatomic pairs using spherical harmonic/irrep structure, and an energy head that maps node embeddings to total energy (Kaniselvan et al., 30 Sep 2025).

The main supervised task is Hamiltonian matrix prediction. The paper reports state-of-the-art performance on OMol_CSH_58k compared to previous baselines, with OMol_all_5k test set error ~2195 $\sim 1000$5. The second task is transfer learning/energy prediction, where the shared embeddings learned during Hamiltonian prediction are reused for scalar-property regression (Kaniselvan et al., 30 Sep 2025).

The paper defines Hamiltonian pretraining as pretraining the network backbone on the Hamiltonian matrix $\sim 1000$6, then transferring to downstream properties such as total energy. The stated rationale is combinatorial: every structure provides $\sim 1000$7 Hamiltonian labels, but only 1 energy and $\sim 1000$8 force labels. Two downstream procedures are given: either freeze the backbone and train a small energy head, or finetune the whole network (backbone + energy head) with a small energy dataset (Kaniselvan et al., 30 Sep 2025).

The reported empirical consequence is that test accuracy in energy prediction is dramatically higher (up to 2× lower MAE) when using Hamiltonian pretraining, especially in low-data regimes. A specific example given is that in ∇²DFT, 2k energy examples + Hamiltonian pretraining outperforms direct energy training even with 10k examples. The embedding geometry is also described: pretraining on $\sim 1000$9 yields clear element clusters and better irrep utilization in UMAP plots, whereas energy-only training mostly learns C/H clusters (Kaniselvan et al., 30 Sep 2025).

The broader interpretation advanced in the paper is that the electronic Hamiltonian encodes all possible 1-electron physics, and therefore contains information relevant to energies, forces, densities, multipole moments, and related observables. This suggests that Hamiltonian supervision can serve as a high-bandwidth intermediate target for universal molecular representation learning.

6. Position among adjacent datasets and disambiguation

OMol_CSH_58k is best understood in relation to broader molecular electronic-structure datasets and to similarly named but substantively unrelated resources. A useful comparison is OE62, which provides 61,489 unique organic molecules extracted from crystal structures, with PBE geometries, PBE and PBE0 orbital eigenvalues and total energies for the full set, PBE0 in water for 30,876 molecules, and $$\psi_i(\mathbf{r}) = \sum_\mu C_{\mu i} \chi_\mu(\mathbf{r}),$$0@PBE0 quasiparticle energies for 5,239 molecules (Stuke et al., 2020). OE62 is a large multi-fidelity spectroscopy dataset, but its content is organized around geometries, orbital energies, charges, and GW benchmarks rather than full Hamiltonian matrices. OMol_CSH_58k differs in making the Hamiltonian itself the primary supervised object and in extending to 58 elements and up to 150 atoms (Kaniselvan et al., 30 Sep 2025).

The name can also be confused with domains outside molecular ML. It is unrelated to the ExoMol line-list work on eight isotopologues of carbon monosulphide, which concerns comprehensive vibration-rotation line lists for CS in the ground electronic states and temperatures up to 3000 K (Paulose et al., 2015). It is likewise unrelated to the high-pressure study of C-S-H compounds up to 143 GPa, which concerns I4/mcm, C2/c, and Pnma crystal structures in compressed carbon–sulfur–hydrogen materials (Bykova et al., 2020). In OMol_CSH_58k, “Hamiltonian matrix dataset” is the correct categorical description.

A common misconception is that OMol_CSH_58k covers the entire periodic table. It does not: it covers 58 elements, explicitly excluding Sc–Zn and Ce–Lu. Another misconception is that Hamiltonian pretraining eliminates the need for downstream property supervision. The paper instead describes a pretrain-then-transfer workflow, with either frozen-backbone or full-finetuning variants. These clarifications are important because the dataset’s significance lies not in unrestricted chemical coverage or in replacing conventional learning objectives, but in making electronic interactions as a rich and transferable data source for representing chemical space (Kaniselvan et al., 30 Sep 2025).