---
title: 'OMol_CSH_58k: Hamiltonian Matrix Dataset'
url: https://www.emergentmind.com/topics/omol_csh_58k
type: topic
---

# OMol_CSH_58k: Hamiltonian Matrix Dataset

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 [2510.00224].

## 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** [2510.00224].

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** [2510.00224].

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** [2510.00224].

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** [2510.00224].

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`)** [2510.00224].

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** [2510.00224].

The total DFT energy is written in terms of the density matrix \(P\),
\[
P = 2C_{occ}C_{occ}^\dagger,
\]
and
\[
E = \frac{1}{2}\mathrm{Tr}[P(H_{core} + F)] + E_{XC} - \mathrm{Tr}[PV_{XC}] + E_{NN},
\]
with
\[
E_{NN} = \sum_{i<j} Z_i Z_j / r_{ij}.
\]
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 [2510.00224].

The dataset further resolves orbital interactions into angular-momentum channels. Each block of \(\mathbf{H}\) for orbital interaction \((l_1,l_2)\) is decomposed as
\[
l_1 \otimes l_2 = \bigoplus_{L=|l_1 - l_2|}^{l_1 + l_2} L.
\]
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:
\[
|L M\rangle = \sum_{m_1, m_2} C^{L M}_{l_1 m_1, l_2 m_2} |l_1 m_1\rangle |l_2 m_2\rangle.
\]
For OMol_CSH_58k, the reported **features per node/edge** are  
**$114\times 0e+229\times1e+239\times2e+161\times3e+73\times4e+24\times5e+4\times6e$**, corresponding to **4096 elements total per node/edge for targets** [2510.00224].

## 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 **$\omega$B97M-V** and **$\omega$B97X-D** for various benchmarks. Within OMol_CSH_58k itself, the defining stored object is the **def2-TZVPD** Hamiltonian matrix rather than only scalar observables [2510.00224].

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 **\(|H_{ij}| < 10^{-5}\)**, 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 [2510.00224].

The dataset also uses **special normalization for the scalar (\(l=0\)) 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 \((\mathbf{H})\)**, the **total electronic energy \((E)\)**, 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 [2510.00224]. 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 \(\mathbf{H}\) blocks for all intra- and interatomic pairs using spherical harmonic/irrep structure, and an **energy head** that maps node embeddings to total energy [2510.00224].

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 \(\mu E_h\)**. The second task is **transfer learning/energy prediction**, where the shared embeddings learned during Hamiltonian prediction are reused for scalar-property regression [2510.00224].

The paper defines **Hamiltonian pretraining** as pretraining the network backbone on the Hamiltonian matrix \(\mathbf{H}\), then transferring to downstream properties such as total energy. The stated rationale is combinatorial: **every structure provides \(\mathcal{O}(N^2)\) Hamiltonian labels, but only 1 energy and \(\mathcal{O}(N)\) 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** [2510.00224].

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 \(\mathbf{H}\) yields **clear element clusters** and **better irrep utilization in UMAP plots**, whereas **energy-only training mostly learns C/H clusters** [2510.00224].

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 **\(G_0W_0\)@PBE0** quasiparticle energies for **5,239 molecules** [2001.08954]. 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** [2510.00224].

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** [1507.01275]. 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 [2012.10528]. 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** [2510.00224].

Source: https://www.emergentmind.com/topics/omol_csh_58k