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arxiv: 2605.29622 · v1 · pith:EK6JMGQKnew · submitted 2026-05-28 · 💻 cs.LG · physics.chem-ph

M\=oLe-{Λ}: Learning the Coupled-Cluster Response State for Energies, Gradients, and Properties

Andreas Burger , Luca Thiede , Abdulrahman Aldossary , Jorge A. Campos-Gonzalez-Angulo , Alex Zook , J\'er\^ome Florian Gonthier , Al\'an Aspuru-Guzik This is my paper

Pith reviewed 2026-06-29 08:25 UTC · model grok-4.3

classification 💻 cs.LG physics.chem-ph
keywords coupled-cluster theorymachine learningquantum chemistryresponse propertiesmolecular orbitalsCCSD amplitudesequivariant models
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The pith

A neural network jointly predicts the T and Λ amplitudes of CCSD to obtain energies, forces, and response properties from Hartree-Fock orbitals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces MōLe-Λ as an extension of Molecular Orbital Learning that learns both the right-hand T-amplitudes and left-hand Λ-amplitudes of CCSD. This joint prediction allows the model to deliver not only accurate energies and forces but also a range of response properties including dipoles, quadrupoles, polarizabilities, electron density, and pair density. The architecture maintains the equivariant orbital encoder, odd sign-equivariant decoding, locality, and size-extensivity of the original model. By doing so, it provides a faster surrogate for full CCSD calculations while accessing more observables than energy and gradient alone.

Core claim

MōLe-Λ predicts the full ground-state coupled-cluster singles and doubles (CCSD) response state by jointly learning right-hand amplitudes (T1, T2) and left-hand amplitudes (Λ1, Λ2) from localized Hartree-Fock molecular orbitals. The model yields accurate CC-quality energies and forces while simultaneously recovering dipoles, quadrupoles, polarizabilities, the electron density, and 2-electron observables such as the pair density, extending the speed advantage of MōLe over full CCSD.

What carries the argument

The MōLe-Λ model with additional Λ1 and Λ2 readouts mirroring the symmetry constraints of the T1 and T2 heads, preserving the equivariant orbital encoder and size-extensivity.

If this is right

  • Accurate CC-quality energies and forces are obtained alongside multiple response properties.
  • The model recovers the electron density and pair density at CCSD level.
  • It extends the speed advantage of the original MōLe over full CCSD while expanding accessible properties.
  • Provides a route to wavefunction-level surrogate models for correlated quantum chemistry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the approach transfers across chemical space, it could support property calculations on molecules larger than those feasible with direct CCSD.
  • The joint learning of T and Λ might enable consistent treatment of properties that depend on both amplitudes without separate models.
  • Size-extensivity preserved by the architecture suggests applicability to extended systems.

Load-bearing premise

That training on localized Hartree-Fock molecular orbitals is sufficient to recover the full CCSD response state with transferability across chemical space without post-hoc corrections or molecule-specific adjustments.

What would settle it

A direct comparison showing that MōLe-Λ predicted polarizabilities or pair densities deviate substantially from CCSD reference values on molecules outside the training distribution.

Figures

Figures reproduced from arXiv: 2605.29622 by Abdulrahman Aldossary, Al\'an Aspuru-Guzik, Alex Zook, Andreas Burger, J\'er\^ome Florian Gonthier, Jorge A. Campos-Gonzalez-Angulo, Luca Thiede.

Figure 1
Figure 1. Figure 1: Given a molecule, a Hartree-Fock calculation provides the molecular orbitals (MOs) represented by their coefficient matrix C. The coefficient vector is padded for each atom to ensure that all atoms have the same number of basis coefficients, and embedded in an equivariant neural network. The model then alternates (a) message passing to mix information within the MOs and (b) attention layers to mix informat… view at source ↗
Figure 2
Figure 2. Figure 2: Scaling of the energy, force, and dipole MAE of MoLe- ¯ Λ and MLIPs with increasing data. Dashed lines predict CCSD labels, solid lines predict the delta between CCSD and MP2. from the predicted amplitudes. Instead of learning each property independently, the model learns the underlying coupled-cluster representation. Coupled-cluster property reconstruction The CCSD correlation energy depends only on the r… view at source ↗
Figure 3
Figure 3. Figure 3: Mean absolute error for dipoles, quadrupoles, and polarizability on the QM7 test set. Plotted on a log scale. train–test split. For each geometry in the dataset, we com￾pute restricted Hartree–Fock orbitals and CCSD response amplitudes (T1, T2,Λ1,Λ2) at the CCSD/def2-SVP level of theory (Weigend & Ahlrichs, 2005). All amplitudes are transformed to the separately localized occupied and vir￾tual orbital gaug… view at source ↗
Figure 6
Figure 6. Figure 6: Empirical scaling timed on a H100 for alkane chains of increasing size. CCSD, MP2, and HF use GPU4PySCF (Li et al., 2025a). The Λ-equations and localization, are timed on CPU [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 4
Figure 4. Figure 4: Out-of-equilibrium tests of the force MAE along a dihedral scan of the centre bond in a butane molecule (top) and the dipole MAE of methanol stretched along the C–O bond (bottom). predictions. The ∆-MP2 Mace performs better on the di￾hedral scan, but still worse than our method, and shows erratic dipole errors on the bond stretch. MoLe- ¯ Λ consis￾tently improves over MP2 and both baselines. This suggests … view at source ↗
Figure 5
Figure 5. Figure 5: Difference in the electron density ρ(r) of MP2, MoLe- ¯ XCCSD, and MoLe- ¯ Λ compared to CCSD on ethylhexylglycerin, PubChem CID 9859093, C11H24O3. Oxygen is coloured red, nitrogen blue, carbon dark grey, and hydrogen light grey. Electron density via the 1-RDM We then evaluate the real-space electron density reconstructed from the Λ-state 1-RDM [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: Error of the on-top pair-density Π, obtained from the 2-RDM, of MP2 and MoLe- ¯ Λ compared to CCSD. target we set the atomic references to zero. All baseline runs are trained on a single NVIDIA A100 80GB GPU for 3000 epochs and follow the standard train/validation splits used throughout our QM7 experiments. MACE: For MACE, we use three interaction blocks, hidden irreps 128x0e + 128x1o, cutoff radius rmax =… view at source ↗
read the original abstract

Coupled-cluster (CC) theory is often considered the gold standard of quantum chemistry, but its high computational cost limits routine access to accurate energies, forces and response properties. While the right-hand $T$-amplitudes determine the correlated wavefunction, many practically important observables additionally require the left-hand $\Lambda$-amplitudes. We introduce M\=oLe-$\Lambda$, an extension of Molecular Orbital Learning (M\=oLe) that predicts the full ground-state coupled-cluster singles and doubles (CCSD) response state by jointly learning right-hand amplitudes $(T_1,T_2)$ and left-hand amplitudes $(\Lambda_1,\Lambda_2)$ from localized Hartree--Fock molecular orbitals. Architecturally, M\=oLe-$\Lambda$ extends M\=oLe with $\Lambda_1$ and $\Lambda_2$ readouts that mirror the symmetry constraints of the $T_1$ and $T_2$ heads, while preserving the original equivariant orbital encoder, odd sign-equivariant decoding, locality and size-extensivity. The resulting model yields accurate CC-quality energies and forces, while simultaneously recovering dipoles, quadrupoles, polarizabilities, the electron density, and 2-electron observables such as the pair density. We show that M\=oLe-$\Lambda$ further extends the speed advantage of M\=oLe over full CCSD while substantially expanding the accessible properties, providing a route to wavefunction-level surrogate models for correlated quantum chemistry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces MōLe-Λ as an extension of the MōLe framework that jointly learns the right-hand CCSD amplitudes (T1, T2) and left-hand amplitudes (Λ1, Λ2) from localized Hartree-Fock molecular orbitals. It claims that the resulting equivariant, size-extensive model delivers CC-quality energies and forces while also recovering dipoles, quadrupoles, polarizabilities, the electron density, and the pair density, thereby expanding the set of accessible properties beyond the original MōLe model.

Significance. If the numerical results and transferability claims hold, the work would be significant because it supplies a single learned surrogate for the full CCSD response state (both T and Λ), enabling efficient access to a broad suite of one- and two-electron properties that normally require separate left- and right-hand solutions. This would constitute a concrete step toward wavefunction-level machine-learning models that preserve the formal structure of coupled-cluster theory.

major comments (2)
  1. [Abstract] Abstract: the central multi-property claim (accurate CC-quality energies, forces, dipoles, quadrupoles, polarizabilities, density, and pair density) is asserted without any numerical error metrics, training-set statistics, loss-function details for the Λ heads, or out-of-distribution test results, rendering the transferability assertion impossible to evaluate from the provided text.
  2. [Abstract] The weakest load-bearing assumption—that a single model trained solely on localized HF MOs can recover both T and Λ amplitudes with sufficient accuracy and transferability across chemical space without molecule-specific corrections or canonical-orbital baselines—is stated but not accompanied by the ablation or extrapolation tests required to substantiate it.
minor comments (2)
  1. [Abstract] The notation MōLe-Λ and the description of the Λ1/Λ2 readouts mirroring the T heads would benefit from an explicit equation or diagram showing how the odd sign-equivariant decoding is applied to the left-hand amplitudes.
  2. [Abstract] The phrase 'substantially expanding the accessible properties' should be accompanied by a direct comparison table (energies/forces vs. full response set) once numerical results are added.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and indicate the revisions planned for the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central multi-property claim (accurate CC-quality energies, forces, dipoles, quadrupoles, polarizabilities, density, and pair density) is asserted without any numerical error metrics, training-set statistics, loss-function details for the λ heads, or out-of-distribution test results, rendering the transferability assertion impossible to evaluate from the provided text.

    Authors: We agree that the abstract would benefit from quantitative support. In the revised manuscript we will add representative error metrics (e.g., MAE for energies, forces, dipoles and polarizabilities), training-set size, and a brief statement on the λ-head loss functions. Out-of-distribution results already appear in Section 4.3; we will add an explicit cross-reference in the abstract. revision: yes

  2. Referee: [Abstract] The weakest load-bearing assumption—that a single model trained solely on localized HF MOs can recover both T and λ amplitudes with sufficient accuracy and transferability across chemical space without molecule-specific corrections or canonical-orbital baselines—is stated but not accompanied by the ablation or extrapolation tests required to substantiate it.

    Authors: The manuscript already reports size-extensive transferability and out-of-distribution performance on molecules of varying size and composition (Sections 4.2 and 4.4) without molecule-specific corrections. We acknowledge, however, that dedicated ablations versus canonical orbitals are absent. We will add a short discussion of this point and, if space allows, a supplementary ablation table in the revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity: trained surrogate model on external CCSD data

full rationale

The paper introduces a machine learning model (MōLe-Λ) that learns T and Λ amplitudes from localized HF MOs, trained on external CCSD computations. No equations or steps in the abstract or description reduce the claimed outputs (energies, forces, properties) to fitted inputs by construction or self-definition. The model is presented as a data-driven surrogate with independent validation implied by training on CCSD targets. No load-bearing self-citations or uniqueness theorems are invoked to force the results; the central claims rest on the learned mapping rather than tautological redefinitions. This is a standard non-circular ML surrogate setup.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The central claim rests on the existence of a sufficiently large and diverse CCSD training set computed on localized HF orbitals; the neural-network weights constitute the fitted parameters that map orbitals to amplitudes.

free parameters (1)
  • neural network weights
    All model parameters are fitted to CCSD reference data; their values are not derived from first principles.

pith-pipeline@v0.9.1-grok · 5857 in / 1052 out tokens · 24756 ms · 2026-06-29T08:25:08.116977+00:00 · methodology

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Reference graph

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