Coupled-cluster molecular properties across the main group that extrapolate beyond training size

Wenhao He; Xu Chen; Noah Song; Haowei Xu; Tim S. Hindges; Bohan Li; Zihan Lin; Yu Yao

physics.chem-phAug 18, 2026

Coupled-cluster molecular properties across the main group that extrapolate beyond training size

Authors: Wenhao He, Xu Chen, Noah Song, Haowei Xu, Tim S. Hindges, Bohan Li, Zihan Lin, Yu Yao, +5 more

Organizations: Center for Computational Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · Department of Chemistry, Emory University, Atlanta, GA 30322, USA · Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · Honda Research Institute USA, San Jose, CA 95134, USA

Abstract

Coupled-cluster theory defines the accuracy standard for molecular electronic-structure properties but scales too steeply for routine application, whereas density-functional theory is affordable yet systematically biased. We resolve this trade-off with a single equivariant network, HARP (Hamiltonian Read-out for Properties), that predicts an effective one-electron Hamiltonian from one inexpensive B3LYP/def2-SVP calculation and derives a broad suite of properties from it (energy, optical gap, dipole, quadrupole, polarizability, Mulliken atomic charges, and Mayer bond orders) at coupled-cluster accuracy across nine main-group elements, including the under-served phosphorus, sulfur, and chlorine chemistries. The model is trained on a new in-house dataset of multi-property labels computed at the CCSD(T) level for all nine elements. On a held-out test set, it reduces the error of every property by a factor of 3.8 to 270 relative to semi-local, hybrid, and double-hybrid DFT (referenced to composite CCSD(T)/cc-pVTZ), while adding only ~0.1 s wall time per molecule, delivering coupled-cluster-quality predictions at the cost of a single DFT calculation. Critically, deriving every property from a predicted Hamiltonian rather than pooling per-atom features builds the correct size-scaling into the model architecture: on pi-conjugated oligothiophenes it matches finite-field CCSD polarizability to ~1% and the EOM-CCSD optical gap to ~3% at the largest sizes where those references remain affordable (44 and 37 atoms, where a single CCSD field point already costs ~500x the model's entire inference) and extrapolates the corrected trends to 58-atom chains, a regime where pooling-based architectures fail by construction. Accurate extrapolation is therefore set by the model's inductive bias rather than by the training data.

Figures & tables

Figure 1: Model overview. From a molecule graph, an equivariant network (input embedding followed by

K

E(3)-convolution layers) produces node and edge features that feed three correction heads:

\Delta E_{\theta}

\Delta\mathbf{H}_{\theta}

, and

\Delta\mathbf{T}_{\theta}

. Each correction is added to the corresponding baseline (

E_{\mathrm{DFT}}

\mathbf{H}_{\mathrm{DFT}}

\mathbf{T}_{\mathrm{DFT}}

) from the cheap (TD)DFT (B3LYP/def2-SVP) reference to give the total energy

E

, the effective Hamiltonian

\mathbf{H}

, and the screening matrix

\mathbf{T}

. Properties are then read off by the defining quantum-mechanical operator rather than pooled from atomic features: the energy from

E

; the dipole, quadrupole, Mulliken atomic charges, Mayer bond orders, and optical gap from

\mathbf{H}

; and the polarizability from the screening matrix

\mathbf{T}

. The dashed path denotes gradients (response properties obtained by differentiation). The full neural inference adds only

\sim

0.1 s (GPU) on top of the DFT baseline, so coupled-cluster-quality properties are obtained at the cost of one DFT single point.

Figure 2: Dataset composition and coverage. a Elemental composition of the dataset across the nine main-group elements included in this work (periods 1–3). Shaded as a heatmap to show the number of molecules in the dataset containing a given element. b Distribution of molecular size, measured as atoms per structure, with median size $14$ . c UMAP embedding of the CCSD(T) property space, colored by heavy-atom count. d Distributions of the seven CCSD(T) ground-truth properties across the dataset: total ground state energy $E$ , the optical gap $E_{\mathrm{gap}}$ , the dipole $|\boldsymbol{\mu}|$ , the quadrupole $\|\mathbf{Q}\|$ , the isotropic polarizability $\alpha_{\mathrm{iso}}$ , the per-atom Mulliken charge $q$ , and the per-bond Mayer bond order (dashed lines mark medians).

property	BP86	B3LYP	DSD-PBEP86	HARP (this work)
$E$ (kcal mol -1 )	63.0	13.2	15.1	0.23
$E_{\mathrm{gap}}$ (eV)	1.32	0.61	4.84	0.067
$\|\mu\|$ (Debye)	0.160	0.159	0.109	0.022
$\\|Q\\|$ (a.u.)	0.389	0.347	0.233	0.044
$\alpha_{\mathrm{iso}}$ (a.u.)	4.71	2.79	1.18	0.121
$q$ (e)	0.0205	0.0256	0.0220	0.0053

Table 1: Global mean absolute error (MAE) versus the composite CCSD(T)/cc-pVTZ ground truth (Methods) on the 959-molecule test set, per property and method. Lower is better; best in bold. Units:

E

in kcal mol -1 ,

E_{\mathrm{gap}}

in eV,

|\mu|

in Debye;

\|Q\|

and

\alpha_{\mathrm{iso}}

in a.u.; charge in

e

. Each method is evaluated at the basis set of the property’s coupled-cluster reference: cc-pVTZ for

E

(compared as an atomization energy),

|\mu|

\|Q\|

q

and bond order, and cc-pVDZ for

\alpha_{\mathrm{iso}}

and

E_{\mathrm{gap}}

(finite-field CCSD and EOM-CCSD, respectively). DSD-PBEP86 is the D3(BJ)-corrected double hybrid.

Figure 3: Bulk accuracy and cost. a Per-molecule wall time per single-point evaluation versus atom count (log–log) with power-law fits; the model shares its DFT baseline’s shallow scaling, while CCSD(T) and the double-hybrid scale steeply. b Per-property error radar on a shared log scale (inner $\to$ smaller error vs. CCSD(T)). c Per-element MAE across the nine supported elements, per-atom averaged; units: Mulliken charge in $e$ , $E$ in kcal mol -1 /atom, $E_{\mathrm{gap}}$ in eV, $\mu$ in D/atom, $Q$ and $\alpha_{\mathrm{iso}}$ in a.u./atom. As in Table 1 , each functional is evaluated at the basis of the property’s coupled-cluster reference (cc-pVTZ for $E$ , $|\mu|$ , $\|Q\|$ , $q$ and bond order; cc-pVDZ for $\alpha_{\mathrm{iso}}$ and $E_{\mathrm{gap}}$ ), all referenced to composite CCSD(T)/cc-pVTZ (Methods), finite-field CCSD/cc-pVDZ and EOM-CCSD/cc-pVDZ; panel a times the cc-pVTZ single point of each functional against DLPNO-CCSD(T)/cc-pVTZ, the cc-pVTZ component of the composite reference.

Electronic-structure (Hamiltonian) read-out
		Output properties
Method	Elements	$E$	$E_{\mathrm{g}}$	$\mu$	$Q$	$\alpha$	$q$	BO	Reference	Public
HARP (this work)	9 (9/9)	✓	✓	✓	✓	✓	✓	✓	CCSD(T) (all)	✓
QHNet [ 12 ]	5 (5/9)	✓	✓	✓	✓	✓	✓	✓	B3LYP	✓
HELM [ 13 ]	58 (9/9)	✓	✓	✓	✓	✓	✓	✓	$\omega$ B97M-V	$\times$
Machine-learned density functional

Table 2: What released machine-learning models can deliver on this benchmark’s chemistry and property suite. A capability/coverage map, not a head-to-head error table (for accuracy see Table 1 ): across the three families of ML approaches to molecular properties, no released model simultaneously spans the nine-element chemistry, outputs the full property suite, and targets a correlated reference. Elements states each released model’s own coverage—several exceed this work’s scope—with, in parentheses, how many of this benchmark’s nine elements (H, C, N, O, F, Si, P, S, Cl) it spans: QHNet lacks Si, P, S, Cl; the released SpookyNet (QM7-X) lacks F, Si, P; MACE-OFF23 lacks Si. Symbols: ✓ validated output of the cited model; ✓ obtainable from the model’s predicted electronic-structure objects (Hamiltonian, density, or charges) but not demonstrated in the cited work; – not accessible. Properties: total/atomization energy

E

, HOMO–LUMO gap

E_{\mathrm{g}}

, dipole

\mu

, quadrupole

Q

, polarizability

\alpha

, atomic charge

q

, bond order (BO); this work reports Mulliken charges and Mayer bond orders, whereas the population convention differs among the other entries (SpookyNet, e.g., predicts its own learned partial charges). HELM’s weights are announced but, as of this writing, not public (its dataset is); MACE-OFF23 dipoles are validated only in its separate

-\mu

variant; SpookyNet dipoles follow from its predicted partial charges but are not benchmarked. The Reference column is the correlated level the tabulated properties are fit to: for the machine-learned functionals (DM21, Skala) only the energy targets a correlated reference (

E

), whereas HARP targets one for every property (all).

Figure 4: Agreement with experiment. Predicted versus measured (NIST CCCBDB) gas-phase dipole magnitudes for twelve molecules spanning the model’s element scope; the line is

y=x

. Mean absolute error

0.064

Debye (RMSE

0.079

Debye), compared with the B3LYP baseline and

0.046

Debye for a canonical CCSD(T)/def2-TZVPP reference. SiH 4 is at the origin by

T_{d}

symmetry.

Figure 5: Out-of-distribution size scaling on oligothiophenes

\mathrm{T}_{n}

(

n=1

–

8

9

–

58

atoms). The direct read-out ablation (blue) shares the backbone architecture and dataset of HARP but predicts each property directly, as a correction to the BP86 baseline; a Hamiltonian read-out on the same BP86 baseline does not drift (SI). a Isotropic polarizability

\alpha_{\mathrm{iso}}

versus size: the Hamiltonian read-out of this work (HARP, red) agrees with finite-field CCSD/cc-pVDZ (black, dashed) to within

1.3\%

everywhere CCSD is computable (

n\leq 6

, up to 44 atoms) and continues the trend to 58 atoms, whereas the direct property read-out (blue; the pooling-style

\Delta

-prediction ablation) over-polarizes, drifting toward the BP86 baseline (grey). b Optical gap versus size: HARP (red) reproduces the EOM-CCSD/cc-pVDZ gap where that reference is affordable (

\mathrm{T}_{1}

–

\mathrm{T}_{5}

, black; within

0.03

eV through

\mathrm{T}_{2}

and within

0.13

eV,

3.4\%

, at

\mathrm{T}_{5}

) and saturates with length; a Kohn–Sham (DFT) gap is not an optical gap and is not shown. The mechanism (frontier-orbital delocalization) and the longitudinal

\alpha_{xx}

are analysed in the Supplementary Information.

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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

Λ

-amplitudes. We introduce MōLe-

Λ

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(T_1,T_2)

and left-hand amplitudes

(Λ_1,Λ_2)

from localized Hartree--Fock molecular orbitals. Architecturally, MōLe-

Λ

extends MōLe with

Λ_1

and

Λ_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 MōLe-

Λ

further extends the speed advantage of MōLe over full CCSD while substantially expanding the accessible properties, providing a route to wavefunction-level surrogate models for correlated quantum chemistry.

Abstract

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