NucleoNet: A Nuclear-Aware Determinant Ansatz for Multi-Geometry Neural Variational Monte Carlo

Qiao Gao; Yang Huang; Enze Hou; Yuzhi Liu; Jingrun Chen; Han Wang

physics.chem-phOct 9, 2026

NucleoNet: A Nuclear-Aware Determinant Ansatz for Multi-Geometry Neural Variational Monte Carlo

Authors: Qiao Gao, Yang Huang, Enze Hou, Yuzhi Liu, Jingrun Chen, Han Wang

Organizations: School of Artificial Intelligence and Data Science, University of Science and Technology of China, Hefei 230026, China · University of Science and Technology of China, Hefei 230026, China · Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou 215213, China · Science, Mathematics and Technology Cluster, Singapore University of Technology and Design, 8 Somapah Road, 487372 Singapore ·

Abstract

Neural-network variational Monte Carlo provides an accurate route to solving the electronic Schrödinger equation, but many neural wavefunctions are optimized for a single nuclear geometry, requiring separate calculations to map out a potential energy surface. We introduce NucleoNet, a nuclear-aware determinant ansatz in which electrons and nuclei form a unified particle set propagated through a shared electron--nucleus backbone that jointly evolves one-particle and two-particle representations. Nuclei carry learned representations that evolve alongside the electronic representations and condition orbital construction, while antisymmetry is imposed only by electronic determinants. On equilibrium small-molecule benchmarks, NucleoNet achieves accuracy comparable to established neural-wavefunction baselines with substantially fewer parameters. For each molecular system, a single model optimized jointly across geometries describes the N $_2$ dissociation curve, Li displacement in LiCN, and the ethane and formamide minimum-energy paths for bond dissociation. Sparse-geometry training generalizes to unseen geometries for N $_2$ and LiCN without further optimization, and N $_2$ ablations show that nuclear streams substantially improve this interpolation. These results show that a compact wavefunction with jointly evolved nuclear representations can describe the geometries of a molecular system with a single set of parameters.

Figures & tables

Figure 1: Overview of NucleoNet. (a) Wavefunction construction. Electrons and nuclei are represented in a unified particle set and processed by a shared electron–nucleus backbone. The final electronic representations produce the Jastrow factor and orbital matrices, while the final nuclear representations generate envelope parameters. These quantities are combined to form the Slater matrices, determinant expansion, and final wavefunction. (b) Initialization layer. Fixed geometric features and particle descriptors are mapped to the initial one-particle and two-particle representations. (c) Shared electron–nucleus backbone. The initial representations are propagated through $L$ electron–nucleus layers. (d) Electron–nucleus layer. Pair-gated messages and permutation-invariant group summaries enter the one-particle update, while the two-particle representations are updated through a shared nonlinear map. (e) Mean pooling. Representations are averaged separately over nuclei, spin-up electrons, and spin-down electrons and concatenated to form the group summaries. For simplicity, the figure uses $\parallel$ for concatenation, $\cdot$ for elementwise multiplication, and $\times$ for scalar multiplication.

Figure 2: Energy error versus number of trainable parameters for equilibrium small-molecule benchmarks. The lower-left region corresponds to lower errors and fewer parameters. Results for the same molecule use the same marker style and are connected by gray lines as a visual guide. NucleoNet lies in the low-parameter, low-error region relative to FermiNet, Psiformer, and LapNet baselines. Benchmark geometries and reference energies follow the protocols of Refs. [ 23 , 29 , 20 ] .

Figure 3: Energy errors along the N 2 dissociation curve relative to the MLR

{}_{4}(6,8)

reference. A single NucleoNet model is jointly optimized on 32 bond lengths uniformly spaced between

1.6

and

6.4~a_{0}

. MAEs reported in the legend are computed over the full bond-length range.

Figure 4: Signed energy differences across the five LiCN geometries as a function of the Li–C distance, relative to independently optimized single-geometry NucleoNet energies. The Orbformer results correspond to the model fine-tuned from the LAC checkpoint; the Orbformer and CCSD(T)/CBS energies are taken from Ref. [ 3 ] . The black zero line denotes the single-geometry NucleoNet baseline, and the MSD values in the legend are averaged over the five geometries.

Figure 5: Multi-geometry dissociation benchmarks for ethane and formamide. Panels (a) and (b) show representative molecular geometries along the corresponding minimum-energy paths (MEPs). Panels (c) and (e) show mean-centered relative energies, while panels (d) and (f) show signed energy differences relative to the reference calculations reported in Ref. [ 3 ] . The MSD values are averaged over all geometries along each path. NucleoNet uses a single set of shared parameters for all geometries of each molecular system.

Figure 6: Generalization from sparse geometry sets. Blue triangles mark the training geometries, blue curves show the model optimized on the training geometries and evaluated at all geometries, and red curves show full-geometry joint optimization. The upper-right values give the corresponding benchmark metric (MAE for N 2 and MSD for LiCN), using the same references and units as in the preceding sections. For N 2 , six uniformly spaced bond lengths are used for optimization; for LiCN, only the two endpoint geometries are used.

Figure 7: Energy errors along the N 2 dissociation curve for NucleoNet and its ablation variants. Triangles mark the six training geometries, and the legend reports the MAE on the evaluation grid. The inset shows the full error range, including the large error of NucleoNet-FE-EB near the short-bond region.

Model	Envelope	Propagated streams	NH 3 $\Delta E$	N 2 $\Delta E$	N 2 training/evaluation $\Delta\mathrm{MAE}$
NucleoNet	Conditioned	Electrons+nuclei	0.00	0.00	0.00 / 0.00
NucleoNet-FE	Fixed	Electrons+nuclei	$+$ 0.01	$+$ 0.11	$+$ 0.21 / $+$ 0.23
NucleoNet-FE-EB	Fixed	Electrons only	$+$ 0.05	$+$ 0.22	$+$ 0.92 / $+$ 38.86

Table 1: Diagnostic ablation results on NH 3 single-geometry, N 2 single-geometry, and N 2 multi-geometry benchmarks. Positive values indicate worse performance. The N 2 training MAE is computed on the six training geometries, and the evaluation MAE on the grid of Sec. III.6 . All relative errors are reported in m

E_{h}

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Category	Hyperparameter	Value
SPRING	Learning-rate schedule	$(1+10^{-4}t)^{-1}$
	Initial learning rate	$5\times 10^{-2}$
	Initial damping $\lambda_{0}$	0.001
	History coefficient $\mu$	0.99
	Norm constraint	0.001
Pretraining	Optimizer	Adam

Appendix

Table S1: Default optimizer, pretraining, and VMC hyperparameters used in the reported calculations.

Category	Hyperparameter	Value
Small molecules	One-particle width $h_{1}$	64
	Two-particle width $h_{2}$	16
Multi-geometry systems	One-particle width $h_{1}$	128
	Two-particle width $h_{2}$	32
Shared architecture	Number of electron–nucleus layers	4
	Number of determinants $K$	16

Appendix

Table S2: Architecture hyperparameters used in the reported calculations.

System		Psiformer(small)	Psiformer(large)	FermiNet(small)	FermiNet(large)	LapNet	NucleoNet
LiH	Energy	-8.070528(5)	-8.070536(4)	-8.07050(1)	-8.070515(8)	-8.070523(4)	-8.070544(2)
	# parameters	1.59M	6.37M	0.68M	2.61M	1.88M	0.11M
Li 2	Energy	-14.99486(1)	-14.99485(2)	-14.99480(2)	-14.99484(2)	-14.99485(1)	-14.994913(4)
	# parameters	1.60M	6.40M	0.70M	2.68M	1.89M	0.11M
NH 3	Energy	-56.56367(2)	-56.56381(2)	-56.56347(4)	-56.56387(3)	-56.56359(2)	-56.56389(2)
	# parameters	1.62M	6.47M	0.74M	2.82M	1.93M	0.12M

Appendix

Table S3: Total energies and parameter counts for eight small molecular systems. Energies are reported in Hartree. FermiNet and Psiformer results are taken from [ 29 ] , and LapNet results are taken from [ 20 ] . Energies are reported in Hartree, and values in parentheses denote the statistical uncertainty in the last reported digit. Bold values indicate the lowest reported energy for each system.

System	Number of geometries	Effective batch size
N 2 dissociation	32	$128\times 32$
LiCN displacement	5	$820\times 5$
Ethane dissociation	20	$128\times 20$
Formamide dissociation	20	$128\times 20$

Appendix

Table S4: Batch sizes used in the multi-geometry experiments.

Bond length $L$ ( $a_{0}$ )	Energy ( $E_{h}$ )	Bond length $L$ ( $a_{0}$ )	Energy ( $E_{h}$ )
1.6000	$-109.22020(8)$	4.0774	$-109.19612(6)$
1.7548	$-109.42544(7)$	4.2323	$-109.19126(6)$
1.9097	$-109.51676(7)$	4.3871	$-109.18787(6)$
2.0645	$-109.54171(7)$	4.5419	$-109.18552(6)$
2.2194	$-109.52889(6)$	4.6968	$-109.18367(6)$
2.3742	$-109.49621(6)$	4.8516	$-109.18246(5)$

Appendix

Table S5: N 2 energies on 32 bond lengths uniformly spaced between

1.6

and

6.4~a_{0}

. Energies are reported in Hartree, and values in parentheses denote the statistical uncertainty in the last reported digit.

Geometry ID	Li–C distance ( $a_{0}$ )	Single-geometry NucleoNet ( $E_{h}$ )	Multi-geometry NucleoNet ( $E_{h}$ )
1	3.590479	$-100.38046(3)$	$-100.38045(3)$
2	3.779452	$-100.37951(3)$	$-100.37945(3)$
3	3.968425	$-100.37624(3)$	$-100.37625(3)$
4	4.346370	$-100.36622(3)$	$-100.36614(4)$
5	4.724315	$-100.35389(3)$	$-100.35385(4)$

Appendix

Table S6: Li–C distances and NucleoNet energies for the five LiCN configurations. The C and N nuclei remain fixed across all configurations, while the Li position varies along the molecular axis. The single-geometry energies are obtained from independent optimization at each geometry and serve as the baseline used in the main text. The multi-geometry energies are obtained from a single NucleoNet model optimized jointly across all five configurations. Distances are reported in

a_{0}

, energies in Hartree, and values in parentheses denote the statistical uncertainty in the last reported digit.

Image ID	Ethane ( $E_{h}$ )	Formamide ( $E_{h}$ )	Image ID	Ethane ( $E_{h}$ )	Formamide ( $E_{h}$ )
0	$-79.82448(8)$	$-169.8992(2)$	10	$-79.64326(7)$	$-169.8623(2)$
1	$-79.82254(7)$	$-169.8995(2)$	11	$-79.64816(7)$	$-169.8478(2)$
2	$-79.81594(7)$	$-169.8957(2)$	12	$-79.64788(7)$	$-169.7359(2)$
3	$-79.77018(8)$	$-169.8853(2)$	13	$-79.64827(7)$	$-169.6903(2)$
4	$-79.70948(7)$	$-169.8650(2)$	14	$-79.64937(7)$	$-169.6905(2)$
5	$-79.65531(7)$	$-169.7275(2)$	15	$-79.65006(7)$	$-169.6891(2)$

Appendix

Table S7: NucleoNet energies for the ethane and formamide dissociation benchmarks. Image IDs identify the 20 geometries along each dissociation path. Energies are reported in Hartree, and values in parentheses denote the statistical uncertainty in the last reported digit.

System	Optimization geometries	Effective batch size
N 2	6	$684\times 6$
LiCN	2	$2048\times 2$

Appendix

Table S8: Batch sizes used in the sparse-geometry experiments.

Bond length $L$ ( $a_{0}$ )	Energy ( $E_{h}$ )
1.6000	$-109.22046(5)$
2.4800	$-109.46837(5)$
3.3600	$-109.25831(5)$
4.2400	$-109.19088(5)$
5.1200	$-109.18093(4)$
6.0000	$-109.17897(4)$

Appendix

Table S9: N 2 optimization geometries and corresponding energies for the sparse-geometry experiment. Energies are reported in Hartree, and values in parentheses denote the statistical uncertainty in the last reported digit.

Bond length $L$ ( $a_{0}$ )	Energy ( $E_{h}$ )	Bond length $L$ ( $a_{0}$ )	Energy ( $E_{h}$ )
1.6000	$-109.22032(7)$	4.0774	$-109.19600(6)$
1.7548	$-109.42535(7)$	4.2323	$-109.19123(7)$
1.9097	$-109.51670(7)$	4.3871	$-109.18768(6)$
2.0645	$-109.54159(8)$	4.5419	$-109.18527(6)$
2.2194	$-109.52875(7)$	4.6968	$-109.18351(7)$
2.3742	$-109.49613(7)$	4.8516	$-109.18239(6)$

Appendix

Table S10: N 2 energies on a 32-point bond-length grid uniformly spaced between

1.6

and

6.4~a_{0}

for the sparse-geometry experiment. Energies are reported in Hartree, and values in parentheses denote the statistical uncertainty in the last reported digit.

Geometry ID	Role	Sparse-geometry energy ( $E_{h}$ )
1	Optimization	$-100.38049(3)$
2	Evaluation	$-100.37944(4)$
3	Evaluation	$-100.37626(4)$
4	Evaluation	$-100.36622(3)$
5	Optimization	$-100.35382(3)$

Appendix

Table S11: LiCN energies evaluated with the model optimized on the two endpoint geometries. The model is optimized only on the two endpoint geometries and evaluated on all five configurations without further wavefunction optimization. Energies are reported in Hartree, and values in parentheses denote the statistical uncertainty in the last reported digit.

System	NucleoNet	NucleoNet-FE	NucleoNet-FE-EB
NH 3	-56.56389(2)	-56.56388(2)	-56.56384(2)
N 2	-109.54195(3)	-109.54184(3)	-109.54173(4)

Appendix

Table S12: Absolute variational energies for the NH 3 and N 2 single-geometry ablation benchmarks. Energies are reported in Hartree, and values in parentheses denote the statistical uncertainty in the last reported digit.

Bond length $L$ ( $a_{0}$ )	NucleoNet	NucleoNet-FE	NucleoNet-FE-EB
1.6000	$-109.22046(5)$	$-109.22033(7)$	$-109.21963(9)$
2.4800	$-109.46837(5)$	$-109.46817(7)$	$-109.46775(9)$
3.3600	$-109.25831(5)$	$-109.25818(8)$	$-109.25729(9)$
4.2400	$-109.19088(5)$	$-109.19091(7)$	$-109.19006(9)$
5.1200	$-109.18093(4)$	$-109.18032(6)$	$-109.17995(8)$
6.0000	$-109.17897(4)$	$-109.17877(5)$	$-109.1777(1)$

Appendix

Table S13: Absolute N 2 energies at the six geometries used to optimize the sparse-geometry ablation models. Energies are reported in Hartree, and values in parentheses denote the statistical uncertainty in the last reported digit.

Bond length $L$ ( $a_{0}$ )	NucleoNet	NucleoNet-FE	NucleoNet-FE-EB
1.6000	$-109.22032(7)$	$-109.22020(7)$	$-109.2197(1)$
1.7548	$-109.42535(7)$	$-109.42536(7)$	$-109.4011(5)$
1.9097	$-109.51670(7)$	$-109.51666(7)$	$-108.997(3)$
2.0645	$-109.54159(8)$	$-109.54156(7)$	$-108.896(4)$
2.2194	$-109.52875(7)$	$-109.52881(7)$	$-109.5015(8)$
2.3742	$-109.49613(7)$	$-109.49617(7)$	$-109.4950(2)$

Appendix

Table S14: Absolute N 2 energies on the evaluation grid for the sparse-geometry ablation models. Energies are reported in Hartree, and values in parentheses denote the statistical uncertainty in the last reported digit.

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