One Model, Whole Energy Curve: Nuclei Join the Network So N2 Dissociation Needs No Retraining

In neural variational Monte Carlo for the electronic Schrödinger equation, the wavefunction no longer serves one fixed nuclear geometry: nuclei carry learned representations into the network and are updated layer by layer alongside the electronic ones. Earlier routes either optimized each nuclear geometry separately or let the nuclear geometry condition the electronic wavefunction in one direction, without the nuclear representations being updated in return by the electronic ones. On the N2 dissociation curve, Li displacement in LiCN, and the ethane and formamide minimum-energy paths, one shared parameter set optimized across geometries produces energy profiles consistent with the references.

The network treats electrons and nuclei as one particle set, and a shared backbone uses group averaging over three groups (nuclei, spin-up electrons, spin-down electrons) at every layer to summarize and then update each representation. Orbital matrices are built only from electronic representations and antisymmetry is imposed only by electronic determinants, while nuclear representations generate the parameters of the orbital envelopes. On eight equilibrium small-molecule benchmarks, NucleoNet reaches variational energies comparable to FermiNet, Psiformer, and LapNet baselines with roughly 0.11 to 0.15 million trainable parameters.

Trained on a sparse set of geometries, the model extrapolates directly to configurations not included in the optimization without further wavefunction optimization. Mapping a potential energy surface previously required separate optimization at each geometry, and stochastic errors uncorrelated across geometries could make the surface noisy. N2 trained on six bond lengths and evaluated on 32, and LiCN trained on only the two endpoint configurations and evaluated on three intermediate ones, both give energy profiles close to full-geometry joint optimization.

Ablations separate the roles of envelope conditioning and nuclear streams: envelope conditioning gives a small consistent gain, while nuclear streams matter more for the multi-geometry fit. Removing the nuclear streams and keeping only electron streams raises the N2 multi-geometry training and evaluation mean absolute errors. On NH3 and N2 at equilibrium geometries and on the N2 sparse-geometry benchmark, the evaluation error is concentrated at geometries not included in the optimization.

The method targets geometric variation within a single molecular system, demonstrated on the N2 dissociation curve, Li displacement in LiCN, and the ethane and formamide minimum-energy paths, with sparse-geometry extrapolation shown along the one-dimensional N2 and LiCN coordinates. The authors propose extending the model to predict forces and support molecular dynamics as a natural next step.

Sparse-geometry extrapolation currently holds only along the one-dimensional N2 and LiCN coordinates; the same extrapolation has not been established along the multi-nuclear ethane and formamide paths. All-particle pair interactions in the shared backbone scale quadratically with particle number, so larger systems would need sparse or locality-based interactions. Particle descriptors are fixed rather than learned, and the authors suggest learned or symmetry-aware descriptors may improve generalization along multi-nuclear paths.