Large Electron Model: A Universal Ground State Predictor

Abstract

We introduce Large Electron Model, a single neural network model that produces variational wavefunctions of interacting electrons over the entire Hamiltonian parameter manifold. Our model employs the Fermi Sets architecture, a universal representation of many-body fermionic wavefunctions, which is further conditioned on Hamiltonian parameter and particle number. On interacting electrons in a two-dimensional harmonic potential, a single trained model accurately predicts the ground state wavefunction while generalizing across unseen coupling strengths and particle-number sectors, producing both accurate real-space charge densities and ground state energies, even up to $50$ particles. Our results establish a foundation model method for material discovery that is grounded in the variational principle, while accurately treating strong electron correlation beyond the capacity of density functional theory.

Figures (9)

• Figure 1: Foundation model architecture. A Multilayer Perceptron (MLP) with determinant head (left) constructs a set of $K$ signature encoders formulated as Slater determinants with orbitals represented by the MLP. Electron correlations are captured by a permutation-symmetric function represented by a transformer with symmetric head (right). Conditioning both parts with the parameters $\bm \Lambda_i$ yields the continuous parameterization of the many-body wavefunction $\Psi_{\bm \Lambda}(\{\bm r_i\})$.
• Figure 2: Workflow. The network represents the many-body wavefunction with continuous parameter dependence $\Psi_\mathbf{\Lambda}(\{ \mathbf{r}_i \})$. The loss is computed as the total energy $\sum_i \langle \Psi_{ \mathbf{\Lambda}_i}|H_{ \mathbf{\Lambda}_i}| \Psi_{ \mathbf{\Lambda}_i}\rangle$ accumulated over a fixed set of system parameters $\{ \mathbf{\Lambda}_i\}$, where the individual energy expectation values are computed using Monte-Carlo sampling. From the gradients of the total energy, an optimizer routine determines updates of network weights $\theta$.
• Figure 3: Real space particle density from inference for various parameter values. The first and second row has $N=10$ and $N = 7$, respectively, with the columns for $\lambda=1.0$ (optimized for $N = 10$, predicted for $N = 7$), $5.0$ (optimized for $N = 10$, predicted for $N = 7$), $7.0$ (predicted).
• Figure 4: Train and Predict results across $(\lambda, N)$ particle grid. Model is able to have pinpoint accuracy on out of sample results, able to generate wavefunction and any equal time observable for any parameter, across Hilbert space sectors, at inference time. Energies are competitive with state of the art benchmarks ghosal2007incipient, of which a detailed table is provided in the supplementary material supplementary. The statistical errors on the energies are far smaller than the radius of the dots, and therefore omitted in the figure.
• Figure 5: Train and Predict Results across for $N=50$ over $\lambda$ parameter manifold. Model is able to have pinpoint accuracy on out of sample results, able to generate ground state energies that are even lower than single parameter training and evaluation runs (highlighted in the inset). The statistical errors on the energies are far smaller than the radius of the dots, and therefore omitted in the figure.