Excited Pfaffians: Generalized Neural Wave Functions Across Structure and State

Abstract

Neural-network wave functions in Variational Monte Carlo (VMC) have achieved great success in accurately representing both ground and excited states. However, achieving sufficient numerical accuracy in state overlaps requires increasing the number of Monte Carlo samples, and consequently the computational cost, with the number of states. We present a nearly constant sample-size approach, Multi-State Importance Sampling (MSIS), that leverages samples from all states to estimate pairwise overlap. To efficiently evaluate all states for all samples, we introduce Excited Pfaffians. Inspired by Hartree-Fock, this architecture represents many states within a single neural network. Excited Pfaffians also serve as generalized wave functions, allowing a single model to represent multi-state potential energy surfaces. On the carbon dimer, we match the $O(N_s^4)$-scaling natural excited states while training $>200\times$ faster and modeling 50\% more states. Our favorable scaling enables us to be the first to use neural networks to find all distinct energy levels of the beryllium atom. Finally, we demonstrate that a single wave function can represent excited states across various molecules.

Figures (14)

• Figure 1: Computational scaling with the number of states. While previous works scale superlinearly, our method exhibits near-constant scaling $\mathcal{O}({N_\mathrm{s}}^{0.14})$.
• Figure 2: Illustration of the difference between single-state importance sampling entwistleElectronicExcitedStates2022szaboImprovedPenaltybasedExcitedstate2024schatzleAbinitioSimulationExcitedstate2025 and our Multi-State Importance Sampling (MSIS) estimator for overlaps. Instead of using a single state's samples to estimate its overlaps, all states' samples are reweighted, thereby increasing the effective sample size.
• Figure 3: Overview of wave function architectures for excited states. (a) Our Excited Pfaffian architecture shares a single backbone network across all states, producing shared orbitals $\Phi_{\text{Pf}}$ and state-specific selector matrices $A_s$ that combine via ${\mathrm{Pf}}(\Phi A_s \Phi^T)$. (b) The standard Slater determinant approach requires separate neural network passes and learned orbitals $\Phi_s$ for each state.
• Figure 4: Second-row atoms' (left) and molecules' (right) excitation energies. We trained an Excited Pfaffian for each group and compare to individual models pfauAccurateComputationQuantum2024szaboImprovedPenaltybasedExcitedstate2024. References from NIST sansonetti_handbook_2003 or QUEST veril_questdb_2021.
• Figure 5: Potential energy surfaces for the lowest 12 states of C$_2$. Results compared against SHCI. singlet, triplet.