Spin-Adapted Neural Network Wavefunctions in Real Space

TL;DR

This work introduces the Spin-Adapted Antisymmetrization Method (SAAM) to enforce exact total spin symmetry in real-space antisymmetric wavefunctions, enabling spin-pure neural network quantum Monte Carlo (NNQMC) without hyperparameters. By decoupling spin and spatial components and representing the spatial part with Neural Network Orbitals (NNOs), SAAM yields compact, chemically interpretable wavefunctions that capture both static and dynamic correlation. The approach achieves chemically accurate spin gaps and excitation energies in biradicals and carbon dimers, and provides detailed spin-state energetics for iron–sulfur clusters, demonstrating its effectiveness for strongly correlated systems. SAAM integrates seamlessly with existing NNQMC frameworks, offering a hyperparameter-free, principled pathway to embed physical spin priors into machine-learned wavefunctions and extend ab initio simulations to challenging multireference regimes.

Abstract

Spin plays a fundamental role in understanding electronic structure, yet many real-space wavefunction methods fail to adequately consider it. We introduce the Spin-Adapted Antisymmetrization Method (SAAM), a general procedure that enforces exact total spin symmetry for antisymmetric many-electron wavefunctions in real space. In the context of neural network-based quantum Monte Carlo (NNQMC), SAAM leverages the expressiveness of deep neural networks to capture electron correlation while enforcing exact spin adaptation via group representation theory. This framework provides a principled route to embed physical priors into otherwise black-box neural network wavefunctions, yielding a compact representation of correlated system with neural network orbitals. Compared with existing treatments of spin in NNQMC, SAAM is more accurate and efficient, achieving exact spin purity without any additional tunable hyperparameters. To demonstrate its effectiveness, we apply SAAM to study the spin ladder of iron-sulfur clusters, a long-standing challenge for many-body methods due to their dense spectrum of nearly degenerate spin states. Our results reveal accurate resolution of low-lying spin states and spin gaps in [Fe$_2$S$_2$] and [Fe$_4$S$_4$] clusters, offering new insights into their electronic structures. In sum, these findings establish SAAM as a robust, hyperparameter-free standard for spin-adapted NNQMC, particularly for strongly correlated systems.

Figures (4)

• Figure 1: Overview of SAAM. We use a spin-spatial decoupled ansätz to represent the complicated spin structure in molecular systems. Spatial part $\Psi$ (blue panel): Following the convention of continuous-space NNQMC, electron coordinates are processed by a permutation-equivariant neural network to capture many-body correlations. The network outputs the spatial matrix $M_{\text{space}}$, where rows correspond to $\psi_i$ and columns correspond to electron positions. Spin part $\Theta_S$ (grey panel): The spin structure is assigned based on chemical prior knowledge and decomposed into products of one-body spin functions. For clarity, the decomposition is illustrated using standard spin-up/down functions. The spatial and spin parts are combined and antisymmetrized to yield a spin-adapted neural network wavefunction (bottom right, pink panel). Concretely, the spatial output is arranged into a matrix, in which the lowest $N_c$ rows (core orbitals) are duplicated to represent double occupied orbitals. Matrices derived from the spin decomposition and electron spin coordinates are then multiplied with the matrix from spatial part. Here, we use the dashed matrix in the spin part to represent the rows related to the core orbitals. The final spin-adapted neural network wavefunction is obtained as a weighted sum of determinants of these matrices, with coefficients determined by the spin decomposition.
• Figure 2: Classical quantum chemistry concepts extended to NNOs. NNOs allow natural extension of classical quantum chemistry concepts, including core/active orbitals, state-averaged, and state-specific excited-state calculations. This bridges between chemical prior knowledge with NNQMC algorithms. Compared to one-body molecular orbitals, NNOs are able to capture electron correlations, enabling a compact, chemical-inspired representation of real-world electron wavefunction with fewer configuration state functions (CSF).
• Figure 3: Benchmark of SA-LapNet.a. Absolute deviations of calculated singlet-triplet (S-T) gaps are shown for a diverse set of biradical systems. The reference experimental results are from Ref.biradical_sheebiradical_lee. Our results (solid bars) are compared with $S_{+}$ penalty spluspenalty (hatched bars), and auxiliary-field quantum Monte Carlo biradical_leebiradical_shee (AFQMC, dotted bars). Both NNQMC methods are extrapolated according to Ref.fu2024variance. The shaded region marks the threshold of chemical accuracy (1 kcal/mol). Our method consistently achieves or approaches chemical accuracy across the whole benchmark set, aligned well with the $S_{+}$ penalty results. It is also more efficient than $S_{+}$ penalty, as it avoids additional penalty terms. b. Energy level of the carbon dimer at $r=1.244$ Å. SA-LapNet (blue solid line) provides excitation energies in close agreement with reference methods including semistochastic heat-bath configuration interaction 10.1063/1.4998614 (SHCI, orange dashed line) and natural excited state NES (NES, green dashed line).
• Figure 4: SA-LapNet calculations for Iron-Sulfur Clusters. In this figure, the balls of different colors represent different elements: red is iron, yellow is sulfur, gray is carbon, and white is hydrogen. a. Structure of the [Fe2S2]^2+ cluster. b. Schematic plot of spin coupling for [Fe2S2]^2+. c. Energy ladder obtained from state-averaged, state-specific, and $S=0$ finetuned training schemes. The estimated magnetic coupling constant from state-specific calculation is $J=0.967$ mHa. d. Structure of the [Fe2S2(SCH3)4]^2- complex. e. $S=0$ finetuned energy ladder. The estimated magnetic coupling constant is $J=1.070$ mHa. f. Local spin $\langle S_A^2 \rangle$ of each state compared with FCIQMC-SCF(10e,10o)dobrautz2021spin, FCIQMC-SCF(22e,26o)dobrautz2021spin, DMRG-CI(30e,32o)sharma2014low, and Heisenberg predictions. These results indicate that the NNOs can well capture the electron delocalization to sulfurs. g. Structure of the Fe4S4(SCH3)4 cluster. h. Schematic plot of spin-coupling pathways into the $S=0$ state. i. Spin gap of Fe4S4(SCH3)4 predicted by SA-LapNet compared with results from FCIQMC-CI and FCIQMC-SCF with active space of (20e,20o) dobrautz2021spin. Orbital relaxation refers to the molecular orbital recombination for different spin configurations. The orbital correlation refers to the orbital distortion caused by the many-body correlation. Both orbital relaxation and orbital correlation enlarge the predicted spin gap.