Spin-adapted neural network backflow for strongly correlated electrons

Accurately describing strongly correlated electrons in systems such as transition metal complexes requires strict adherence to spin symmetry, a feature largely absent in modern neural-network-based variational wavefunctions. This deficiency can lead to severe spin contamination in simulating systems with near-degenerate spin states. To resolve this limitation, we present a spin-adapted neural network backflow (SA-NNBF) ansatz, formulated in second quantization for fermionic lattice models and ab initio quantum chemistry. Our approach constructs a fully antisymmetric wavefunction by combining a neural-network backflow spatial component with a spin eigenfunction expressed in a sum-of-products form. To address the computational complexity of spin adaptation, we introduce a tensor compression algorithm for spin eigenfunctions, and a more compact wavefunction representation based on the particle-hole duality in second quantization. These advancements enable variational Monte Carlo calculations using SA-NNBF for challenging molecular systems with more than one hundred electrons, including the FeMo-cofactor (FeMoco) in nitrogenase. Applications to prototypical strongly correlated molecules demonstrate that SA-NNBF consistently outperforms standard NNBF with a similar number of parameters. Furthermore, it surpasses the accuracy of the state-of-the-art spin-adapted density matrix renormalization group (SA-DMRG) algorithm for FeMoco with a significantly reduced computational resource. Our work establishes a foundational framework for exploring fully symmetry-preserving neural-network quantum states for interacting fermion problems.