Classical solution of the FeMo-cofactor model to chemical accuracy and its implications

TL;DR

This work tackles the FeMo-cofactor electronic-structure problem by delivering a classical solution at chemical accuracy for a widely used quantum-resource model (LLDUC) and a QM/MM-augmented extension. By combining spin-unrestricted CC and high-level DMRG with a careful reference-filtering protocol and symmetry considerations, the authors achieve a ground-state energy estimate of $E_0 \,\approx\,-22140.4106$ Hartrees with an uncertainty of about $0.0005$ Ha (≈ $0.31$ kcal/mol). They find near-degeneracy between spin isomers BS7-235 and BS8-237, reveal a dense low-energy manifold, and demonstrate that environmental fluctuations modestly shift relative energies while preserving the qualitative landscape. Beyond the LLDUC model, the study introduces a scalable protocol for extending active spaces and demonstrates oxidation-state calibration and analysis consistent with experiment, outlining a practical path toward high-accuracy, QM/MM-based understanding of FeMo-co electronic structure and potential quantum-computation advantages. The results suggest that simpler single-reference methods, when used with appropriate orbital bases and referencing, can correctly rank low-energy states, enabling realistic reaction-mechanism modeling for nitrogenase with future quantum or ML-augmented approaches.

Abstract

The main source of reduced nitrogen for living things comes from nitrogenase, which converts N2 to NH3 at the FeMo-cofactor (FeMo-co). Because of its role in supporting life, the uncertainty surrounding the catalytic cycle, and its compositional richness with eight transition metal ions, FeMo-co has fascinated scientists for decades. After much effort, the complete atomic structure was resolved. However, its electronic structure, central to reactivity, remains under intense debate. FeMo-co's complexity, arising from many unpaired electrons, has led to suggestions that it lies beyond the reach of classical computing. Consequently, there has been much interest in the potential of quantum algorithms to compute its electronic structure. Estimating the cost to compute the ground-state to chemical accuracy (~1 kcal/mol) within one or more FeMo-co models is a common benchmark of quantum algorithms in quantum chemistry, with numerous resource estimates in the literature. Here we address how to perform the same task using classical computation. We use a 76 orbital/152 qubit resting state model, the subject of most quantum resource estimates. Based on insight into the multiple configuration nature of the states, we devise classical protocols that yield rigorous or empirical upper bounds to the ground-state energy. Extrapolating these we predict the ground-state energy with an estimated uncertainty on the order of chemical accuracy. Having performed this long-discussed computational task, we next consider implications beyond the model. We distill a simpler computational procedure which we apply to reveal the electronic landscape in realistic representations of the cofactor. We thus illustrate a path to a precise computational understanding of FeMo-co electronic structure.

Figures (34)

• Figure 1: Structure, active space model and states of FeMo-co.A Geometry of the FeMo-co core and surroundings, built from PDB 3U7Q spatzal2011evidenceli2019electronic-femoco). B Active space model with 113 electrons and 76 spatial orbitals from 22 atoms li2019electronic-femoco, with transition metals labeled by numbers (1-7 correspond to the crystallographic Fe nomenclature). C 35 broken symmetry initial guesses representing different spin couplings. D Schematic showing the origin of broken symmetry isomers from a tower of pure spin isomers; each broken symmetry isomer further splits due to electron delocalization on the superexchange energy scale. Note that the spin gap in each tower of states is expected to be $< 1$ kcal/mol. E Schematic showing the interaction between potential electron delocalization and spin coupling in spin isomer BS8-237. F DMRG calculations with bond dimension 5000 using localized restricted spatial orbitals (LMO), entanglement minimized restricted spatial orbitals (EMO) li2025entanglement, and using localized unrestricted orbitals (LUO, this work). The estimated ground state total energy $-22140.409107$ Hartrees is used as the energy reference. G Ranking and filtering of low-energy broken-symmetry configurations at different levels of theory.
• Figure 1: DMRG extrapolation error for $\ce{2Fe^{II}-2Fe^{III}}$ cluster, $\ce{P^N}$ cluster, and FeMo-cofactor (data taken from Ref. lee2023evaluating).
• Figure 2: Electronic landscape of FeMo-co active space model.A,B Energies of different spin isomers computed using UHF, UCCSD, UCCSD(T), UCCSDT, UDMRG with bond dimension $D=5000$ and $8000$, for the (113e, 76o) LLDUC model. The estimated ground state total energy $-22140.409107$ Hartrees is used as the energy reference. C Relative energies computed using UHF, UCCSD, and UCCSDT for the LLDUC model, compared with DFT results with B3LYP/def2-TZVPD and TPSS-D3/def2-TZVPD from Ref. cao2018influence. The minimum energy across all isomers for each theory is used as the energy reference. The lowest energy in each BS family is shown as a larger symbol. D Noodleman's schematic for the BS states BS7-346, BS7-235, BS8-237, and BS10-146 showing the spin coupling lovell2001femocao2018influence.
• Figure 2: Energy landscape of FeMo-co LLDUC model computed using GCCSD and GCCSD(T), compared with UHF counterparts. Integers are GHF reference indices based on ranking of GHF energies.
• Figure 3: Ground-state energy estimation of the FeMo-co active space model.A The $\Delta E \sim \exp[-\kappa(\log D)^2]$ fitting of UDMRG energies for the BS8-237 state in the LLDUC model. Note that the $D=18,000$ UDMRG energy from forward schedule is not fully converged, and is not included in the extrapolation. B Composite UCCSDT and FNO-UCCSDTQ energies and extrapolation (using linear fitting for 46o, 50o, and 54o energies) with respect to UCCSD natural orbital occupation cutoff, for the BS7-235 and BS8-237 states in the LLDUC model. C Estimation of the exact UCC energy using correlation energy increment ratios between the Fe(III)-Fe(II) dimer and FeMo-co LLDUC model, for the BS7-235 and BS8-237 states.