Can Phaseless Auxiliary-Field Quantum Monte Carlo with Broken Symmetry Trials Describe Iron–Sulfur Clusters?

Abstract Phaseless auxiliary-field quantum Monte Carlo (AFQMC) has in several cases been found to perform well on strongly correlated systems, including compounds containing transition-metal elements. Here, we benchmark the method for three iron–sulfur clusters ([2Fe-2S], [4Fe-4S], and the FeMo cofactor) using a hierarchy of systematically improved trial states derived from coupled cluster (CC) theory, including up to quadruple excitations relative to the Hartree–Fock (HF) reference, as well as multi-Slater trial states derived from unrestricted density matrix renormalization group wave functions. Our results reveal for these systems that, as the symmetry-broken trial is improved beyond the mean-field level, the phaseless AFQMC energy can become less accurate, and in some cases even less accurate than the underlying trial projected energy itself, displaying an inverted energy pattern that is only corrected once the trial fidelity is sufficiently high. For [2Fe-2S], we show that a decreased accuracy in the AFQMC energy can coincide with a simultaneous improvement in the trial state and the walker ensemble, as measured by their fidelity relative to the exact ground state, a behavior which is allowed by the nonvariational nature of the phaseless AFQMC energy estimator. We further find that this is not solely due to the use of spin-unrestricted trial states, as the inversion persists in [2Fe-2S] when we explicitly break the symmetry of the Hamiltonian by applying a fictitious local spin-Zeeman field. Instead, we find that the energy inversion is related to the choice of measurement trial in the estimator, where using a high-order CC trial state for measurements may introduce errors that are suppressed when the measurement wave function is restricted to lower excitation subspaces. In particular, measuring the energy with the mean-field reference while guiding the walkers with a CC trial removes the inverted pattern and improves the overall accuracy across the iron–sulfur clusters, with a possible exception for [4Fe-4S] where energies with a CCSDT guide may be less accurate than those guided by CCSD for some broken-symmetry families. Taken together, our findings suggest that the relatively accurate energies obtained with an HF trial state in these systems arise from favorable error cancellation, warranting significant caution about the reliability of phaseless AFQMC with such trials for strongly correlated transition-metal systems of this kind.