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This paper benchmarks phaseless auxiliary-field quantum Monte Carlo (AFQMC) on iron-sulfur clusters, using trial states from coupled cluster theory and the density matrix renormalization group. The key finding is that improving the symmetry-broken trial state can paradoxically *decrease* the accuracy of the AFQMC energy, even falling below the trial projected energy, until the trial fidelity reaches a high threshold. The authors trace this energy inversion to the choice of measurement trial, suggesting that high-order CC trial states for measurements can introduce errors, and propose using a mean-field reference for measurements to improve accuracy.
Seemingly counterintuitively, better trial wavefunctions can actually *worsen* the accuracy of phaseless AFQMC for iron-sulfur clusters, revealing a subtle interplay between trial state fidelity and measurement strategy.
Phaseless auxiliary-field quantum Monte Carlo (AFQMC) has in several cases been found to perform well on strongly correlated systems. Here, we benchmark the method for three iron-sulfur clusters ([2Fe-2S], [4Fe-4S], and the FeMo cofactor) using a hierarchy of trial states derived from coupled cluster (CC) theory, including up to quadruple excitations, as well as multi-Slater trial states derived from the density matrix renormalization group. Our results reveal for these systems that, as the symmetry-broken trial is improved, the phaseless AFQMC energy can become less accurate, and in some cases even less accurate than the underlying trial projected energy, displaying an inverted energy pattern that is only corrected once the trial fidelity is sufficiently high. For [2Fe-2S], we show that this can coincide with a simultaneous improvement in the trial state and the walker ensemble. 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 spin-Zeeman field. Instead, we find that the energy inversion is related to the choice of measurement trial, 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 improves the overall accuracy across the iron-sulfur clusters, with a possible exception for [4Fe-4S]. 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.
