Abstract A procedure for defining virtual spaces, and the periodic one-electron and two-electron integrals, for plane-wave second quantized Hamiltonians has been developed, and it was validated using full configuration interaction (FCI) calculations, as well as executions of variational quantum eigensolver (VQE) circuits on Quantinuum’s ion trap quantum computers accessed through Microsoft’s Azure Quantum service. This work is an extension to periodic systems of a new class of algorithms in which the virtual spaces were generated by optimizing orbitals from small pairwise CI Hamiltonians, which we term as correlation optimized virtual orbitals with the abbreviation COVOs. In this extension, the integration of the first Brillouin zone is automatically incorporated into the two-electron integrals. With these procedures, we have been able to derive virtual spaces, containing only a few orbitals, that were able to capture a significant amount of correlation. The focus in this manuscript is on comparing the simulations of small molecules calculated with plane-wave basis sets with large periodic unit cells at the $$\Gamma$$ Γ -point, including images, to results for plane-wave basis sets with aperiodic unit cells. The results for this approach were promising, as we were able to obtain good agreement between periodic and aperiodic results for an LiH molecule. Calculations performed on the Quantinuum H1-1 quantum computer produced surprisingly good energies, in which the error mitigation played a small role in the quantum hardware calculations and the (noisy) quantum simulator results. Using a modest number of circuit runs (500 shots), we reproduced the FCI values for the 1 COVO Hamiltonian with an error of 11 milliHartree, which is expected to improve with a larger number of circuit runs.
Aqueous Fe(II) has been shown to exchange with structural Fe(III) in goethite without any significant phase transformation. It remains unclear, however, whether aqueous Fe(II) undergoes similar exchange reactions with structural Fe(III) in hematite, a ubiquitous iron oxide mineral. Here, we use an enriched 57Fe tracer to show that aqueous Fe(II) exchanges with structural Fe(III) in hematite at room temperature, and that the amount of exchange is influenced by particle size, pH, and Fe(II) concentration. Reaction of 80 nm-hematite (27 m2 g–1) with aqueous Fe(II) at pH 7.0 for 30 days results in ∼5% of its structural Fe(III) atoms exchanging with Fe(II) in solution, which equates to about one surface iron layer. Smaller, 50 nm-hematite particles (54 m2 g–1) undergo about 25% exchange (∼3× surface iron) with aqueous Fe(II), demonstrating that structural Fe(III) in hematite is accessible to the fluid in the presence of Fe(II). The extent of exchange in hematite increases with pH up to 7.5 and then begins to decrease as the pH progresses to 8.0, likely due to surface site saturation by sorbed Fe(II). Similarly, when we vary the initial amount of added Fe(II), we observe decreasing amounts of exchange when aqueous Fe(II) is increased beyond surface saturation. This work shows that Fe(II) can catalyze iron atom exchange between bulk hematite and aqueous Fe(II), despite hematite being the most thermodynamically stable iron oxide.
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