Dual-atom catalysts (DACs) are promising candidates for various catalytic reactions, including electrocatalysis, chemical synthesis, and environmental remediation. However, the high-activity origin and mechanism underlying intrinsic activity enhancement remain elusive, especially for the Fenton-like reaction. Herein, we systematically compared the catalytic performance of dual-atom FeCo–N/C with its single-atom counterparts by activating peroxymonosulfate (PMS) for pollutant abatement. The unusual spin-state reconstruction on FeCo–N/C is demonstrated to effectively improve the electronic structure of Fe and Co in the d orbital and enhance the PMS activation efficiency. Accordingly, the dual-atom FeCo–N/C with an intermediate-spin state remarkably boosts the Fenton-like reaction by almost 1 order of magnitude compared with low-spin Co–N/C and high-spin Fe–N/C. Moreover, the established dual-atom-activated PMS system also exhibits excellent stability and robust resistance against harsh conditions. Combined theoretical calculations reveal that unlike unitary Co atom or Fe atom transferring electrons to the PMS molecule, the Fe atom of FeCo–N/C provides extra electrons to the neighboring Co atom and positively shifts the d band of the Co center, thereby optimizing the PMS adsorption and decomposition into a unique high-valent FeIV–O–CoIV species via a low-energy barrier pathway. This work advances a conceptually novel mechanistic understanding of the enhanced catalytic activity of DACs in Fenton-like reactions and helps to expand the application of DACs in various catalytic reactions.

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This paper considers decisions of bidders with bounded rationality in different thinking levels for a first-price sealed-bid reverse auction. To characterize the nonequilibrium strategic thinking, we construct a mathematical model that incorporates the "level-k decision rule" to iteratively derive closed-form solutions. Then the effects of bidders' heterogeneous beliefs of thinking levels on their bid prices and expected payoffs are investigated. We find that under some assumptions, bidders will exhibit oscillating behavior in their bid prices and expected profits in terms of thinking levels. When the thinking level tends to infinity, the bid prices and expected profits converge to those in the conventional analysis with perfectly rational bidders. An interesting finding from theoretical analysis verified by numerical experiments is that any bidder with a thinking level above two should bid according to level-2 bidder's strategy to achieve the highest expected profit, and in this case the buyer will pay no more than level-2 bidder's bid price.

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