We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ chain is the same as, or very close to, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ chain at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate to eigenstate fluctuations vanish in a power law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω = Eα − E β ({Eα} are the eigenenergies) whose magnitude scales as 1/D, where D is the Hilbert space dimension.

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A dicobalt tetrakis(Schiff base) macrocycle has recently been reported to electrochemically catalyze the reduction of H + to H 2 in an acetonitrile solution. Density functional theory (DFT) calculations using the ωB97X-D functional are shown to produce structural and thermodynamic results in good agreement with the experimental data. A mechanistic model based on thermodynamics is developed that incorporates electrochemical and magnetic details of the complex, accounting for electron-spin reorganization of the metal center after redox steps. The model is validated through a comparison of the predicted electrochemical potentials with the irreversible cyclic voltammogram of [Co 2 LAc] + , which shows redox-coupled spin-crossover (RCSCO) behavior for the Co II/III transitions. Using our model, we predict the thermodynamically favored mechanism of H 2 evolution by [Co 2 L] 2+ to be one of heterolytic proton attack on a [Co II 2 L(μ-H)] + species. Understanding the electronic details and thermodynamically preferred mechanism of this catalyst will aid in improving its efficiency and the future design of bimetallic Co-based H + electrocatalysts. Also, this work will assist in the future DFT modeling of bimetallic RCSCO complexes.