A Flexible and Generalized Constant-Potential Framework in i-PI

Constant-potential molecular dynamics is essential for realistic simulations of electrochemical interfaces under Operando conditions. Although various constant-potential frameworks exist, most are tightly coupled to specific electronic-structure codes or numerical architectures, limiting portability and extensibility─especially for codes constrained to integer electron numbers. Here, we present a flexible constant-potential framework implemented in the i-PI driver, interfacing with multiple density functional theory (DFT) engines and, in principle, extensible to constant-potential machine-learning potentials. The method regulates and samples the electronic chemical potential by introducing an explicit electronic degree of freedom and a dedicated potentiostat module in i-PI. To bypass the integer-electron constraint without modifying the underlying DFT code, we employ a mixed-Hamiltonian interpolation scheme: two adjacent integer-charge clients are run in parallel, and their energies, forces, and electronic chemical potentials (Fermi level/work function) are linearly interpolated to obtain an effective fractional-charge description. We validate the method on a one-dimensional asymmetric double-well model and an Al(111) surface, demonstrating stable potential control and well-behaved charge fluctuations. Finally, we couple constant-potential ab initio molecular dynamics (AIMD) with enhanced sampling to study CO₂ reduction on NiN₄-doped graphene, enabling efficient characterization of potential-dependent reactivity and free-energy landscapes. Overall, this framework provides a portable and scalable platform for conducting rigorous constant-potential simulations across diverse electronic-structure clients and, in principle, machine-learning potentials.