Periodic Constrained Nuclear-Electronic Orbital Density Functional Theory for Nuclear Quantum Effects: Method...

We develop constrained nuclear-electronic orbital density functional theory (CNEO-DFT) with periodic boundary conditions, enabling simultaneous quantum mechanical treatment of both electrons and nuclei in extended systems at computational costs comparable to conventional DFT. Our approach employs the Gaussian-augmented plane wave framework of CP2K for both electrons and nuclei. The quantum nuclei are treated as localized, distinguishable particles, while the collective nuclear distribution satisfies periodicity. When applied to hydrogen adsorption on Pt(111), our method predicts a shift in the preferred binding site from atop (conventional DFT) to fcc hollow (CNEO-DFT), primarily due to zero-point effects. Furthermore, by capturing subtle shallow tunneling effects that enhance hydrogen mobility, CNEO-DFT shows excellent agreement with fully quantum reference calculations for differential entropy across catalytically relevant temperatures (500-800 K). The implementation also includes analytic gradients, enabling geometry optimization and molecular dynamics. This development of periodic CNEO-DFT offers an accurate and efficient framework for treating nuclear quantum effects in surfaces, interfaces, and bulk materials where hydrogen chemistry plays a crucial role.