Constrained Nuclear-Electronic Orbital Density Functional Theory with Periodic Boundary Conditions: Nuclear Quantum Effects in Surface Hydrogen Dynamics

We present an implementation of constrained nuclear-electronic orbital density functional theory (CNEO-DFT) with periodic boundary conditions, enabling quantum mechanical treatment of both electrons and select nuclei. Using the Gaussian-augmented plane wave framework in CP2K, we achieve computational cost comparable to conventional DFT. The position constraint on quantum nuclear densities establishes an energy surface that naturally incorporates nuclear quantum effects. The implementation includes analytic gradients for both classical nuclear coordinates and quantum nuclear expectation positions. For hydrogen on Pt(111), conventional DFT incorrectly predicts atop site as the most stable position, while both CNEO-DFT and DFT with perpendicular harmonic zero-point energy corrections identify the fcc hollow site as energetically favorable, highlighting the importance of nuclear quantum effects in determining correct adsorption geometries. Through calculations of differential H adsorption entropy on Pt(111), we find that at catalytically relevant temperatures, CNEO-DFT predicts entropy values that deviate from quantum results by only 0.2 J/mol/K, compared to classical deviations of 2.0 J/mol/K. At room temperature, CNEO-DFT captures half of the 3.8 J/mol/K deviation between classical and quantum results. This implementation enables efficient inclusion of nuclear quantum effects in periodic systems and is particularly valuable for studying hydrogen-containing materials.