Quantization of Protons in Periodic Electronic Structure Calculation

The nuclear-electronic orbital (NEO) method is a well-established approach for treating nuclei quantum mechanically in molecular systems beyond the usual Born-Oppenheimer approximation. In this work, we present a new strategy to implement the NEO method into periodic electronic structure calculations, particularly focused on density functional theory calculations. We discuss our implementation in an all-electron electronic structure code, FHI-aims, and show agreement with an existing NEO implementation for molecular systems in the Q-Chem code. Proof-of-concept examples are presented to illustrate the effects of quantized protons on the electronic structures of extended systems such as 2D materials and liquid-semiconductor interfaces. In addition to the proton quantization effects for the extended systems, our work shows that the electron-proton correlation has a non-negligible impact on band structures.