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 strategy to implement the NEO method for periodic electronic structure calculations, particularly focused on multicomponent density functional theory (DFT). The NEO-DFT method is implemented in an all-electron electronic structure code, FHI-aims, using a combination of analytical and numerical integration techniques as well as a resolution of the identity scheme to enhance computational efficiency. After validating this implementation, proof-of-concept applications are presented to illustrate the effects of quantized protons on the physical properties of extended systems, such as two-dimensional materials and liquid–semiconductor interfaces. Specifically, periodic NEO-DFT calculations are performed for a trans-polyacetylene chain, a hydrogen boride sheet, and a titanium oxide–water interface. The zero-point energy effects of the protons as well as electron–proton correlation are shown to noticeably impact the density of states and band structures for these systems. These developments provide a foundation for the application of multicomponent DFT to a wide range of other extended condensed matter systems.
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The geometry is optimized with NEO-TDDFT on the first electronic excited state surface. The Cartesian coordinates (in Å) are (−0.123 98, 0.672 955, 0.0) and (0.123 98, −0.672 955, 0.0) for the carbon atoms and (0.694 71, 1.430 146 5, 0.0) and (−0.694 71, −1.430 146 5, 0.0) for the hydrogen atoms.