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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. [ABSTRACT FROM AUTHOR]

The recently developed real-time nuclear–electronic orbital (RT-NEO) approach provides an elegant framework for treating electrons and selected nuclei, typically protons, quantum mechanically in nonequilibrium dynamical processes. However, the RT-NEO approach neglects the motion of the other nuclei, preventing a complete description of the coupled nuclear–electronic dynamics and spectroscopy. In this work, the dynamical interactions between the other nuclei and the electron–proton subsystem are described with the mixed quantum–classical Ehrenfest dynamics method. The NEO-Ehrenfest approach propagates the electrons and quantum protons in a time-dependent variational framework, while the remaining nuclei move classically on the corresponding average electron–proton vibronic surface. This approach includes the non-Born–Oppenheimer effects between the electrons and the quantum protons with RT-NEO and between the classical nuclei and the electron–proton subsystem with Ehrenfest dynamics. Spectral features for vibrational modes involving both quantum and classical nuclei are resolved from the time-dependent dipole moments. This work shows that the NEO-Ehrenfest method is a powerful tool to study dynamical processes with coupled electronic and nuclear degrees of freedom. [ABSTRACT FROM AUTHOR]

Multicomponent density functional theory (DFT) has many practical advantages for incorporating nuclear quantum effects into quantum chemistry calculations. Within the nuclear-electronic orbital (NEO) framework, specified nuclei, typically protons, are treated quantum mechanically on the same level as the electrons. Previously, electron-proton correlation functionals based on the local density approximation (LDA), denoted epc17 and epc18, were developed and shown to provide more accurate proton densities and energies compared to the neglect of electron-proton correlation, but a quantitatively accurate description of both densities and energies simultaneously has remained elusive. Herein, an electron-proton correlation functional that depends on the electron and proton density gradients, as well as the densities, is derived and implemented. Compared to the LDA functionals, the resulting generalized gradient approximation functional, denoted epc19, is able to simultaneously provide accurate proton densities and energies, as well as reproduce the impact of nuclear quantum effects on optimized geometries. In addition, without further parameterization, the NEO-DFT/epc19 method provides accurate densities and energies for deuterium as well as hydrogen. These results demonstrate that the form of the epc19 functional is able to capture the essential aspects of electron-proton correlation and highlights the importance of including gradient terms. This approach will enable the exploration of nuclear quantum effects and isotope effects in a wide range of systems. [ABSTRACT FROM AUTHOR]