Multicomponent density functional theory (DFT) enables the consistent quantum mechanical treatment of both electrons and protons. A major challenge has been the design of electron-proton correlation functionals that produce even qualitatively accurate proton densities. Herein an electron-proton correlation functional, epc17, is derived analogously to the Colle-Salvetti formalism for electron correlation and is implemented within the nuclear-electronic orbital (NEO) framework. The NEO-DFT/epc17 method produces accurate proton densities efficiently and is promising for diverse applications.

In multicomponent quantum chemistry, more than one type of particle is treated quantum mechanically with either density functional theory or wave function based methods. In particular, the nuclear-electronic orbital (NEO) approach treats specified nuclei, typically hydrogen nuclei, on the same level as the electrons. This approach enables the incorporation of nuclear quantum effects, such as nuclear delocalization, anharmonicity, zero-point energy, and tunneling, as well as non-Born–Oppenheimer effects directly into quantum chemistry calculations. Such effects impact optimized geometries, molecular vibrational frequencies, reaction paths, isotope effects, and dynamical simulations. Multicomponent density functional theory (NEO-DFT) and time-dependent DFT (NEO-TDDFT) achieve an optimal balance between computational efficiency and accuracy for computing ground and excited state properties, respectively. Multicomponent wave function based methods, such as the coupled cluster singles and doubles (NEO-CCSD) method for ground states and the equation-of-motion counterpart (NEO-EOM-CCSD) for excited states, attain similar accuracy without requiring any parametrization and can be systematically improved but are more computationally expensive. Variants of the orbital-optimized perturbation theory (NEO-OOMP2) method achieve nearly the accuracy of NEO-CCSD for ground states with significantly lower computational cost. Additional approaches for computing excited electronic, vibrational, and vibronic states include the delta self-consistent field (NEO-ΔSCF), complete active space SCF (NEO-CASSCF), and nonorthogonal configuration interaction methods. Multireference methods are particularly important for describing hydrogen tunneling processes. Other types of multicomponent systems, such as those containing electrons and positrons, have also been studied within the NEO framework. The NEO approach allows the incorporation of nuclear quantum effects and non-Born–Oppenheimer effects for specified nuclei into quantum chemistry calculations in an accessible and computationally efficient manner.

The nuclear-electronic orbital (NEO) method treats all electrons and specified nuclei, typically protons, quantum mechanically on the same level with molecular orbital techniques. This approach directly includes nuclear delocalization, anharmonicity, and zero point energy contributions of the quantum nuclei in the self-consistent-field procedure for solving the time-independent Schrodinger equation. Herein the multicomponent wave function based methods configuration interaction singles and doubles (CISD) and coupled cluster singles and doubles (CCSD) are implemented within the NEO framework and are applied to molecular systems. In contrast to the NEO-HF (Hartree−Fock) and NEO-CISD methods, which produce proton densities that are much too localized, the NEO-CCSD method produces accurate proton densities in reasonable agreement with a grid-based reference. Moreover, the NEO-CCSD method also predicts accurate proton affinities in agreement with experimental measurements for a set of 12 molecules. An advantage of the NEO-CCSD method is its ability to include nuclear quantum effects, such as proton delocalization and zero point energy, during geometry optimizations and nuclear dynamics simulations. The NEO-CCSD method is a promising, parameter free approach for including nuclear quantum effects in high-level electronic structure calculations of molecular systems.

The quantum mechanical treatment of both electrons and protons in the calculation of excited state properties is critical for describing nonadiabatic processes such as photoinduced proton-coupled electron transfer. Multicomponent density functional theory enables the consistent quantum mechanical treatment of more than one type of particle and has been implemented previously for studying ground state molecular properties within the nuclear-electronic orbital (NEO) framework, where all electrons and specified protons are treated quantum mechanically. To enable the study of excited state molecular properties, herein the linear response multicomponent time-dependent density functional theory (TDDFT) is derived and implemented within the NEO framework. Initial applications to FHF and HCN illustrate that NEO-TDDFT provides accurate proton and electron excitation energies within a single calculation. As its computational cost is similar to that of conventional electronic TDDFT, the NEO-TDDFT approach is promising for diverse applications, particularly nonadiabatic proton transfer reactions, which may exhibit mixed electron-proton vibronic excitations.

Communication: A novel implementation to compute MP2 correlation energies without basis set superposition errors and complete basis set extrapolation