### Nuclear-electronic orbital (NEO) method

We have developed the nuclear-electronic orbital (NEO) method for the incorporation of nuclear quantum effects into electronic structure calculations. In the NEO approach, specified nuclei are treated quantum mechanically on the same level as the electrons, and mixed nuclear-electronic wavefunctions are calculated variationally with molecular orbital methods. Both electronic and nuclear molecular orbitals are expanded in Gaussian basis sets, and the energy is minimized with respect to all molecular orbitals, as well as the centers of the nuclear basis functions. Correlation among electrons and nuclei can be included with multiconfigurational, explicitly correlated, perturbation theory, and density functional theory approaches. The advantages of the NEO approach are that nuclear quantum effects are incorporated during the electronic structure calculation, the Born-Oppenheimer separation of electrons and nuclei is avoided, nonadiabatic effects are included, excited vibrational-electronic states may be calculated, and its accuracy may be improved systematically.

For hydrogen transfer and hydrogen bonding systems, typically the hydrogen nuclei and all electrons are treated quantum mechanically. Electron-proton dynamical correlation is highly significant because of the attractive electrostatic interaction between the electron and the proton. We have formulated an explicitly correlated Hartree-Fock scheme to incorporate explicit electron-proton correlation directly into the variational self-consistent-field framework with Gaussian-type geminal functions. We have also formulated a multicomponent density functional theory and have developed electron-proton functionals based on the explicitly correlated electron-proton pair density. Initial applications illustrate that these new methods significantly improve the description of the nuclear densities, thereby leading to more accurate calculations of molecular properties such as geometries and frequencies. This approach also provides fundamental insight into the coupling between electronic and nuclear motions.

#### NEO methods developed or under development (in GAMESS)

- NEO-HF: Hartree-Fock,
^{54}analytical gradients and numerical Hessians^{57} - NEO-MP2: 2nd-order perturbation theory corrections for electron-electron and electron-proton correlation
^{70} - NEO-CI: configuration interaction
^{54} - NEO-MCSCF: multiconfigurational self-consistent-field
^{54} - NEO-NOCI: nonorthogonal CI,
^{75}tunneling splittings and vibronic couplings - NEO-vibronic coupling theory: electrons and transferring H treated with NEO while other modes treated with vibronic coupling theory
^{111} - NEO-FMO: fragment molecular orbital method
^{123} - NEO-XCHF: explicitly correlated Hartree-Fock
^{86, 103, 108, 142}with Gaussian geminals - NEO-DFT: electron-electron correlation with available electronic functionals,
^{96, 154}electron-proton correlation with new functionals based on explicitly correlated electron-proton pair density^{105, 145, 154}

#### NEO-related projects

- Developed Fourier Grid Hamiltonian MCSCF method for calculating multidimensional proton vibrational wavefunctions
^{38} - Analyzed hydrogen tunneling and hydrogen transfer with NEO
^{62, 66, 72} - Investigated geometric isotope effects with NEO
^{73} - Examined nuclear quantum effects on hydrogen bonding
^{76, 95} - Extended NEO to positrons
^{100, 112, 155} - Used NEO-vibronic coupling theory to calculate tunneling splitting in malonaldehyde
^{111} - Derived properties of the exact university functional in multicomponent DFT
^{118} - Developed localized Hartree product treatment of multiple protons in NEO
^{125}

#### Current NEO and NEO-related projects

- Developing nonadiabatic grid-based methods
- Developing extensions of NEO-XCHF to enhance efficiency
- Designing electron-proton functionals for NEO-DFT
- Applying NEO-XCHF, NEO-DFT, and grid-based methods to hydrogen bonding and hydrogen tunneling systems, geometric isotope effects