A method that includes explicit electron-proton correlation directly into the nuclear-electronic orbital self-consistent-field framework is presented. This nuclear-electronic orbital explicitly correlated Hartree-Fock (NEO-XCHF) scheme is formulated using Gaussian basis functions for the electrons and the quantum nuclei in conjunction with Gaussian-type geminal functions. The NEO approach is designed for the quantum treatment of a relatively small number of nuclei, such as the hydrogen nuclei involved in key hydrogen bonding interactions or hydrogen transfer reactions. The conventional nuclear-electronic-orbital-based methods produce nuclear wave functions that are too localized, leading to severe overestimations of hydrogen vibrational frequencies, as well as inaccuracies in geometries, isotope effects, couplings, and tunneling splittings. The application of the NEO-XCHF approach to a model system illustrates that the description of the nuclear wave function is significantly improved by the inclusion of explicit electron-proton correlation. In contrast to the NEO-HF method, the NEO-XCHF method leads to hydrogen vibrational stretch frequencies that are in excellent agreement with those calculated from grid-based methods. This approach is computationally practical for many-electron systems because only a relatively small number of nuclei are treated quantum mechanically and only electron-proton correlation is treated explicitly. Electron-electron dynamical correlation can be included with density functional theory or perturbation theory methods.

Electronic structure calculations are used to examine the nuclear motions responsible for ultrafast internal conversion in the benzylidene malononitriles DMN (4-N,N-dimethylaminobenzylidenemalononitrile) and JDMN (julolidinemalononitrile). Gas-phase B3LYP and RI-CC2 calculations using triple-ζ valence polarized basis sets reproduce the structural features measured in the crystalline state and the solution-phase dipole moments of these molecules in their ground states with reasonable accuracy. Most properties of the vertical S0 → S1 transition, which is well separated from other transitions, are also reasonably reproduced by both types of calculation. The large change in dipole moment (8−9 D) upon excitation is grossly underestimated by TDDFT calculations, despite the fact that such calculations predict the transition energies to within experimental uncertainties. Exploration of the S1 potential energy surface of DMN using DFT, RI-CC2, and CASSCF methods indicates that the internal conversion pathway is double-bond isomerization, not the TICT process often assumed. Preliminary classical molecular dynamics simulations of DMN in acetonitrile using the ab initio S1 surface support this assignment.

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The structural impact of nuclear quantum effects is investigated for a set of bihalides, [XHX](-), X = F, Cl, and Br, and the hydrogen fluoride dimer. Structures are calculated with the vibrational self-consistent-field (VSCF) method, the second-order vibrational perturbation theory method (VPT2), and the nuclear-electronic orbital (NEO) approach. In the VSCF and VPT2 methods, the vibrationally averaged geometries are calculated for the Born-Oppenheimer electronic potential energy surface. In the NEO approach, the hydrogen nuclei are treated quantum mechanically on the same level as the electrons, and mixed nuclear-electronic wave functions are calculated variationally with molecular orbital methods. Electron-electron and electron-proton dynamical correlation effects are included in the NEO approach using second-order perturbation theory (NEO-MP2). The nuclear quantum effects are found to alter the distances between the heavy atoms by 0.02-0.05 A for the systems studied. These effects are of similar magnitude as the electron correlation effects. For the bihalides, inclusion of the nuclear quantum effects with the NEO-MP2 or the VSCF method increases the X-X distance. The bihalide X-X distances are similar for both methods and are consistent with two-dimensional grid calculations and experimental values, thereby validating the use of the computationally efficient NEO-MP2 method for these types of systems. For the hydrogen fluoride dimer, inclusion of nuclear quantum effects decreases the F-F distance with the NEO-MP2 method and increases the F-F distance with the VSCF and VPT2 methods. The VPT2 F-F distances for the hydrogen fluoride dimer and the deuterated form are consistent with the experimentally determined values. The NEO-MP2 F-F distance is in excellent agreement with the distance obtained experimentally for a model that removes the large amplitude bending motions. The analysis of these calculations provides insight into the significance of electron-electron and electron-proton correlation, anharmonicity of the vibrational modes, and nonadiabatic effects for hydrogen-bonded systems.

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