Nuclear–Electronic Orbital Coupled Cluster Theory with the Nuclear Hartree Product Representation for Multiple Quantum Nuclei and Application to Geometric Isotope Effects

Abstract Nuclear–electronic orbital (NEO) methods treat specified nuclei, typically protons, quantum mechanically on the same level as electrons within quantum chemical calculations. This approach allows the computationally efficient inclusion of nuclear quantum effects such as nuclear delocalization, anharmonic zero-point energy, and tunneling into energy calculations, geometry optimizations, and dynamics. NEO coupled cluster (NEO-CC) methods recover correlation to an extent that enables prediction of accurate ground-state properties for moderately sized systems. Herein, NEO-CCSD(T), which includes all relevant single and double electron and nucleus excitations, as well as perturbative triple excitations, is formulated with the nuclear Hartree product representation for an arbitrary number of quantum nuclei. Compared to the nuclear Slater determinant representation, iterative convergence of the t-amplitude equations, as well as the preceding NEO Hartree–Fock procedure, is significantly faster for multiple quantum protons. Moreover, perturbative corrections in methods such as NEO second-order Møller–Plesset perturbation theory and NEO-CCSD(T) are more accurate using the nuclear Hartree product representation, most likely due to more physically reasonable virtual orbitals. Electronic frozen natural orbital truncation is implemented to extend the accessible system size while maintaining accuracy. These combined methods are used to calculate interaction energies of systems with multiple hydrogen bonds, as well as energies of water clusters with up to 24 quantum protons. In addition, NEO-CCSD(T) with the nuclear Hartree product representation is used to predict hydrogen/deuterium geometric isotope effects, incorporating the necessary anharmonic zero-point energy, for FHF–, a water dimer, and HF clusters of up to five molecules. These examples show that NEO-CCSD(T) with the nuclear Hartree product representation is an efficient and accurate method for incorporating nuclear quantum effects into ground-state calculations.