The constrained nuclear–electronic orbital (CNEO) framework for nuclear quantum effects in quantum chemistry and molecular simulations

Nuclear quantum effects play crucial roles in determining the structure, dynamics, and reactivity of hydrogen-containing systems. Incorporating these effects into routine quantum chemistry calculations and molecular simulations remains challenging, in part because widely available treatments that combine accuracy, efficiency, and straightforward workflows are still limited. The constrained nuclear–electronic orbital (CNEO) framework addresses this challenge by treating some or all nuclei quantum mechanically while preserving the conventional molecular-structure picture through constraints on nuclear position expectation values. This formulation leads to the CNEO effective potential energy surface, which directly includes zero-point energy, nuclear quantum delocalization, and shallow tunneling effects. Molecular dynamics (MD) on this surface, termed CNEO-MD, has demonstrated systematic improvements in predicted vibrational frequencies for hydrogen-involving motions, including both terminal hydrogen vibrations and hydrogen-bonded vibrations. Applications to reaction kinetics reveal substantial rate enhancements arising from shallow tunneling, and CNEO transition state theory provides accurate predictions without empirical scaling factors or separate tunneling corrections. Recent extensions to excited-state calculations and nonadiabatic dynamics enable studies of photochemical processes in which nuclear quantum effects influence both energetics and dynamics, while integration with solvation models, QM/MM schemes, and periodic boundary conditions extends the capability of the CNEO framework to complex environments. The computational costs of these approaches are comparable with those of conventional electronic-structure methods or ab initio MD. Overall, the CNEO framework provides an efficient route for the quantum-mechanical treatment of nuclei that is suitable for broad use in quantum chemistry calculations and molecular simulations.