Comparison of Born–Oppenheimer approximation and electron-nuclear correlation

The electronic structure of molecules is routinely examined for static arrangements of point nuclei, which is a highly useful premise but not without certain limitations. Here, the electron-nuclear correlation energy is analysed in the context of the quantised nuclear energy associated with the chemical bond vibrations. To this end, accurate calculations of the ground state of a hydrogen atom (including basis functions in the energy continuum) whose nucleus is confined by a quadratic potential well, are performed for all values of the spring constant. The numerically exact ground state energy is compared to the results obtained within the Born–Oppenheimer approximation and within the uncorrelated product wavefunction ansatz. The electron-nuclear correlation energy is found to be smaller in magnitude than the electron correlation in two-electron atomic systems, in the limit of weak nuclear confinement, and is further reduced by at least an order of magnitude in the limit of strong confinement. In all cases, however, the electron-nuclear correlation energy is found to be significantly larger than the error of the Born–Oppenheimer approximation. Moreover, the error of the Born–Oppenheimer approximation is found to persist within the limit of infinite nuclear confinement. Insight into the electron-nuclear correlation energy from this study will help incorporate the nuclear quantum effects into the electronic structure theories.