Nuclear–electronic orbital second-order coupled cluster for excited states

Excited-state methods within the nuclear–electronic orbital (NEO) framework have the potential to capture vibrational, electronic, and vibronic transitions in a single calculation. In the NEO approach, specified nuclei, typically protons, are treated quantum mechanically at the same level of theory as the electrons. Affordable excited-state NEO methods, such as time-dependent density functional theory, are limited to capturing the subset of excitations with single-excitation character, whereas existing methods that capture the full spectrum are limited in applicability due to their high computational cost. Herein, we introduce the excited-state variant of NEO coupled cluster with approximate second-order doubles (NEO-CC2) and its scaled-opposite-spin variant with electron–proton correlation scaling (NEO-SOS′-CC2). We benchmark this method for positronium hydride, where the electrons and positron are treated quantum mechanically, and find that NEO-CC2 deviates from exact results, but NEO-SOS′-CC2 can achieve near-quantitative accuracy by increasing the electron–positron correlation. Benchmarking NEO-CC2 and NEO-SOS′-CC2 on four different triatomic molecules with a quantum proton, we find that NEO-CC2 captures qualitatively correct vibrational features such as overtones and combination bands, as well as mixed electron–proton double excitations. Electron–proton correlation scaling that increases the excited-state correlation relative to the ground-state correlation improves the accuracy across all the molecular systems tested. Quantitative accuracy is not achieved due to a combination of finite basis set effects and incomplete description of excited-state electron–proton correlation. Nevertheless, NEO-SOS′-CC2 can describe single and mixed protonic and electronic excitations with accuracy approaching that of much more computationally intensive methods.