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The quantum theory of atoms in molecules (QTAIM) is based on the clampednucleus paradigm and solely working with the electronic wavefunctions, so doesnot include nuclear vibrations in the AIM analysis. On the other hand, therecently extended version of the QTAIM, called the multi-component QTAIM(MC-QTAIM), incorporates both electrons and quantum nuclei, i.e. those nucleitreated as quantum waves instead of clamped point charges, into the AIManalysis using non-adiabatic wavefunctions. Thus, the MC-QTAIM is the naturalframework to incorporate the role of nuclear vibrations into the AIM analysis.In this study, within the context of the MC-QTAIM, the formalism of includingnuclear vibrational energy in the atomic basin energy is developed in detailand its contribution is derived analytically using the recently proposednon-adiabatic Hartree product nuclear wavefunction. It is demonstrated thatwithin the context of this wavefunction the quantum nuclei may be conceivedpseudo-adiabatically as quantum oscillators and both isotropic harmonic andanisotropic anharmonic oscillator models are used to compute the zero-pointnuclear vibrational energy contribution to the basin energies explicitly.Inspired by the results gained within the context of the MC-QTAIM analysis, aheuristic approach is proposed within the context of the QTAIM to includenuclear vibrational energy in the basin energy from the vibrationalwavefunction derived adiabatically. The explicit calculation of the basincontribution of the zero-point vibrational energy using uncoupled harmonicoscillator model leads to results consistent with those derived from theMC-QTAIM.

Incorporating nuclear vibrational energies into the -atom in molecules- analysis: An analytical study