  The quantum theory of atoms in molecules (QTAIM) is based on the clamped
nucleus paradigm and solely working with the electronic wavefunctions, so does
not include nuclear vibrations in the AIM analysis. On the other hand, the
recently extended version of the QTAIM, called the multi-component QTAIM
(MC-QTAIM), incorporates both electrons and quantum nuclei, i.e. those nuclei
treated as quantum waves instead of clamped point charges, into the AIM
analysis using non-adiabatic wavefunctions. Thus, the MC-QTAIM is the natural
framework to incorporate the role of nuclear vibrations into the AIM analysis.
In this study, within the context of the MC-QTAIM, the formalism of including
nuclear vibrational energy in the atomic basin energy is developed in detail
and its contribution is derived analytically using the recently proposed
non-adiabatic Hartree product nuclear wavefunction. It is demonstrated that
within the context of this wavefunction the quantum nuclei may be conceived
pseudo-adiabatically as quantum oscillators and both isotropic harmonic and
anisotropic anharmonic oscillator models are used to compute the zero-point
nuclear vibrational energy contribution to the basin energies explicitly.
Inspired by the results gained within the context of the MC-QTAIM analysis, a
heuristic approach is proposed within the context of the QTAIM to include
nuclear vibrational energy in the basin energy from the vibrational
wavefunction derived adiabatically. The explicit calculation of the basin
contribution of the zero-point vibrational energy using uncoupled harmonic
oscillator model leads to results consistent with those derived from the
MC-QTAIM.
