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

TL;DR

This paper extends the quantum theory of atoms in molecules (QTAIM) to include nuclear vibrational energies by developing an analytical formalism within the multi-component QTAIM framework, allowing for more comprehensive atomic basin energy analysis.

Contribution

It introduces a formalism to incorporate nuclear vibrational energies into AIM analysis using non-adiabatic wavefunctions and proposes a heuristic approach compatible with traditional QTAIM.

Findings

Analytical expression for nuclear vibrational energy contribution to basin energies.

Consistent results between MC-QTAIM and heuristic harmonic oscillator models.

Demonstration of nuclear vibrations as quantum oscillators within the framework.

Abstract

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.