On the Probability Density of the Nuclei in a Vibrationally Excited Molecule

Abstract

For localized and oriented vibrationally excited molecules, the qualitative features of the one-body probability density of the nuclei (one-nucleus density) are investigated. Like the familiar and widely used one-electron density that represents the probability of finding an electron at a given location in space, the one-nucleus density represents the probability of finding a nucleus at a given position in space independent of the location of the other nuclei and independent of their type. In contrast to the electrons, however, the nuclei are comparably localized. Due to this localization of the individual nuclei, the one-nucleus density provides a quantum-mechanical representation of the “chemical picture” of the molecule as an object that can largely be understood in a three-dimensional space, even though its full nuclear probability density is defined on the high-dimensional configuration space of all the nuclei. We study how the nodal structure of the wavefunctions of vibrationally excited states translates to the one-nucleus density. It is found that nodes do not necessarily lead to visible changes in the one-nucleus density: Already for relatively small molecules, only certain vibrational excitations change the one-nucleus density qualitatively compared to the ground state. It turns out that there are simple rules for predicting the shape of the one-nucleus density from the normal mode coordinates. A Python module for the computation of the one-nucleus density is provided at https://gitlab.com/axelschild/mQNMc

Anharmonic quantum nuclear densities from full dimensional vibrational eigenfunctions with application to protonated glycine

“…In conclusion, the present method allows for the calculation of high dimensional quantum eigenfunctions of vibrational ground and excited states for molecules of moderate size, and provides an immediately informative real space nuclear density representation of molecular vibrations, overcoming the major limitations of the normal-mode approach and going beyond the quantum harmonic picture 53 .…”

“…In this work, we introduce the calculation of one-nucleus marginal densities by Monte Carlo integration of the anharmonic eigenfunctions obtained with MC SCIVR simulations. Specifically, we employ the eigenfunction modulus square to weight a Monte Carlo Cartesian space sampling at each molecular conformation, and we obtain the final molecular nuclear density as a direct sum of each nucleus contribution 53 , (see Supplementary Methods). This quantity is the nuclear analogue of electron density in Density Functional Theory for electronic structure calculations.…”

Orientational decoherence within molecules and emergence of the molecular shape

“…The present study of spatial (de)coherence within the molecule adds a missing bit to our earlier understanding [25][26][27][28][29][30] that followed a path proposed by Claverie and Diner [5]. According to them, elements of the molecular structure can be recognized as nuclear configurations for which the particle density is large.…”

Representing Molecular Ground and Excited Vibrational Eigenstates with Nuclear Densities obtained from Semiclassical Initial Value Representation Molecular Dynamics

“…) which corresponds to the marginal i-th one-nucleus density, 36 that is the nuclear analogue of electron density of Density Functional Theory for electronic structure calculations. 70 Due to the larger mass of the nuclei as compared to electrons, the one-nucleus densities are sufficiently localized so that the overlap of densities of different nuclei in the molecule is negligible.…”

“…the probability of finding each nucleus in a molecule at a given position in space independently of the location of the others, has been proposed as a tool to get information about molecular normal modes from the wavefunction. 36 In that work, harmonic one-nucleus densities were computed by analytic integration of the harmonic eigenfunctions. The focus was on how the wavefunction nodal structure of the vibrationally excited states is reflected in the one-nucleus density.…”