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

## TL;DR

This paper investigates the one-nucleus probability density in vibrationally excited molecules, revealing how vibrational states influence nuclear distributions and how these can be predicted and potentially observed experimentally.

## Contribution

It introduces a quantum-mechanical analysis of the one-nucleus density in vibrationally excited molecules, providing rules to predict vibrational excitation visibility in experiments.

## Key findings

- Nodes are not always visible in the one-nucleus density.
- Certain vibrational excitations significantly alter the density.
- Simple rules can predict the shape of the density from normal modes.

## Abstract

For localized and oriented vibrationally excited molecules, the one-body probability density of the nuclei (one-nucleus density) is studied. 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. In contrast to the full many-dimensional nuclear probability density, the one-nucleus density contains less information and may thus be better accessible by experiment, especially for large molecules. It also provides a quantum-mechanical view of molecular vibrations that can easily be visualized. We study how the nodal structure of the wavefunctions of vibrationally excited states translates to the one-nucleus density. It is found that nodes are not necessarily visible: 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 some simple rules for predicting the shape of the one-nucleus density from the normal mode coordinates, and thus for predicting if a vibrational excitation is visible in a corresponding experiment.