What is the number of electrons in a spatial domain?

TL;DR

This paper investigates the probabilistic nature of assigning electron counts to spatial regions in molecules, revealing significant fluctuations and the challenges in defining a precise electron number for chemical regions.

Contribution

It introduces a probabilistic framework for quantifying electron numbers in spatial domains and demonstrates the inherent fluctuations and ambiguity in such assignments.

Findings

Maximizing probability domains yields regions with higher chemical relevance.

Electron number in a spatial region fluctuates significantly, even in optimal domains.

The concept of a fixed electron count in a region is often ill-posed.

Abstract

We like to attribute a number of electrons to spatial domains (atoms, bonds, ...). However, as a rule, the number of electrons in a spatial domain is not a sharp number. We thus study probabilities for having any number of electrons (between 0 and the total number of electrons in the system) in a given spatial domain. We show that by choosing a domain that maximizes a chosen probability (or is close to it), one obtains higher probabilities for chemically relevant regions. The probability to have a given electronic arrangement, - for example, by attributing a number of electrons to an atomic shell - can be low. It remains so even in the "best" case, i.e, if the spatial domain is chosen to maximize the chosen probability. In other words, the number of electrons in a spatial region significantly fluctuates. The freedom of choosing the number of electrons we are interested in shows that a "chemical" question is not always well-posed. We show it using as an example the KrF2 molecule.