# What's the difference between charge density and the electron localization function (ELF)?

The following figure is the electron localization function (ELF) for monolayer LaBr$$_2$$.

The ELF is calculated to reflect the bonding character in SL LaBr$$_2$$. As shown in the figure, ELF shows two localization areas: one is located around La (brown color) and the other around Br (grey color) atoms. Almost no electron is localized between La and Br atoms, suggesting the typical ionic bonding, with La atoms donating electrons to Br atoms.

However, this bond character can be understood from the charge density plot. So what's the difference between charge density and the electron localization function (ELF)? What else can we read from the ELF plot?

In simple terms, the charge density gives you the probability of finding an electron at a given point, while the electron localization function (ELF) gives you the probability of finding another electron at that same point, therefore it is very sensitive to open shell / shared electron interactions.

The ELF equals 0.5 for a metallic-like electron density, and goes to 1 for highly-localized electrons (lone pairs, covalent bonds).

More formally, the ELF is related to the kinetic energy density, and can be thought of as the excess kinetic energy density as a result of Pauli exclusion.

There are many methods to try to obtain a chemically intuitive picture of bonding from the outputs of a Density Functional Theory (DFT) calculation, such as the ELF, but also QTAIM/Bader methods. These methods are only indicators, there is no single unambiguous way to back out a chemically intuitive picture of bonding from a DFT calculation. Each has its own trade-offs and their own advocates.

This chapter is a nice reference for further reading, and this page also has a nice summary.

• +1. If this answer came two days earlier the question would have become a Hot Network Question, but the formula for that drops off exponentially with the number hours since the question is asked. – Nike Dattani Nov 12 '20 at 20:44
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