# From Local Atomic Environments to Molecular Information Entropy

## TL;DR

This paper explores how molecular complexity can be measured using information entropy derived from atomic environments, showing its usefulness in comparing molecules.

## Contribution

The paper introduces a novel method to compute molecular entropy based on similarity matrices derived from atomic environments.

## Key findings

- Two similarity definitions (SMILES and SOAP kernel) produce comparable entropy measures when sensitivity is adjusted.
- Entropy gain from mixing molecules can serve as a similarity measure, aligning with other kernel-based methods.

## Abstract

The similarity of
local atomic environments is an important concept
in many machine learning techniques, which find applications in computational
chemistry and material science. Here, we present and discuss a connection
between the information entropy and the similarity matrix of a molecule.
The resulting entropy can be used as a measure of the complexity of
a molecule. Exemplarily, we introduce and evaluate two specific choices
for defining the similarity: one is based on a SMILES representation
of local substructures, and the other is based on the SOAP kernel.
By tuning the sensitivity of the latter, we can achieve good agreement
between the respective entropies. Finally, we consider the entropy
of two molecules in a mixture. The gain of entropy due to the mixing
can be used as a similarity measure of the molecules. We compare this
measure to the average and best-match kernel. The results indicate
a connection between the different approaches and demonstrate the
usefulness and broad applicability of the similarity-based entropy
approach.