# Atoms in molecules from alchemical perturbation density functional theory

## TL;DR

This paper introduces a novel approach to defining atoms within molecules using alchemical perturbation density functional theory, providing clear and unambiguous atomic energies and densities through thermodynamic integration.

## Contribution

It presents a new method for atoms in molecules based on APDFT with a uniform electron gas reference, enabling precise atomic energy and density definitions across various molecules.

## Key findings

- Exact and unambiguous atomic energies and densities obtained
- Applicable to diverse molecules including diatomics and doped benzene
- Provides new insights into molecular energies and electron densities

## Abstract

Based on thermodynamic integration we introduce atoms in molecules (AIM) using the orbital-free framework of alchemical perturbation density functional theory (APDFT). Within APDFT, atomic energies and electron densities in molecules are arbitrary because any arbitrary reference system and integration path can be selected as long as it meets the boundary conditions. We choose the uniform electron gas as the most generic reference, and linearly scale up all nuclear charges, situated at any query molecule's atomic coordinates. Within the approximations made when calculating one-particle electron densities, this choice affords exact and unambiguous definitions of energies and electron densities of AIMs We illustrate the approach for neutral iso-electronic diatomics (CO, N$_2$, BF), various small molecules with different electronic hybridisation states of carbon (CH$_4$, C$_2$H$_6$, C$_2$H$_4$, C$_2$H$_2$, HCN), and for all the possible BN doped mutants connecting benzene to borazine (C$_{2n}$B$_{3-n}$N$_{3-n}$H$_6$, $0 \le n \le 3$). Analysis of the numerical results obtained suggests that APDFT based AIMs enable meaningful and new interpretations of molecular energies and electron densities.