# Toward a thermodynamic stability order of the phosphorus allotropes

## TL;DR

This paper uses quantum calculations to determine which form of phosphorus is the most stable under different conditions.

## Contribution

The study provides new insights into the thermodynamic stability of various phosphorus allotropes using advanced computational methods.

## Key findings

- Violet phosphorus is the most stable at 0 K and 298 K according to DFT and Spin-Component Scaled p-LMP2 calculations.
- Black phosphorus is suggested as the most stable by pure p-LMP2, but this may be due to its narrow band gap.
- The stability differences between violet and black phosphorus are very small.

## Abstract

In this work, we investigate through quantum–mechanical calculations the relative stability of white-γ, white-β, fibrous red, violet, and orthorhombic black phosphorus allotropes, a longstanding yet challenging problem. We performed DFT-D3 calculations with the CRYSTAL code as well as periodic local second-order Møller–Plesset perturbation theory (p-LMP2) calculations. DFT and Spin-Component Scaled p-LMP2 place violet phosphorus as the thermodynamically most stable allotrope both at 0 K and at 298 K, yet within a tiny margin from the black phosphorus. Pure p-LMP2 suggests that black phosphorus is the most stable, although its accuracy may be affected by the narrow band gap in this material.

In this work, we investigate through quantum–mechanical calculations the relative stability of white-γ, white-β, fibrous red, violet, and orthorhombic black phosphorus allotropes, a longstanding yet challenging problem.