{% raw %}
Title: Create a Markdown Blog Post Integrating Research Details and a Featured Paper
====================================================================================
This task involves generating a Markdown file (ready for a GitHub-served Jekyll site) that integrates our research details with a featured research paper. The output must follow the exact format and conventions described below.
====================================================================================
Output Format (Markdown):
------------------------------------------------------------------------------------
---
layout: post
title: "A foundation model for atomistic materials chemistry"
date: 2023-12-29
categories: papers
---
Content generated by [gemini-2.5-pro](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/2023-12-29-2401.00096.txt).
Image generated by [imagen-3.0-generate-002](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/2023-12-29-2401.00096.txt).
------------------------------------------------------------------------------------
====================================================================================
Please adhere strictly to the following instructions:
====================================================================================
Section 1: Content Creation Instructions
====================================================================================
1. **Generate the Page Body:**
- Write a well-composed, engaging narrative that is suitable for a scholarly audience interested in advanced AI and astrophysics.
- Ensure the narrative is original and reflective of the tone and style and content in the "Homepage Content" block (provided below), but do not reuse its content.
- Use bullet points, subheadings, or other formatting to enhance readability.
2. **Highlight Key Research Details:**
- Emphasize the contributions and impact of the paper, focusing on its methodology, significance, and context within current research.
- Specifically highlight the lead author ({'name': 'Ilyes Batatia'}). When referencing any author, use Markdown links from the Author Information block (choose academic or GitHub links over social media).
3. **Integrate Data from Multiple Sources:**
- Seamlessly weave information from the following:
- **Paper Metadata (YAML):** Essential details including the title and authors.
- **Paper Source (TeX):** Technical content from the paper.
- **Bibliographic Information (bbl):** Extract bibliographic references.
- **Author Information (YAML):** Profile details for constructing Markdown links.
- Merge insights from the Paper Metadata, TeX source, Bibliographic Information, and Author Information blocks into a coherent narrative—do not treat these as separate or isolated pieces.
- Insert the generated narrative between the HTML comments:
and
4. **Generate Bibliographic References:**
- Review the Bibliographic Information block carefully.
- For each reference that includes a DOI or arXiv identifier:
- For DOIs, generate a link formatted as:
[10.1234/xyz](https://doi.org/10.1234/xyz)
- For arXiv entries, generate a link formatted as:
[2103.12345](https://arxiv.org/abs/2103.12345)
- **Important:** Do not use any LaTeX citation commands (e.g., `\cite{...}`). Every reference must be rendered directly as a Markdown link. For example, instead of `\cite{mycitation}`, output `[mycitation](https://doi.org/mycitation)`
- **Incorrect:** `\cite{10.1234/xyz}`
- **Correct:** `[10.1234/xyz](https://doi.org/10.1234/xyz)`
- Ensure that at least three (3) of the most relevant references are naturally integrated into the narrative.
- Ensure that the link to the Featured paper [2401.00096](https://arxiv.org/abs/2401.00096) is included in the first sentence.
5. **Final Formatting Requirements:**
- The output must be plain Markdown; do not wrap it in Markdown code fences.
- Preserve the YAML front matter exactly as provided.
====================================================================================
Section 2: Provided Data for Integration
====================================================================================
1. **Homepage Content (Tone and Style Reference):**
```markdown
---
layout: home
---
The Handley Research Group stands at the forefront of cosmological exploration, pioneering novel approaches that fuse fundamental physics with the transformative power of artificial intelligence. We are a dynamic team of researchers, including PhD students, postdoctoral fellows, and project students, based at the University of Cambridge. Our mission is to unravel the mysteries of the Universe, from its earliest moments to its present-day structure and ultimate fate. We tackle fundamental questions in cosmology and astrophysics, with a particular focus on leveraging advanced Bayesian statistical methods and AI to push the frontiers of scientific discovery. Our research spans a wide array of topics, including the [primordial Universe](https://arxiv.org/abs/1907.08524), [inflation](https://arxiv.org/abs/1807.06211), the nature of [dark energy](https://arxiv.org/abs/2503.08658) and [dark matter](https://arxiv.org/abs/2405.17548), [21-cm cosmology](https://arxiv.org/abs/2210.07409), the [Cosmic Microwave Background (CMB)](https://arxiv.org/abs/1807.06209), and [gravitational wave astrophysics](https://arxiv.org/abs/2411.17663).
### Our Research Approach: Innovation at the Intersection of Physics and AI
At The Handley Research Group, we develop and apply cutting-edge computational techniques to analyze complex astronomical datasets. Our work is characterized by a deep commitment to principled [Bayesian inference](https://arxiv.org/abs/2205.15570) and the innovative application of [artificial intelligence (AI) and machine learning (ML)](https://arxiv.org/abs/2504.10230).
**Key Research Themes:**
* **Cosmology:** We investigate the early Universe, including [quantum initial conditions for inflation](https://arxiv.org/abs/2002.07042) and the generation of [primordial power spectra](https://arxiv.org/abs/2112.07547). We explore the enigmatic nature of [dark energy, using methods like non-parametric reconstructions](https://arxiv.org/abs/2503.08658), and search for new insights into [dark matter](https://arxiv.org/abs/2405.17548). A significant portion of our efforts is dedicated to [21-cm cosmology](https://arxiv.org/abs/2104.04336), aiming to detect faint signals from the Cosmic Dawn and the Epoch of Reionization.
* **Gravitational Wave Astrophysics:** We develop methods for [analyzing gravitational wave signals](https://arxiv.org/abs/2411.17663), extracting information about extreme astrophysical events and fundamental physics.
* **Bayesian Methods & AI for Physical Sciences:** A core component of our research is the development of novel statistical and AI-driven methodologies. This includes advancing [nested sampling techniques](https://arxiv.org/abs/1506.00171) (e.g., [PolyChord](https://arxiv.org/abs/1506.00171), [dynamic nested sampling](https://arxiv.org/abs/1704.03459), and [accelerated nested sampling with $\beta$-flows](https://arxiv.org/abs/2411.17663)), creating powerful [simulation-based inference (SBI) frameworks](https://arxiv.org/abs/2504.10230), and employing [machine learning for tasks such as radiometer calibration](https://arxiv.org/abs/2504.16791), [cosmological emulation](https://arxiv.org/abs/2503.13263), and [mitigating radio frequency interference](https://arxiv.org/abs/2211.15448). We also explore the potential of [foundation models for scientific discovery](https://arxiv.org/abs/2401.00096).
**Technical Contributions:**
Our group has a strong track record of developing widely-used scientific software. Notable examples include:
* [**PolyChord**](https://arxiv.org/abs/1506.00171): A next-generation nested sampling algorithm for Bayesian computation.
* [**anesthetic**](https://arxiv.org/abs/1905.04768): A Python package for processing and visualizing nested sampling runs.
* [**GLOBALEMU**](https://arxiv.org/abs/2104.04336): An emulator for the sky-averaged 21-cm signal.
* [**maxsmooth**](https://arxiv.org/abs/2007.14970): A tool for rapid maximally smooth function fitting.
* [**margarine**](https://arxiv.org/abs/2205.12841): For marginal Bayesian statistics using normalizing flows and KDEs.
* [**fgivenx**](https://arxiv.org/abs/1908.01711): A package for functional posterior plotting.
* [**nestcheck**](https://arxiv.org/abs/1804.06406): Diagnostic tests for nested sampling calculations.
### Impact and Discoveries
Our research has led to significant advancements in cosmological data analysis and yielded new insights into the Universe. Key achievements include:
* Pioneering the development and application of advanced Bayesian inference tools, such as [PolyChord](https://arxiv.org/abs/1506.00171), which has become a cornerstone for cosmological parameter estimation and model comparison globally.
* Making significant contributions to the analysis of major cosmological datasets, including the [Planck mission](https://arxiv.org/abs/1807.06209), providing some of the tightest constraints on cosmological parameters and models of [inflation](https://arxiv.org/abs/1807.06211).
* Developing novel AI-driven approaches for astrophysical challenges, such as using [machine learning for radiometer calibration in 21-cm experiments](https://arxiv.org/abs/2504.16791) and [simulation-based inference for extracting cosmological information from galaxy clusters](https://arxiv.org/abs/2504.10230).
* Probing the nature of dark energy through innovative [non-parametric reconstructions of its equation of state](https://arxiv.org/abs/2503.08658) from combined datasets.
* Advancing our understanding of the early Universe through detailed studies of [21-cm signals from the Cosmic Dawn and Epoch of Reionization](https://arxiv.org/abs/2301.03298), including the development of sophisticated foreground modelling techniques and emulators like [GLOBALEMU](https://arxiv.org/abs/2104.04336).
* Developing new statistical methods for quantifying tensions between cosmological datasets ([Quantifying tensions in cosmological parameters: Interpreting the DES evidence ratio](https://arxiv.org/abs/1902.04029)) and for robust Bayesian model selection ([Bayesian model selection without evidences: application to the dark energy equation-of-state](https://arxiv.org/abs/1506.09024)).
* Exploring fundamental physics questions such as potential [parity violation in the Large-Scale Structure using machine learning](https://arxiv.org/abs/2410.16030).
### Charting the Future: AI-Powered Cosmological Discovery
The Handley Research Group is poised to lead a new era of cosmological analysis, driven by the explosive growth in data from next-generation observatories and transformative advances in artificial intelligence. Our future ambitions are centred on harnessing these capabilities to address the most pressing questions in fundamental physics.
**Strategic Research Pillars:**
* **Next-Generation Simulation-Based Inference (SBI):** We are developing advanced SBI frameworks to move beyond traditional likelihood-based analyses. This involves creating sophisticated codes for simulating [Cosmic Microwave Background (CMB)](https://arxiv.org/abs/1908.00906) and [Baryon Acoustic Oscillation (BAO)](https://arxiv.org/abs/1607.00270) datasets from surveys like DESI and 4MOST, incorporating realistic astrophysical effects and systematic uncertainties. Our AI initiatives in this area focus on developing and implementing cutting-edge SBI algorithms, particularly [neural ratio estimation (NRE) methods](https://arxiv.org/abs/2407.15478), to enable robust and scalable inference from these complex simulations.
* **Probing Fundamental Physics:** Our enhanced analytical toolkit will be deployed to test the standard cosmological model ($\Lambda$CDM) with unprecedented precision and to explore [extensions to Einstein's General Relativity](https://arxiv.org/abs/2006.03581). We aim to constrain a wide range of theoretical models, from modified gravity to the nature of [dark matter](https://arxiv.org/abs/2106.02056) and [dark energy](https://arxiv.org/abs/1701.08165). This includes leveraging data from upcoming [gravitational wave observatories](https://arxiv.org/abs/1803.10210) like LISA, alongside CMB and large-scale structure surveys from facilities such as Euclid and JWST.
* **Synergies with Particle Physics:** We will continue to strengthen the connection between cosmology and particle physics by expanding the [GAMBIT framework](https://arxiv.org/abs/2009.03286) to interface with our new SBI tools. This will facilitate joint analyses of cosmological and particle physics data, providing a holistic approach to understanding the Universe's fundamental constituents.
* **AI-Driven Theoretical Exploration:** We are pioneering the use of AI, including [large language models and symbolic computation](https://arxiv.org/abs/2401.00096), to automate and accelerate the process of theoretical model building and testing. This innovative approach will allow us to explore a broader landscape of physical theories and derive new constraints from diverse astrophysical datasets, such as those from GAIA.
Our overarching goal is to remain at the forefront of scientific discovery by integrating the latest AI advancements into every stage of our research, from theoretical modeling to data analysis and interpretation. We are excited by the prospect of using these powerful new tools to unlock the secrets of the cosmos.
Content generated by [gemini-2.5-pro-preview-05-06](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/index.txt).
Image generated by [imagen-3.0-generate-002](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/index.txt).
```
2. **Paper Metadata:**
```yaml
!!python/object/new:feedparser.util.FeedParserDict
dictitems:
id: http://arxiv.org/abs/2401.00096v2
guidislink: true
link: http://arxiv.org/abs/2401.00096v2
updated: '2024-03-01T07:19:05Z'
updated_parsed: !!python/object/apply:time.struct_time
- !!python/tuple
- 2024
- 3
- 1
- 7
- 19
- 5
- 4
- 61
- 0
- tm_zone: null
tm_gmtoff: null
published: '2023-12-29T23:08:59Z'
published_parsed: !!python/object/apply:time.struct_time
- !!python/tuple
- 2023
- 12
- 29
- 23
- 8
- 59
- 4
- 363
- 0
- tm_zone: null
tm_gmtoff: null
title: A foundation model for atomistic materials chemistry
title_detail: !!python/object/new:feedparser.util.FeedParserDict
dictitems:
type: text/plain
language: null
base: ''
value: A foundation model for atomistic materials chemistry
summary: 'Machine-learned force fields have transformed the atomistic modelling
of
materials by enabling simulations of ab initio quality on unprecedented time
and length scales. However, they are currently limited by: (i) the significant
computational and human effort that must go into development and validation of
potentials for each particular system of interest; and (ii) a general lack of
transferability from one chemical system to the next. Here, using the
state-of-the-art MACE architecture we introduce a single general-purpose ML
model, trained on a public database of 150k inorganic crystals, that is capable
of running stable molecular dynamics on molecules and materials. We demonstrate
the power of the MACE-MP-0 model - and its qualitative and at times
quantitative accuracy - on a diverse set problems in the physical sciences,
including the properties of solids, liquids, gases, chemical reactions,
interfaces and even the dynamics of a small protein. The model can be applied
out of the box and as a starting or "foundation model" for any atomistic system
of interest and is thus a step towards democratising the revolution of ML force
fields by lowering the barriers to entry.'
summary_detail: !!python/object/new:feedparser.util.FeedParserDict
dictitems:
type: text/plain
language: null
base: ''
value: 'Machine-learned force fields have transformed the atomistic modelling
of
materials by enabling simulations of ab initio quality on unprecedented time
and length scales. However, they are currently limited by: (i) the significant
computational and human effort that must go into development and validation
of
potentials for each particular system of interest; and (ii) a general lack
of
transferability from one chemical system to the next. Here, using the
state-of-the-art MACE architecture we introduce a single general-purpose ML
model, trained on a public database of 150k inorganic crystals, that is capable
of running stable molecular dynamics on molecules and materials. We demonstrate
the power of the MACE-MP-0 model - and its qualitative and at times
quantitative accuracy - on a diverse set problems in the physical sciences,
including the properties of solids, liquids, gases, chemical reactions,
interfaces and even the dynamics of a small protein. The model can be applied
out of the box and as a starting or "foundation model" for any atomistic system
of interest and is thus a step towards democratising the revolution of ML
force
fields by lowering the barriers to entry.'
authors:
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Ilyes Batatia
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Philipp Benner
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Yuan Chiang
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Alin M. Elena
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: "D\xE1vid P. Kov\xE1cs"
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Janosh Riebesell
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Xavier R. Advincula
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Mark Asta
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Matthew Avaylon
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: William J. Baldwin
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Fabian Berger
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Noam Bernstein
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Arghya Bhowmik
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Samuel M. Blau
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: "Vlad C\u0103rare"
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: James P. Darby
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Sandip De
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Flaviano Della Pia
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Volker L. Deringer
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: "Rokas Elijo\u0161ius"
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Zakariya El-Machachi
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Fabio Falcioni
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Edvin Fako
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Andrea C. Ferrari
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Annalena Genreith-Schriever
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Janine George
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Rhys E. A. Goodall
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Clare P. Grey
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Petr Grigorev
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Shuang Han
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Will Handley
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Hendrik H. Heenen
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Kersti Hermansson
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Christian Holm
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Jad Jaafar
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Stephan Hofmann
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Konstantin S. Jakob
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Hyunwook Jung
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Venkat Kapil
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Aaron D. Kaplan
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Nima Karimitari
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: James R. Kermode
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Namu Kroupa
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Jolla Kullgren
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Matthew C. Kuner
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Domantas Kuryla
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Guoda Liepuoniute
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Johannes T. Margraf
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: "Ioan-Bogdan Magd\u0103u"
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Angelos Michaelides
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: J. Harry Moore
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Aakash A. Naik
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Samuel P. Niblett
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Sam Walton Norwood
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Niamh O'Neill
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Christoph Ortner
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Kristin A. Persson
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Karsten Reuter
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Andrew S. Rosen
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Lars L. Schaaf
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Christoph Schran
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Benjamin X. Shi
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Eric Sivonxay
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: "Tam\xE1s K. Stenczel"
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Viktor Svahn
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Christopher Sutton
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Thomas D. Swinburne
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Jules Tilly
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Cas van der Oord
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Eszter Varga-Umbrich
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Tejs Vegge
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: "Martin Vondr\xE1k"
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Yangshuai Wang
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: William C. Witt
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Fabian Zills
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: "G\xE1bor Cs\xE1nyi"
author_detail: !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: "G\xE1bor Cs\xE1nyi"
author: "G\xE1bor Cs\xE1nyi"
arxiv_comment: 119 pages, 63 figures, 37MB PDF
links:
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
href: http://arxiv.org/abs/2401.00096v2
rel: alternate
type: text/html
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
title: pdf
href: http://arxiv.org/pdf/2401.00096v2
rel: related
type: application/pdf
arxiv_primary_category:
term: physics.chem-ph
scheme: http://arxiv.org/schemas/atom
tags:
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
term: physics.chem-ph
scheme: http://arxiv.org/schemas/atom
label: null
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
term: cond-mat.mtrl-sci
scheme: http://arxiv.org/schemas/atom
label: null
```
3. **Paper Source (TeX):**
```tex
\documentclass[10pt]{article}
\usepackage{scicite}
\usepackage[letterpaper, margin=1in]{geometry}
\usepackage{graphicx}
\usepackage{subcaption}
\usepackage[range-phrase={--},per-mode = single-symbol]{siunitx}
\usepackage{chemformula}
\usepackage[version=4]{mhchem}
\usepackage{hyperref}
\usepackage{cleveref}
\usepackage{authblk}
\usepackage{varwidth}
\usepackage{xcolor}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[normalem]{ulem}
\usepackage{adjustbox}
\Crefname{figure}{Figure}{Figures}
\crefname{figure}{Fig.}{Figs.}
\newcommand{\MLFF}{MACE-MP-0} %
\newenvironment{sciabstract}{%
\begin{quote} \bf}
{\end{quote}}
\newcommand{\suggest}[1]{{\color{red} #1}}
\newcommand{\done}[1]{{\color{green} #1}}
\DeclareSIUnit\Angstrom{\textup{\AA}} %
\newcommand\longvar[1]{\mathchardef\UrlBreakPenalty=100\mathchardef\UrlBigBreakPenalty=100\url{#1}}
\begin{document}
\title{A foundation model for atomistic materials chemistry}
\renewcommand*{\thefootnote}{\fnsymbol{footnote}}
\footnotetext[2]{\normalsize These authors, ordered alphabetically, contributed equally. All others, except for the corresponding author, are also ordered alphabetically.}
\footnotetext[1]{\normalsize Corresponding author: \href{mailto:gc121@cam.ac.uk}{gc121@cam.ac.uk}}
\renewcommand*{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\author[1]{Ilyes Batatia$^\dagger$}
\author[2]{Philipp Benner$^\dagger$}
\author[3,4]{Yuan Chiang$^\dagger$}
\author[17]{Alin M. Elena$^\dagger$} %
\author[1]{D\'avid P. Kov\'acs$^\dagger$}
\author[4,13]{Janosh Riebesell$^\dagger$}
\author[12,13]{Xavier R. Advincula}
\author[3,4]{Mark Asta}
\author[30]{Matthew Avaylon}
\author[1]{William J. Baldwin}
\author[12]{Fabian Berger} %
\author[11]{Noam Bernstein}
\author[25]{Arghya Bhowmik}
\author[10]{Samuel M. Blau}
\author[1,13]{Vlad C\u{a}rare}
\author[1]{James P. Darby}
\author[18]{Sandip De}
\author[12]{Flaviano Della Pia}
\author[16]{Volker L. Deringer}
\author[1]{Rokas Elijošius}
\author[16]{Zakariya El-Machachi}
\author[31]{Fabio Falcioni}
\author[18]{Edvin Fako}
\author[26]{Andrea C. Ferrari}
\author[12]{Annalena Genreith-Schriever}
\author[2,6]{Janine George}
\author[15]{Rhys E. A. Goodall} %
\author[12]{Clare P. Grey}
\author[27]{Petr Grigorev}
\author[18]{Shuang Han}
\author[13,19]{Will Handley}
\author[9]{Hendrik H. Heenen}
\author[23]{Kersti Hermansson}
\author[22]{Christian Holm}
\author[1]{Stephan Hofmann}
\author[1]{Jad Jaafar}
\author[9]{Konstantin S. Jakob}
\author[9]{Hyunwook Jung}
\author[12, 21]{Venkat Kapil}
\author[4]{Aaron D. Kaplan}
\author[20]{Nima Karimitari}
\author[28]{James R. Kermode}
\author[13,19,1]{Namu Kroupa}
\author[23]{Jolla Kullgren}
\author[3,4]{Matthew C. Kuner}
\author[12]{Domantas Kuryla}
\author[1,26]{Guoda Liepuoniute}
\author[8]{Johannes T. Margraf}
\author[24]{Ioan-Bogdan Magd\u{a}u}
\author[12]{Angelos Michaelides}
\author[1]{J. Harry Moore}
\author[2,6]{Aakash A. Naik}
\author[12]{Samuel P. Niblett}
\author[25]{Sam Walton Norwood}
\author[12,13]{Niamh O'Neill}
\author[5]{Christoph Ortner}
\author[3,4,7]{Kristin A. Persson}
\author[9]{Karsten Reuter}
\author[3,4]{Andrew S. Rosen} %
\author[1]{Lars L. Schaaf}
\author[13]{Christoph Schran}
\author[12]{Benjamin X. Shi} %
\author[10]{Eric Sivonxay}
\author[1]{Tam\'as K. Stenczel}
\author[23]{Viktor Svahn}
\author[20]{Christopher Sutton}
\author[27]{Thomas D. Swinburne}
\author[31]{Jules Tilly}
\author[1]{Cas van der Oord}
\author[29]{Santiago Vargas}
\author[1]{Eszter Varga-Umbrich}
\author[25]{Tejs Vegge}
\author[8,9]{Martin Vondrák}
\author[5]{Yangshuai Wang}
\author[14]{William C. Witt}
\author[22]{Fabian Zills}
\author[1]{G\'abor Cs\'anyi$^*$}
\affil[1]{Engineering Laboratory, University of Cambridge, Trumpington St and JJ Thomson Ave, Cambridge, UK}
\affil[2]{Federal Institute of Materials Research and Testing (BAM), Berlin, Germany}
\affil[3]{Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA}
\affil[4]{Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA}
\affil[5]{Mathematics Department, University of British Columbia, 1984 Mathematics Rd, Vancouver, BC V6T 1Z2, Canada}
\affil[6]{Institute of Condensed Matter Theory and Solid State Optics, Friedrich Schiller University Jena, Germany}
\affil[7]{Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA}
\affil[8]{University of Bayreuth, Bavarian Center for Battery Technology (BayBatt), Bayreuth, Germany}
\affil[9]{Fritz-Haber-Institute of the Max-Planck-Society, Berlin, Germany}
\affil[10]{Energy Technologies Area, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA}
\affil[11]{U.~S. Naval Research Laboratory, Washington DC 20375, USA}
\affil[12]{Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, UK}
\affil[13]{Cavendish Laboratory, University of Cambridge, J. J. Thomson Ave, Cambridge, UK}
\affil[14]{Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, CB3 0FS, Cambridge, United Kingdom}
\affil[15]{Chemix, Inc., Sunnyvale, CA 94085, USA}
\affil[16]{Inorganic Chemistry Laboratory, Department of Chemistry, University of Oxford, Oxford OX1 3QR, UK}
\affil[17]{Scientific Computing Department, Science and Technology Facilities Council, Daresbury Laboratory, Keckwick Lane, Daresbury WA4 4AD, UK}
\affil[18]{BASF SE, Carl-Bosch-Stra{\ss}e 38, 67056 Ludwigshafen, Germany}
\affil[19]{Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK}
\affil[20]{Department of Chemistry and Biochemistry, University of South Carolina, South Carolina 29208, USA}
\affil[21]{Lennard-Jones Centre, University of Cambridge, Trinity Ln, Cambridge, CB2 1TN, UK}
\affil[22]{Institute for Computational Physics, University of Stuttgart, 70569 Stuttgart, Germany}
\affil[23]{Department of Chemistry--Ångström, Uppsala University, Box 538, S-751 21, Uppsala, Sweden}
\affil[24]{School of Natural and Environmental Science, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK}
\affil[25]{Department of Energy Conversion and Storage, Technical University of Denmark, Anker Engelunds Vej 301, 2800 Kgs. Lyngby, Denmark}
\affil[26]{Cambridge Graphene Centre, University of Cambridge, Cambridge, CB3 0FA, UK}
\affil[27]{Aix-Marseille Universit\'{e}, CNRS, CINaM UMR 7325, Campus de Luminy, 13288 Marseille, France}
\affil[28]{Warwick Centre for Predictive Modelling, School of Engineering, University of Warwick, Coventry CV4 7AL, United Kingdom}
\affil[29]{Department of Chemistry and Biochemistry, University of California -- Los Angeles, 607 Charles E. Young Drive East, Los Angeles, CA, 90095 USA}
\affil[30]{Computing Sciences Area, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA}
\affil[31]{InstaDeep, London, W2 1AY, United Kingdom}
\date{\today}
\maketitle
\begin{sciabstract}
Machine-learned force fields have transformed the atomistic modelling of materials by enabling simulations of {\em ab initio} quality on unprecedented time and length scales.
However, they are currently limited by: (i) the significant computational and human effort that must go into development and validation of potentials for each particular system of interest; and (ii) a general lack of transferability from one chemical system to the next.
Here, using the state-of-the-art MACE architecture we introduce a single general-purpose ML model, trained on a public database of 150k inorganic crystals, that is capable of running stable molecular dynamics on molecules and materials.
We demonstrate the power of the \MLFF{} model — and its qualitative and at times quantitative accuracy — on a diverse set problems in the physical sciences, including the properties of solids, liquids, gases, chemical reactions, interfaces and even the dynamics of a small protein.
The model can be applied out of the box and as a starting or “foundation model” for any atomistic system of interest and is thus a step towards democratising the revolution of ML force fields by lowering the barriers to entry.
\end{sciabstract}
\addtocontents{toc}{\protect\setcounter{tocdepth}{0}} %
\section{Introduction}
Atomic-scale simulation based on density functional theory (DFT) is an enormously successful component of materials modeling\cite{KohnDFT,qe2020,kuhne2020cp2k,VASP1, Hasnip2014Density,Jain2016Computational,Neugebauer2013Density}.
However, the computational cost of such {\em ab initio} methods, which use electronic structure theory directly, becomes prohibitive for many important cases (e.g., amorphous solids, condensed phase liquids, nanostructured materials, and more). Although fast analytical models in the form of empirical interatomic potentials (or force fields) have existed for decades, having varying levels of accuracy and applicability\cite{finnis2003interatomic}, they generally fail to achieve DFT accuracy, particularly when describing reactive events and phase transitions. As a result, they have been unable to displace DFT for most applications. More recently, machine learning (ML)-based interatomic potentials, custom-trained for a particular material or system, have improved the achievable accuracy considerably, albeit at a moderate increase in cost relative to empirical force fields \cite{behler2007, gap, THOMPSON2015316, schnet, deringer_machine_2019, drautz2019, lilienfeld2020, batzner2022, ko2023recent}. Yet, such custom-trained potentials require significant computational and human effort for the generation of DFT training data, as well as model training and validation\cite{deringer_chemrev2021}.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figs/mp0v6compressed-lowres.pdf}
\caption{
{\bf A foundation model for materials modelling.}
Trained only on Materials Project data \cite{jain2013commentary} which consists primarily of inorganic crystals and is skewed heavily towards oxides, \MLFF{} is capable of molecular dynamics simulation across a wide variety of chemistries in the solid, liquid and gaseous phases.}
\label{fig:mp}
\end{figure}
A pinnacle achievement of ML potentials would be to accurately describe the potential energy surface (PES) across all possible chemical and structural spaces without incurring the high computational cost of {\em ab initio} electronic structure methods. By enabling robust, accurate molecular dynamics (MD) simulations for any material, such a potential would enable immediate study of arbitrary systems at a scale currently inaccessible even via the largest available computational resources.
(Here, by robustness we mean that the trajectory should not irreversibly end up in unphysical configurations, a frequently observed behaviour for current-generation ML potentials, especially in long multi-nanosecond simulations\cite{Stocker2022robust}.)
Particularly desirable applications would include complex chemical reaction processes in both solid and liquid phases, at solid-fluid interfaces, or under pressure.
A key advance towards this goal was made by the MEGNet~\cite{megnet2019} model, which provided property prediction for inorganic crystals, and was trained on minimum energy configurations in the Materials Project (MP)~\cite{jain2013commentary} that includes most elements of the periodic table (89) and electronic structure calculations using the Perdew-Burke-Enzerhof (PBE) exchange-correlation functional\cite{perdew1996generalized}.
More recently, models using graph neural network architectures with the capacity to compute forces were also trained on MP-based datasets, including M3GNet~\cite{chen_universal_2022} and CHGNet~\cite{deng_chgnet_2023}, which were
both trained on snapshots from DFT relaxations of MP structures, with CHGNet using the MPtrj dataset introduced at the same time\cite{deng_chgnet_2023}.
The ALIGNN-FF model\cite{choudhary2023unified} was also trained on a database of inorganic crystals, JARVIS-DFT\cite{jarvis}, that covers 89 elements and uses the optB88vdW exchange-correlation functional\cite{optB88vdW}.
The proprietary GNoME \cite{merchant_scaling_2023} (NequIP architecture \cite{batzner2022}) model also starts from MP, but uses a complex active learning workflow to generate and train on a dataset of inorganic crystals nearly two orders of magnitude larger than MPtrj.
The above models were created primarily for the purpose of ``materials discovery'', i.e. predicting thermodynamic stability of hypothetical inorganic crystals. In addition, they were capable of molecular dynamics for such crystals, and indeed both CHGNet and GNoME were used to study alkali metal ion diffusion in battery materials.
The DPA models (DPA-1\cite{dpa1} and DPA-2\cite{dpa2}) were trained to a wide variety of datasets (with 56 and 73 elements, respectively), a combination of some previously available and some released with the models (altogether 4M configurations). The second paper reports MD results for versions of the baseline model fine-tuned separately to specific systems (e.g. water, solid state electrolytes, ferroelectric oxide).
To date, the most general and transferable force field for molecular dynamics is the PFP model \cite{takamoto2022towards} (TeaNet architecture \cite{takamoto2022teanet}), also proprietary (including its training set that covered 45 elements originally, recently updated to 72 elements\cite{Takamoto2023}, and is significantly larger than MP and also covers molecules and surfaces). PFP was demonstrated for running simulations on solid-state ionic conductors, and a molecular adsorption and a heterogeneous catalysis example.
There are also ML force fields specialized for organic molecules (with a much more limited number of elements)
such as the ANI (and later AimNET) series of models\cite{smith_ani-1_2017, Smith2019,aimnet2019} and the MACE-OFF23 model\cite{kovacs2023maceoff}, as well as for metal alloys\cite{lopanitsyna2023modeling}.
However, there has yet to be a comprehensive demonstration that a single ML potential can describe solid, liquid, and gaseous systems of materials and molecules across the periodic table and well beyond the distribution of the underlying training set.
Here, we present \MLFF{}, a new interatomic potential using the MACE architecture \cite{batatia_mace_2023} that is trained just on the MPtrj dataset, and demonstrate its capabilities on an unprecedented range of qualitative and quantitative examples drawn from computational chemistry and materials science, including running stable molecular dynamics simulations in a wide variety of chemistries, predicting phonon spectra, calculating activation energies for point defect and dislocation motion, simulating solvent mixtures, combusting hydrogen gas, modelling a complete rechargeable battery cell, and much more; several of these are illustrated in \cref{fig:mp}.
We find that this pre-trained {\em foundation model} shows remarkable out-of-distribution performance.
The MACE architecture, which extends the atomic cluster expansion (ACE)\cite{drautz2019, dusson2022atomic, lysogorskiy2021performant, witt2023acepotentials, batatia2022design}, was designed to keep only what appear to be essential components of equivariant graph neural networks\cite{batatia2022design}: the element embedding with tensor decomposition\cite{Darby2023Trace} and the higher order equivariant messages constructed through the tensor product operation. Its unique innovations are that (i) it uses high body order equivariant features in each layer (4-body in the present case), and consequently only two layers of message passing are sufficient; (ii) it is only mildly nonlinear, as the only nonlinear activations are in the radial basis and the final readout layer, hence its classification as a graph tensor network. Its computational cost for evaluation is broadly in line with other graph neural networks, presently allowing simulations of around a thousand atoms for nanoseconds per day on a GPU.
In the following, we highlight three classes of application examples: solid and liquid water, heterogeneous catalysis, and metal--organic frameworks.
The SI contains additional examples in 30 separate sections demonstrating the wide-ranging
transferability of \MLFF{} in predicting properties and dynamic processes of both molecules and materials, as well as benchmarks and graphical exploration of the training data.
\clearpage
\section{Applications}
\subsection{Water and aqueous systems}\label{sec:water_results}
\begin{figure}[!htbp]
\centering\includegraphics[width=0.9\textwidth,,keepaspectratio]{figures_water/water-main.pdf}
\caption{\textbf{\MLFF{} performance for aqueous systems.}
(a) Oxygen--oxygen radial distribution function for bulk water (experimental result from Ref.~\cite{Skinner2013/10.1063/1.4790861}) and ice Ih.
(b) Experimental (Ref.~\cite{Bertie1996,Moberg2017}) and computed infrared spectra of bulk water and ice Ih.
(c) Free energy profiles as a function of the proton transfer barrier for a hydroxide ion and excess proton in ice Ih at \SI{250}{\kelvin} and bulk water at \SI{330}{\kelvin}. Snapshots at the top show the simulation cells.
(d) Performance of \MLFF{} (blue squares) on the relative lattice energies of the DMC-ICE13 dataset, compared to the reference method, PBE-D3\cite{grimme2010consistent} (black circles).
(e) Dissolution of a $4 \times 4 \times 4$ unit-cell \ch{NaCl} nanocrystal in water at \SI{400}{\kelvin}, monitoring the extent of dissolution over the simulation time via the crystal size. Performance of the \MLFF{} (blue line) is compared to a neural network potential\cite{oneill2022crumbling} trained explicitly to capture \ch{NaCl} dissolution (black dashed line).
(f) \ch{SiO2}/water interface simulation showing density modulations and dissociative water adsorption, with an inset highlighting the deprotonation of water as indicated by a shoulder in the water density plot. \ch{H3O+} defects in the liquid are highlighted in green.
(g) The free energy profile of the \ce{O-H} distance in the superionic phase of monolayer water in a confining potential. The inset shows a snapshot of the monolayer superionic phase with lines indicating the \SI{50}{\pico\second}-long trajectory of randomly chosen hydrogen atoms with ``$\times$'' indicating their initial positions.}
\label{fig:water}
\end{figure}
Water is ubiquitous in nature and technology and has long been a major focus of computational work. Driven by the delicate balance between directional hydrogen bonding and primarily non-directional van der Waals interactions, aqueous systems remain a challenge for simulations~\cite{gillan2016perspective}.
For example, the study of proton transfer in water, a fundamental process characterized by the continuous breaking and forming of covalent bonds, has long required using {\em ab initio} molecular dynamics for detailed atomistic insight~\cite{Marx1999/10.1038/17579,Tuckerman2002/10.1038/nature00797,Agmon2016/10.1021/acs.chemrev.5b00736}. We demonstrate in this section how \MLFF{} describes various aqueous systems.
We start by examining the structure of liquid water and hexagonal ice (ice Ih). The oxygen--oxygen radial distribution function, depicted in \cref{fig:water}a, shows reasonable agreement with reference simulations. The infra-red vibrational spectra of both phases, shown in panel \cref{fig:water}b, align well with experimental observations, albeit with a notable red shift in the stretching vibrations indicating a softer description of the \ch{O-H} bond as is well-known for PBE-D3~\cite{gillan2016perspective}. In panel \cref{fig:water}d, the relative stabilities of 12 ice polymorphs with respect to ice Ih, used in a recent benchmark\cite{dmcice13}, show excellent agreement with respect to PBE-D3 with a MAE of around \SI{4}{meV}. Proton defects (\ch{OH^-} and \ch{H3O^+}) in ice Ih and liquid water were simulated, revealing robust descriptions of proton transfer, as shown in \cref{fig:water}c. The proton transfer barrier for hydroxide is higher than for hydronium in liquid water, consistent with experimental diffusion trends.
Next, we evaluate \MLFF{} for describing solid--liquid interfaces.
First, we focus on \ch{NaCl} in water in two cases: a \ch{NaCl}(001) interface in contact with water and a small nanocrystal surrounded by water.
Simulations were performed at \SI{400}{\kelvin} to promote dissolution, and compared to simulations with a custom-trained ML potential based on revPBE-D3 from Ref.~\cite{oneill2022crumbling}.
As expected, for the flat surface the model predicts no dissolution events on the timescale of the simulation (\SI{0.5}{\nano\second}).
Meanwhile, for the nanocrystal surrounded by water, \MLFF{} captures a dissolution mechanism resembling that in Ref.~\cite{oneill2022crumbling} as shown in \cref{fig:water}e. The dissolution proceeds via a crumbling mechanism, where an initial steady loss of ions is followed by the rapid disintegration of the crystal. As ions dissolve from the crystal, they are hydrated by water. The dissolution process is stochastic, leading to an intrinsic variation between independent simulations. The final structure of the dissolved ions in water also displays the expected orientation of the water molecules with respect to the ions.
We then model the \ch{SiO2}/water interface, \cref{fig:water}f, revealing the expected density modulations in the first few contact layers. As before, the liquid phase is found to be overstructured, a common characteristic of the PBE functional \cite{gillan2016perspective} used by MP and therefore by \MLFF{}. \ch{SiO2} is known for its dissociative water adsorption, which we observe in our simulations. Deprotonation of water is evidenced by the shoulder in the water density plot and can also be seen in the inset of a snapshot of this system in \cref{fig:water}f.
Finally, we investigate nanoconfined water in graphene-like nanocapillaries~\cite{Algara-Siller2015/10.1038/nature14295,Fumagalli2018/10.1126/science.aat4191}, which exhibits dramatically different properties from bulk water. \MLFF{} proved robust in simulating nanoconfined water. Stable simulations were conducted at \SI{4}{\giga\pascal} and \SI{600}{\kelvin}, conditions under which a superionic phase with high ionic conductivity was previously predicted~\cite{kapil_first-principles_2023} using a custom-trained ML potential. The \MLFF{} model accurately captured the dynamical characteristics of this phase, including extensive proton transfer on the ten pico-seconds timescale, as illustrated in the inset of \cref{fig:water}g. Comparing the free energy profile associated with the \ce{O-H} distance [Fig.~\ref{fig:water}g] against the PBE-D3 reference, \MLFF{} shows an overall good description, albeit underestimating the proton transfer barrier by \num{1}-\SI{2}{kcal/mol}. This tendency towards autoprotolysis is consistent with the soft description of the \ce{O-H} bond observed in bulk phases.
\subsection{Catalysis}\label{sec:catalysis}
\begin{figure}[!htbp]
\centering\includegraphics[width=0.9\textwidth,keepaspectratio]{figs/Fig_catalysis_v5-lowres.pdf}
\caption{\textbf{\MLFF{} performance for catalytic applications.}
(a) Pourbaix diagrams of \ch{CuO} bulk systems constructed with \MLFF{} (left) and Materials Project reference data (right).
(b) {\MLFF{}}+D3-calculated Pt(111) surface Pourbaix diagram, in overall good agreement with the literature\cite{Hansen2008-zz}.
(c) The relative adsorption energy scaling relation between \ch{O} and \ch{OH} on transition metal surfaces is captured correctly by \MLFF{}+D3, as is the lack of linear scaling between \ch{C} and \ch{O}~\cite{norskov2022nonlinear}. Metals are colored according to rows in the Periodic Table as 3d, 4d and 5d.
(d) Reaction profile of multistep electrochemical \ch{CO} oxidation on \ch{Cu}. \ce{CO-OH} coupling and dehydrogenation reactions are characterised in the upper and lower panel, respectively. Energy profiles from \MLFF{}+D3 nudged elastic band (NEB) calculations, along with PBE+D3 single-point calculations and independent BEEF-vdW profiles from a previous study \cite{Tiwari2020}.
(e) \MLFF{} reaction profile for a key reaction step (\ch{CH2O2 -> CH2 + O}) in the \ch{CO2}-to-methanol conversion on \ch{In2O3}\cite{schaaf2023accurate}.
(f) Comparison of the atomic environments in the training data (blue) and in the \ch{In2O3} NEB images (red) in the form of a UMAP plot~\cite{mcinnes2018umap, rokasel_2023_10426282}. Insets show local environments with similar MACE features (inset frames in blue for training data and in red for NEB configurations), exemplifying which bulk training environments influence predictions for the out-of-domain catalytic test case.}
\label{fig:cat_main}
\end{figure}
The study of heterogeneous \cite{Nrskov2009Towards,Medford2015From,Bruix2019First-principles-based} and electrocatalysis \cite{Qin2023Cation-Coordinated, Man2011Universality, Auer2020Self-activation} is another major area where DFT excels. It provides atomistic insight into the underlying reaction mechanisms and enables the prediction of the properties of new catalytic materials,\cite{Wang2021Ternary} including reaction barriers and rates, in turn used to predict turnover frequencies\cite{Norskov2014}. The latter is essential for the computational discovery of new solid catalysts for overcoming the dependence on rare and toxic elements and improving the efficiency of critical processes for energy conversion. However, the computational cost of DFT is a serious impediment. Empirical interatomic potentials are typically inadequate for catalysis applications as they rarely describe chemical reactions accurately. Machine learning has already had strong impact in computational catalysis\cite{Margraf2023Exploring,schaaf2023accurate, Yang2023Neural}, \textit{e.g.}, enabling fast screening of materials spaces\cite{Tran2018Active,Foppa_Sandip_2022,Khatamirad2023}, and free energy calculations beyond the harmonic approximation\cite{schaaf2023accurate,Stocker2023Estimating,Tran2023Open}. However, developing such accurate potentials from scratch still requires significant human and computational effort.
We now test the performance of \MLFF{} for different catalysis applications and summarise the results in \cref{fig:cat_main}.
Potential--pH Pourbaix diagrams are central to understanding the aqueous stability of solid materials in an electrochemical environment\cite{pourbaix1966,pourbaix1973}, and thus allow predicting the active phase of an electrocatalyst under given conditions. Within the computational hydrogen electrode (CHE) framework \cite{Norskov2004-ow}, these diagrams can be computed without an explicit electrostatic model. \Cref{fig:cat_main}a--b show the \MLFF{}+D3-calculated Pourbaix diagrams for bulk \ch{CuO} and a \ch{Pt}(111) surface. The Pourbaix diagrams are constructed via the formalism described in \cite{Persson2012-kg,Singh2017-uu}, where only the energies of the relevant solids are calculated while corrected experimentally-derived energies are used for the aqueous ions. In both cases, the \MLFF{} results show remarkably good agreement with DFT \cite{Hansen2008-zz}, predicting the correct sequence of stable phases (with the exception of a very narrow region of \ch{Cu2O} stability) and corresponding pH and potential ranges. While this accuracy may be expected for the bulk \ch{CuO} system that is represented in the training set, the electrosorption at the \ch{Pt}(111) surface is also well described despite being out of domain.
In \cref{fig:cat_main}c, adsorption energy scaling relations between atomic and hydrogenated adsorbates on transi\-tion-metal surfaces are shown for \MLFF{}+D3 and PBE+D3 (see SI for more examples). Such scaling relations are central to understanding the activity of heterogeneous catalysts \cite{norskov2007linear, Lopez2019lsr}. \MLFF{}+D3 captures these trends well, and the slopes of the linear fits are in reasonable agreement with DFT (\textit{e.g.} \num{0.6} for \ch{O} vs.\ \ch{OH}, compared to \num{0.64} for PBE+D3).
Importantly, the lack of correlation between \ch{O} and \ch{C} adsorption energies is also captured, indicating that the model is not merely sorting metals according to their general reactivity\cite{norskov2022nonlinear, GarciaMuelas2019redox}. \Cref{fig:cat_main}d--e show reaction energy profiles for \ch{CO} oxidation on \ch{Cu}
\cite{Tiwari2020} and a key step in \ch{CO2} conversion to methanol on \ch{In2O3}\cite{dang2020rationally, schaaf2023accurate}, respectively. While these are not quantitatively accurate when compared to DFT, \MLFF{} nevertheless captures the location and magnitude of the barriers surprisingly well. \Cref{fig:cat_main}f illustrates how \MLFF{} generalizes to out-of-domain catalysis tasks from bulk training configurations. To this end, the high-dimensional \MLFF{} features are projected to 2D using a Uniform Manifold Approximation Projection (UMAP)\cite{mcinnes2018umap}, with local atomic environments in the training set shown in blue and those found in the \ch{In2O3} transition path shown in red. Representative environments with similar \MLFF{} features are highlighted, indicating that the internal representation of the atomic environments in the NEB configurations is similar to the representation of under-coordinated environments and metal--organic systems in the training set.
While \MLFF{} is not always quantitatively accurate for the most challenging catalysis applications, its stability in MD and exploring reactive pathways is remarkable and provides a starting point for further optimisations. Relevant configurations or phase space regions thus identified may subsequently be validated either by first-principles calculations or serve to initiate active-learning for refining the model. Even at its current foundation level, \MLFF{} already allows a statistical sampling far beyond the present DFT-based state of the art which is still largely thermochemistry-centered, whereas \MLFF{} will pave the way for true kinetic modeling by explicit evaluations of reaction profiles and the reactive flux along them.
\subsection{Metal--organic frameworks}\label{sec:mofs}
\begin{figure}[http!]
\centering
\includegraphics[width=0.87\textwidth,keepaspectratio]{figures_mofs/mace_mofs-lowres.pdf}
\caption{\textbf{\MLFF{} applied to MOFs.} (a) Comparison between \MLFF{} and DFT (PBE) energies on 20,375 relaxed structures taken from the QMOF database \cite{rosen2021machine,rosen2022high}. (b) Element-wise mean absolute error (MAE) of \MLFF{} predicted energies with respect to PBE energies from the QMOF database. The absolute energy error per atom of each structure is distributed over all constituent elements (see SI). (c) Mg-MOF-74 structure with chemisorbed \ch{CO2} optimized with \MLFF{}. Color key: \ch{Mg} (orange), \ch{O} (red), \ch{C} (brown), \ch{H} (white). (d) Left: free energy landscape of \ch{CO2} in Mg-MOF-74. Middle: free energy landscape from Ref.~\cite{zeng2023deepmd} using a custom-trained DeePMD ML force field. Right: free energy landscape using the UFF classical force field \cite{rappe1992uff} with DDEC6 charges \cite{manz2016introducing} for the framework and TraPPE for \ch{CO2} \cite{potoff2001vapor}. (e) Free energy maps of 91 hypothetical MOF-74 analogues, with the QMOF ID of the parent \ch{Mg}-containing frameworks indicated at the bottom of each column and the transition metal to the left of each row.}
\label{fig:mof}
\end{figure}
Metal--organic frameworks (MOFs) are a class of nanoporous materials comprised of metal cations or clusters connected by organic linkers arranged in a periodic lattice \cite{yaghi2019introduction}. Due to their large surface areas, tunable building blocks, and permanent porosity, MOFs hold substantial promise for various applications, including but not limited to catalysis, energy storage, gas adsorption and separations, and optoelectronic devices \cite{yaghi2019introduction}.
We tested our pre-trained model directly against version 14 of the Quantum MOF (QMOF) database, which contains DFT-computed properties at several levels of theory for 20,000+ MOFs and structurally related coordination polymers \cite{rosen2021machine, rosen2022high}. \MLFF{} was not trained on any data from the QMOF database, making this a challenging test of its transferability to largely unseen chemistries.
As shown in \cref{fig:mof}a, \MLFF{} performs very well in predicting the absolute energies of MOFs, achieving an MAE of \SI{0.033}{eV/atom} despite the pronounced difference between the inorganic crystals of the MPtrj training set and the MOF structures that make up the QMOF database. This accuracy spans most of the periodic table, as demonstrated in \Cref{fig:mof}b. When the energy prediction is distributed on a per-atom basis, we note a strong elemental dependence of predicted energy error. The higher-than-average errors for certain elements (\ch{Mo}, \ch{V}, \ch{Gd}, \ch{Yb} \textit{etc.}) are due to a difference in the chosen pseudopotentials between the MP and QMOF datasets (see SI \ref{sec:mofs}).
To validate the use of \MLFF{} for capturing dynamic processes, we investigate \ch{CO2} adsorption in a prototypical MOF known as Mg-MOF-74. The MOF-74 family, including the Mg-containing version, has been extensively studied for the selective adsorption of \ch{CO2} \cite{britt2009highly,queen2014comprehensive, choe2021mof}. Of particular note, the coordinatively unsaturated metal sites \cite{kokccam2020coordinatively} of Mg-MOF-74 enable chemical bonding interactions between the metal and \ch{CO2} adsorbate \cite{britt2009highly} that cannot be captured from classical force fields alone. We directly compare the adsorption dynamics against the results presented in Ref.\cite{zheng2023quantum}, which considered the same system using a custom-trained ML force field generated using DeePMD-Kit \cite{zeng2023deepmd} and PBE-D3 calculations in CP2K \cite{kuhne2020cp2k}.
\MLFF{} accurately and efficiently captures the \ch{CO2} adsorption process in Mg-MOF-74.
As shown in \cref{fig:mof}c, the \ch{CO2} adsorbate favorably binds to the Mg center in a tilted configuration that is in agreement with both experimental neutron diffraction data \cite{valenzano2010computational,queen2014comprehensive} and the previous custom-trained ML model \cite{zheng2023quantum}. The mean bond distance between the \ch{Mg} center and \ch{CO2} adsorbate is predicted to be \SI{2.38}{\Angstrom} from \MLFF{} (\Cref{fig:mof_si_a}), in close agreement with the experimental value of \SI{2.27}{\Angstrom} \cite{queen2014comprehensive} and the value of \SI{2.23}{\Angstrom} from the custom ML model in Ref. \cite{zheng2023quantum}. The mean \ce{Mg-O-C} bond angle is predicted to be \ang{133.7} from \MLFF{} (\cref{fig:mof_si_a}), substantially closer to the experimentally determined bond angle of 131$^\circ$ \cite{queen2014comprehensive} than the \ang{118.6} value from the ML model in Ref.~\cite{zheng2023quantum}. The projected density map for the \ch{CO2} adsorption site (\cref{fig:mof}b) is, again, in excellent agreement with prior work \cite{valenzano2010computational,zheng2023quantum} and shows how the adsorbed \ch{CO2} molecules are mobile but largely confined to the vicinity of the Mg binding site due to chemisorption.
To showcase an example of how one might use \MLFF{} in a high-throughput setting, we considered 91 hypothetical MOF-74 analogues derived from those in Ref.\ \cite{witman2016silico} based on 13 (out of 58) different frameworks and seven different metal cations (M) that have been used to synthesize M-MOF-74 \cite{queen2014comprehensive}. \Cref{fig:mof}e shows the resulting free energy maps, comprising over \SI{357}{\nano\second} of simulation altogether, displaying diverse and dynamic behaviour of the \ch{CO2} adsorbate across the range of hypothetical MOF-74 analogues.
Given the nature of our foundation model, we anticipate many additional application areas where \MLFF{} (or one of its future variants) could be of value in the MOF field. Based on the \ch{CO2} adsorption example, we envision applications in capturing dynamic processes, particularly those that cannot be accurately modeled using classical force fields and are prohibitively expensive to carry out with {\em ab initio} MD given the large unit-cell size required to describe most MOFs. Foundation models are promising for modeling competitive multi-component physisorption and chemisorption processes, especially across many families of compositionally different MOFs and combinations of gas mixtures, for which training a system-specific, on-the-fly active learning model would be expensive or even prohibitive.
In addition to the compositional diversity relevant to high-throughput screening, not all MOFs can be described via a static picture and based on an ideal crystalline structure: in fact, there has been recent interest in liquid and amorphous MOFs \cite{bennett2018liquid,castel2022atomistic}, and the dynamic behavior of crystalline frameworks \cite{evans2020four} --- such as in the so-called ``flexible'' and ``breathing'' MOFs --- has been leveraged for highly selective separation processes \cite{taylor2018near}. This dynamic behavior cannot be completely captured from static DFT calculations alone, and accurate and easily accessible interatomic potentials are expected to accelerate the modeling of spatio-temporal processes in future studies \cite{van_speybroeck_towards_2021}.
\subsection{Further applications and Supplementary Information}
In the Supplementary Information in 32 subsections, we provide further application examples. We also give the results of a comprehensive set of benchmarks, including the performance on calculating phonon dispersions, bulk and shear moduli of crystals, atomisation energies and lattice constants of elemental solids, the cohesive energies of the S66 set\cite{S66} of molecular dimers and the X23 set\cite{x23_rt} of molecular crystals, the CRBH20 set\cite{crbh20} of reaction barrier heights, and the homonuclear diatomic binding curves. The full set of heteronuclear diatomic curves is provided in the Supplementary Materials.
We also give more details on the training protocol, and a graphical exploration of the data, including histograms of energies, forces, stresses, magnetic moments, and element and composition counts.
\clearpage
\section{Current limitations and future outlook}
The DFT-quality simulation and stable MD propagation for a wide range of materials across the periodic table that we have shown here are landmark achievements for a single machine-learned interatomic potential.
Yet there are a number of limitations of the current version of the \MLFF{} model.
The exchange-correlation functional used in the MPtrj dataset is PBE~\cite{perdew1996generalized}, which must be augmented with Hubbard $U$ terms
to improve electronic correlations for particular element combinations (introducing inconsistencies in the PES that must be
compensated~\cite{Jain2016Computational}), and dispersion corrections, such as D3\cite{grimme2010consistent}. Recent developments in DFT are beginning to supersede it by achieving improved accuracy at comparable computational cost~\cite{Furness2020Accurate,Kingsbury2022Performance}, and methods beyond DFT such as hybrid functionals~\cite{henderson_accurate_2011} and the random phase approximation~\cite{harl_assessing_2010} improve
upon this even further but at much larger computational cost.
Refitting the model to a more modern functional is expected to increase its predictive power, and will reduce the need for system-dependent
corrections such as the use of Hubbard $U$ terms and dispersion.
The MACE model that we used to fit the data does not contain explicit long range interactions (beyond the \SI{12}{\Angstrom} receptive field afforded by two steps of message passing), nor does it take into account magnetic or spin degrees of freedom. Despite the success in describing many different chemistries demonstrated herein, there will be observables, particularly in the context of dilute solutions and at interfaces, that cannot be calculated with a short-range model. There are several approaches to incorporating explicit electrostatic interactions into ML models in the literature\cite{Ghasemi2015Interatomic,Grisafi2019Incorporating,Ko2021fourth-generation,Vondrak2023q-pac}, as well as spin degrees of freedom\cite{mtpmagnetic2022,rinaldi2023noncollinear, deng_chgnet_2023}. A subsequent version of our model will undoubtedly benefit from such an extension.
Considering the results for the diverse systems shown in the SI, there are two broad areas where the model clearly needs improvement: (i) describing intermolecular interactions, (ii) high pressure simulations. While the overarching goal of MD stability is achieved for ambient conditions, for many systems there is room for improvement in a quantitative sense.
In some cases, e.g.\ low pressure simulations of ethanol (section~\cref{sec:ethwat}), these small quantitative deviations lead to qualitative
errors by shifting an important phase transition's temperature or pressure, thereby changing the equilibrium phase at ambient conditions. In
other cases, namely atoms that approach each other at very close distances in random structure search (section~\cref{sec:rss}) or high pressure hydrogen (section~\cref{sec:h2}), the energy errors are large, and the simulations become trapped in anomalously low energy, unphysical geometries.
These errors are not specific to certain systems, and both can be addressed straightforwardly by extending the existing training data to lower and higher pressures\cite{magduau2023machine} for the former, and using repulsive pair potentials\cite{byggmastar2019} for the latter. In most cases, it is not yet wise to solely rely on ML potentials for all chemical or physical predictions without further validation \cite{morrow_how_2023}, and the same is true for \MLFF{}.
Several possible factors may be limiting the model's accuracy: the size of the model in terms of number of free parameters and the limits this places on its expressivity, the total amount and type of configurations in the training set, or inconsistencies in the quantum-mechanically computed data labels.
Exploration of possible improvements to the model and its training data is ongoing to determine which of these are responsible for current limitations. The results will determine in what ways the model and its improved versions will be used in the future.
The most pessimistic, but we think unlikely, possibility is that using \MLFF{} as a foundation model that must be fine-tuned to give quantitative accuracy for specific systems will require very large amounts of data and/or training time, and that the pre-trained model will not provide a significant shortcut compared to training models from scratch.
In this case, the ability of \MLFF{} to produce {\em reasonable} trajectories will still make it useful as an efficient source of configurations for system-specific fitting databases, perhaps augmented by further active learning.
A more likely scenario is that the current model will at least be able to serve as a starting point that can be efficiently
fine-tuned for any particular system.
It remains to be seen how much additional data would be needed for such refinement, but based on previous experience we are optimistic;
pre-training with cheaply generated data and subsequent fine-tuning has been shown to improve accuracy and stability of ML potentials \cite{Gardner2023, dpa2}, and transfer-learning approaches can enable such models to fit higher quality reference data\cite{Smith2019}.
This type of refinement will definitely be required for systems where the level of theory that was used to calculate the currently used training dataset are considered to be inadequate.
There is good evidence that reaching higher levels of electronic structure theory from a DFT baseline and beyond requires significantly less data than fitting to DFT itself\cite{bartok_water_2013,Smith2019,dral2020}
Finally, we may find that adding only a moderate number of additional configurations computed with essentially the same methodology will be sufficient to achieve quantitative agreement with the target level of theory across the full range of chemistry and structure. If this turns out to be true, future versions of \MLFF{} may truly provide a universal model for carrying out material simulations.
\clearpage
\section{Methods}
\subsection{Model}
\paragraph{MACE} All models trained in the paper use the MACE~\cite{batatia_mace_2023} architecture implemented in PyTorch~\cite{Paszke2019PyTorchAI} and employing the {\em e3nn} library~\cite{geiger2022e3nn}. The MACE training and evaluation codes are distributed via GitHub under the MIT license, available at \url{https://github.com/ACEsuit/mace/}. The models used in this paper are available at \url{https://github.com/ACEsuit/mace-mp/}.
MACE is an equivariant message-passing graph tensor network where each layer encodes many-body information of atomic geometry.
At each layer, many-body messages are formed using a linear combination of a tensor product basis~\cite{batatia2022design, Darby2023Trace}.
This is constructed by taking tensor products of a sum of two-body permutation-invariant polynomials, expanded in a spherical basis.
The final output is the energy contribution of each atom to the total potential energy.
For a more detailed description of the architecture, see Refs.~\cite{batatia_mace_2023} and \cite{KovacsBenchmark2023}.
\paragraph{Hyper-parameters} All models referred to in this work use two MACE layers, a spherical expansion of up to $l_\text{max}=3$, and 4-body messages in each layer (correlation order 3).
All models use a \num{128}-channel dimension for tensor decomposition.
We use a radial cutoff of \SI{6}{\Angstrom} and expand the interatomic distances into 10 Bessel functions multiplied by a smooth polynomial cutoff function to construct radial features, in turn fed into a fully-connected feed-forward neural network with three hidden layers of \num{64} hidden units and SiLU non-linearities.
We fit three different size models, which only differ by the maximal message equivariance, $L=0,1,2$ for the small, medium and large models, respectively, and provide different compromises between computational cost and fitting accuracy.
The irreducible representations of the messages have alternating parity (in {\em e3nn} notation, $\texttt{128x0e}$ for the small model, $\texttt{128x0e + 128x1o}$ for the medium model, and $\texttt{128x0e + 128x1o + 128x2e}$ for the large model).
All application examples in this paper are run with the medium $L=1$ model as it offers a good cost-accuracy trade-off.
\paragraph{Normalization}
To ensure internal normalization of the weights, we divide the atomic basis in each layer by the average number of neighbors in the training dataset, as proposed in~\cite{batatia2022design}.
This number is fixed at $\approx 62$. The node energy $\epsilon_{a}$ of atom $a$ is shifted by the mean of the atomic energies.
Therefore, the prediction of the energy for the whole structure is constructed as
\begin{equation*}
\hat{E} = \sum_{a=1}^{N} \left[\sigma \left(\sum_{k=1}^K \epsilon_a^{(k)}\right) + \mu_{Z_a}\right]
\end{equation*}
where $K$ denotes the total number of message passing layers and $\epsilon_a^{(k)}$ is the energy of atom $a$ at layer $k$.
$\mu$ and $\sigma$ are the mean atomic energies and the mean square of the atomic forces computed on the training set.
The predicted forces and stresses are computed as derivatives of the total energy with respect to the atomic positions and the strain tensor, respectively.
\paragraph{Training loss}
The models were trained using a weighted sum of Huber losses of energy, forces, and stress:
\begin{equation}
\begin{aligned}
\mathcal{L} & = \frac{\lambda_E}{N_b} \sum_{b=1}^{N_b}\mathcal{L}_\text{Huber}\biggl(\frac{\hat{E}_b}{N_a}, \frac{E_b}{N_a}, \delta_E\biggr) \\
&\hphantom{=}+ \frac{\lambda_F}{3\sum_{b=1}^{N_b}N_a} \sum_{b=1}^{N_b} \sum_{a=1}^{N_a}\sum_{i=1}^{3}\mathcal{L}^\star_\text{Huber}\biggl(\frac{\partial\hat{E}_b}{\partial {r}_{b,a,i}}, F_{b,a,i}, \delta_F\biggr) \\
& \hphantom{=}+ \frac{\lambda_\sigma}{9N_b} \sum_{b=1}^{N_b}\sum_{i=1}^3\sum_{j=1}^3\mathcal{L}_\text{Huber}\biggl(\frac{1}{V_b}\frac{\partial\hat{E}_b}{\partial {\varepsilon}_{b,ij}}, \sigma_{b,ij}, \delta_\sigma\biggr),
\end{aligned}
\label{eq:loss}
\end{equation}
where $\lambda_E, \lambda_F, \lambda_\sigma$ are predetermined weights of energy ($E$), forces ($F$), and stress ($\sigma$) losses, the symbols under a hat correspond to predicted values, and $N_b$ and $N_a$ are the batch size and the number of atoms in each structure. In the last term involving the stress, $\varepsilon_b$ and $\sigma_b$ correspond to the strain and stress tensors, respectively.
We used $(\lambda_E, \lambda_F, \lambda_\sigma) = (1, 10, 100)$ and Huber deltas of $\delta_E = 0.01, \delta_F = 0.01, \delta_\sigma = 0.01$.
We use a conditional Huber loss $\mathcal{L}^\star_\text{Huber}$ for forces, where the Huber delta $\delta_F$ is adaptive to the force magnitude on each atom.
The Huber delta $\delta_F$ decreases step-wise by a factor from \num{1.0} to \num{0.1} as the atomic force increases from \num{0} to \SI{300}{eV\per\Angstrom}.
For more details, see the section~\ref{sec:trainingprotocol} in the SI.
\paragraph{Optimization} The models are trained with the AMSGrad~\cite{Reddi2019} variant of Adam~\cite{Kingma2014} with default parameters $\beta_1 = 0.9$, $\beta_2 = 0.999$, and $\epsilon=10^{-8}$.
We use a learning rate of 0.001 and a exponential moving average (EMA) learning scheduler with decaying factor of 0.995.
We employ a gradient clipping of 100. The training curves for small ($L=0$), medium ($L=1$), and large ($L=2$) models are presented in \cref{fig:metrics} in the SI.
Models are trained for 200 epochs on \numrange{40}{80} NVIDIA A100 GPUs across \numrange{10}{20} nodes.
Training the medium-sized model took approx.\ 2,600 GPU hours. We find that \MLFF{} achieves an energy MAE of \SI{20}{\milli\eV\per atom} and a force MAE of \SI{45}{\milli\eV\per\Angstrom} for the medium model. After fine-turning with higher weights for energies for an additional 50 epochs, the small model is able to achieve an energy MAE of \SI{13}{\milli\eV\per atom} (see SI \ref{sec:trainingprotocol}).
\paragraph{Performance}
The speed of evaluation of the \MLFF{} model depends on the atomic density, hardware, floating point precision, size of model, \textit{etc.} (see section SI~\ref{sec:timings} for details), but a rough guide is that on a single NVIDIA A100 GPU with 80GB of RAM, it can do several nanoseconds per day for 1000 atoms. When run in parallel using domain decomposition, weak scaling at \SI{0.1}{\nano\second\per day} is perfect up to 32,000 atoms and 64 GPUs for a dense metallic alloy.
\paragraph{Training data}
The \MLFF{} model was trained on the \texttt{MPtrj} dataset which was compiled originally for CHGNet~\cite{deng_chgnet_2023}. This dataset consists of a large number of static calculations and structural optimization trajectories from the Materials Project (MP)~\cite{jain2013commentary}.
These include approx.\ $1.5$M configurations (roughly ten times the approx. $150$k unique MP structures), mainly small periodic unit cells (90\% under \num{70} atoms) describing inorganic crystals with some molecular components. The DFT calculations use the PBE exchange-correlation functional with Hubbard $U$ terms applied to some transition metal oxide systems, but no additional dispersion correction~\cite{MP_calc_details}.
Since the potential we fit calculates the energy based only on structural information, ideally we would like to use consistent electronic calculation parameters and the lowest energy electronic state for each configuration. One significant source of inconsistency is the application of Hubbard $U$, which is used in MP calculations only when \ch{O} or \ch{F} are present together with any of 8 transition metals (\ch{Co}, \ch{Cr}, \ch{Fe}, \ch{Mn}, \ch{Mo}, \ch{Ni}, \ch{V}, \ch{W})~\cite{MP_Hubbard_U}.
The application of $U$ leads to a shift in energy correlated with the value of $U$, \textit{i.e.} a few \unit{eV}, not explicitly accounted for in our fit. Thus, energies from calculations using those \num{8} elements with and without \ch{O} or \ch{F} are inconsistent (in the sense that the energy along a continuous deformation path that removes the \ch{O} or \ch{F} atoms from around these metals would be discontinuous).
The pre-trained CHGNet fit to MPtrj used energies corrected to account for the presence or absence of $U$~\cite{Jain_PRB_2011}.
In our fit, this shift only occurs between structures with different compositions and for any given composition the energies should be consistent.
As a result, we expect configurations that include local regions of these metals with very different \ch{O} or \ch{F} content, \textit{e.g.} an interface between a metal and an oxide, may be poorly described.
In addition, the current fitting database includes a variety of magnetic orders generated as part of a systematic search for the magnetic ground state~\cite{Horton_npjCM_2019}, chosen from the full database only based on calculation type (``GGA Static'' and ``GGA Structure Optimization'') and energy-difference criteria~\cite{deng_chgnet_2023}.
To quantify the effect of this additional and unaccounted-for degree of freedom, we classify the magnetic order associated with each calculation task into one of four categories: 1) no atomic magnetic moment listed, 2) moment converged to zero on all atoms, 3) converged to ferromagnetic order, and 4) converged to another magnetic order.
Of the approx. $150$k MP-IDs present, about $48$k have more than one magnetic order present in the fitting database.
In the vast majority of cases, this includes a calculation where the moments are {\em unknown} (\textit{i.e.} not recorded) and a single other magnetic order, and we can hope that they are actually consistent.
However, for \num{5186} MP-IDs we find multiple non-trivial magnetic orders.
To quantify the effect on the fitting quantities, we calculate the minimum energies of each magnetic order for each material, and
analyze the range of minima values seen for each material (distribution is plotted in SI \cref{fig:E_magmom_range}).
While the vast majority of materials have negligible variation, there are hundreds with variation \SI{>100}{meV/atom} (\textit{i.e.} an order of magnitude larger than the energy error on the validation set), and a few that vary by \SI{<0.5}{eV/atom}.
\paragraph{Long-range dispersion corrections}
Dispersion interactions, sometimes called van der Waals interactions, are crucial for describing the weak, long-range interactions between electrons.
Common approximations in DFT, such as PBE \cite{perdew1996generalized}, cannot capture such long-ranged interactions, motivating the use of additive non-local corrections, such as DFT-D3 \cite{grimme2010consistent} or rVV10 \cite{sabatini2013rVV10}.
Inclusion of a dispersion correction to DFT is necessary to describe the dynamics of liquid water \cite{lin2012water}, the geometries and binding energies of layered solids \cite{terentjev2018layered}, and stability of metal--organic frameworks \cite{formalik2018MOFvdW}, among many other examples.
Additive dispersion corrections typically employ a physical model for dispersion interactions with empirical parameters optimized to cut off the correction at interatomic distances where approximate DFT is reliable.
DFT-D3 is an interatomic potential which uses tabulated values of atomic polarizabilities to describe two-body and, optionally, three-body Axilrod--Teller \cite{axilrod1943threebody} dispersion interactions.
As \MLFF{} is trained to PBE energies, forces, and stresses, it inherits PBE's lack of long-range dispersion interactions.
An optional, additive DFT-D3 dispersion correction can be applied to \MLFF{}.
The PyTorch implementation of DFT-D3 used in this work is described in Ref. \cite{takamoto2022towards}.
The same parameters used in PBE-D3(BJ), i.e., DFT-D3 with a Becke-Johnson damping function \cite{grimme2011effect}, are used in the D3 correction to \MLFF{}.
\section*{Author contributions}
\textbf{Model training:} YC, PB, IB; \textbf{Data/Model analysis:} PB, YC, JR, NB, RE, MCK, ES; \textbf{MACE code:} IB, YC, SWN, DPK, PB, WCW, MA, SV, ES; \textbf{Application examples:} WJB (\ch{CsPbI3}, \cref{sec:pero_ino}); LLS (catalysis: \ch{In2O3}, \cref{sec:catalysis,sec;in2o3}); IB (a-C quenches, \cref{sec:C-mq}); ZEM (a-C graphitisation, \cref{sec:C-gra}); NK (Si interstitials, \cref{sec:si_si}); EVU, XRA, NON (aqueous interfaces, \cref{sec:water_results,sec:aqueous_interfaces}); YC (molten salts, \cref{sec:molten_salts}); CSc, VK, FDP, XRA (water and ice, \cref{sec:water_results,sec:water-bulk-nano}); SPN and AGS (\ch{LiNiO2}, \cref{sec:LNO}); SWN (lithiated graphite, \cref{sec:LiC}); AME (zeolites \cref{sec:zeo}); JJ (transition metal dichalcogenides, \cref{sec:tmd}); JHM (ethanol/water, \cref{sec:ethwat}, trialanine, \cref{sec:ala}); GL (a-Si \cref{sec:si_amo}); DK (carborane, \cref{sec:carborane}, ammonia-borane, \cref{sec:amonia_borane}); VC (S polymerisation, \cref{sec:S}); JR, JG and AAN (phonons, \cref{sec:phonons}); JR, REAG (materials discovery: formation energy, \cref{sec:formation}); KSJ (materials discovery: stoichiometric substitutions, \cref{sec:element-substitution}); ADK (materials discovery: highly-coordinated structures, \cref{sec:coordination}); ASR, YC and AME (MOFs, \cref{sec:mofs,sec:si_mofs}); MV (solvent mixtures, \cref{sec:mixtures}); DPK, ES, SMB (hydrogen combustion, \cref{sec:h_comb}); NKa, CSu (HOIPs, \cref{sec:pero_organic}); FF, JT (protein folding and stability \cref{sec:protein_folding}); PG, YW, TDS, JRK and CO (point and extended defects in BCC metals, \cref{sec:bcc_defects}); BXS, FB (molecule-surface interactions, \cref{sec:molecule_surf_interactions}) ;WCW (HEA, \cref{sec:timings}); EF, SD (catalysis: linear scaling relationships, \cref{sec:catalysis,sec:lsr}); HJ, HHH (catalysis: \ch{CO} oxidation on \ch{Cu}, \cref{sec:catalysis,sec:oxi}); SH, SD (catalysis: Pourbaix diagrams, \cref{sec:catalysis,sec:pour}); MCK (benchmarks: bulk and shear moduli, \cref{sec:bulk_moduli}); FDP (benchmarks: cohesive energies and lattice constants of solids, \cref{sec:lattice-energies}, atomization energies \cref{sec:atomization-energies}, and reaction barrier heights, \cref{sec:reaction-barrier}); TKS (\ch{Al2O3}, \cref{sec:alumina}, diatomics, \cref{sec:dia}); JPD (Arsenic random structure search, \cref{sec:rss}); IBM (high-pressure hydrogen, \cref{sec:h2}); IBM, CvdO (electrode-electrolyte interface / battery system, \cref{sec:battery}); JK, VS and KH (\ch{CeO2}, \cref{sec:ceria}); FZ (ionic liquids, \cref{sec:ionic_liquids}) \textbf{Supervision:} AB, ACF, AM, ASR, CH, CO, CPG, CSu, GC, HHH, JG, JK, JTM, KAP, KH, KR, MA, SD, SMB, TV, VLD, WH, WJB; \textbf{Drafted manuscript:} IB, NB, YC, GC, SD, HHH, MCK, JR, ASR, CSc, JTM; \textbf{Edited manuscript:} IB, NB, YC, GC, VLD, JG, REAG, JR, MCK, KAP, ASR, LLS, JTM, AM, CO, AME, WCW.
\section*{Acknowledgments}
Model training made use of resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under contract no. DE-AC02-05CH11231 using awards BES-ERCAP0023528 and BES-ERCAP0022838.
Part of this work was performed using the Cambridge Service for Data-Driven Discovery (CSD3), part of which is operated by the University of Cambridge Research Computing on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The DiRAC component of CSD3 was funded by BEIS capital funding via STFC capital grants ST/P002307/1 and ST/R002452/1 and STFC operations grant ST/R00689X/1. DiRAC is part of the National e-Infrastructure.
The work of YC, JR, ADK, MCK, MA and KAP was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under contract no. DE-AC02-05-CH11231 (Materials Project program KC23MP).
We could not have done this work without the DFT relaxation trajectories freely provided by the Materials Project and carefully curated into the MPtrj training set by Bowen Deng \cite{deng_chgnet_2023}.
JR acknowledges support from the German Academic Scholarship Foundation (Studienstiftung).
YC acknowledges financial support from UC Berkeley and Taiwan-UC Berkeley Fellowship from the Ministry of Education in Taiwan.
MCK acknowledges support by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-2146752. Any opinions, findings, and conclusions or recommendations expressed in this work are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
NB was supported by fundamental-research base-program funding from the U.S. Naval Research Laboratory.
ASR acknowledges support via a Miller Research Fellowship from the Miller Institute for Basic Research in Science, University of California, Berkeley.
LLS acknowledges support from the EPSRC Syntech CDT with grant reference EP/S024220/1.
AM and XRA acknowledge support from the European Union under the ``n-AQUA" European Research Council project (Grant no. 101071937). SWN, AB, TV acknowledge support from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Actions (Grant Agreement 945357) as part of the DESTINY PhD program. GC, CPG, TV, AB and SWN acknowledge support from the European Union's Horizon 2020 research and innovation program under Grant Agreement 957189 (BIG-MAP). VK acknowledges support from the Ernest Oppenheimer Early Career Fellowship and the Sydney Harvey Junior Research Fellowship, Churchill College, University of Cambridge. V.K. acknowledges computational support from the Swiss National Supercomputing Centre under project s1209.
ZEM acknowledges support from the EPSRC Centre for Doctoral Training in Theory and Modeling in Chemical Sciences (TMCS), under grant EP/L015722/1.
VLD acknowledges support from the John Fell OUP Research Fund.
CH and FZ acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in the framework of the priority program SPP 2363, “Utilization and Development of Machine Learning for Molecular Applications - Molecular Machine Learning” Project No. 497249646 as well as further funding though the DFG under Germany's Excellence Strategy - EXC 2075 - 390740016 and the Stuttgart Center for Simulation Science (SimTech).
AME's work used the DiRAC Extreme Scaling service (Tursa) at the University of Edinburgh, which is part of the STFC DiRAC HPC Facility (www.dirac.ac.uk) and scarf cluster (www.scarf.rl.ac.uk/) maintained by Scientific Computing Department STFC. AME's access to DiRAC resources was granted through a Director’s Discretionary Time allocation in 2023/24, under the auspices of the UKRI-funded DiRAC Federation Project. AME's work was also supported by Ada Lovelace centre at STFC (https://adalovelacecentre.ac.uk/), Physical Sciences Databases Infrastructure (https://psdi.ac.uk) and EPSRC under grants EP/W026775/1 and EP/V028537/1.
IB, RE and NK were supported by the Harding Distinguished Postgraduate Scholarship.
HJ gratefully acknowledges support from the Alexander-von-Humboldt (AvH) Foundation. HHH, JTM and KR acknowledge support from the German Research Foundation (DFG) through DFG CoE e-conversion EXC 2089/1.
FB acknowledges the Alexander von Humboldt Foundation for a Feodor Lynen Research Fellowship and the Isaac Newton Trust for an Early Career Fellowship. BXS acknowledges support from the EPSRC Doctoral Training Partnership (EP/T517847/1).
IB, DPK, XRA, WJB, FDP, RE, VK, DK, GL, NON, LLS, CSc, TKS, CvdO, EVU, WCW acknowledge access to CSD3 GPU resources through a University of Cambridge EPSRC Core Equipment Award (EP/X034712/1).
We acknowledge project/application support by the Max Planck Computing and Data Facility.
KH, JK and VS acknowledge the Swedish Research Council (Vetenskapsrådet,
project number 2021-06757) and the National Strategic
e-Science program eSSENCE for funding, as well as the Swedish National Infrastructure for Computing
(SNIC/NAISS) for providing computer resources used in this project.
SW, AB and TV acknowledge the Pioneer Center for Accelerating P2X Materials discovery (CAPeX), DNRF Grant number P3.
YW acknowledges support from the Shanghai Jiao Tong University.
WCW acknowledges support from the EPSRC (Grant EP/V062654/1).
CO acknowledges support from NSERC (Discovery Grant GR019381) and NFRF (Exploration Grant GR022937).
JG and AN would like to acknowledge the Gauss Centre for Supercomputing e.V. (https://www.gauss-centre.eu) for funding workflow-related developments by providing generous computing time on the GCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre (www.lrz.de) (Project pn73da).
SH and JJ acknowledge funding from EPSRC (EP/T001038/1, EP/S022953/1 ).
ADK acknowledges the Savio computational cluster resource provided by the Berkeley Research Computing program at the University of California, Berkeley (supported by the UC Berkeley Chancellor, Vice Chancellor for Research, and Chief Information Officer). ACF acknowledges funding from EU Graphene Flagship, ERC grants Hetero2D, GIPT, EU grants Graph-X, CHARM, EPSRC grants EP/K01711X/1, EP/K017144/1, EP/N010345/1, EP/L016087/1, EP/V000055/1, EP/X015742/1.
WB, CSu, and CG thank the US AFRL for partial funding of this project through grant FA8655-21-1-7010. JPD, JRK and GC acknowledge funding from the NOMAD Centre of Excellence (European Commission grant agreement ID 951786). PG and TDS acknowledge the support from the Cross-Disciplinary Program on Numerical Simulation of CEA, the French Alternative Energies and Atomic Energy Commission. PG and TDS used access to the HPC resources of IDRIS under the allocation A0120913455 attributed by GENCI.
GC is grateful to \'Agnes Borsz\'eki for help with graphics.
\bibliographystyle{ieeetr}
\bibliography{references}
\clearpage
\appendix
\section*{Supplementary Information}
\label{sec:si}
The following sections contain a diverse set of examples where the \MLFF\ foundation model is applied
to a variety of material and chemical systems with each subsection containing one application with one or more related examples.
\subsubsection*{Similarity statement}
Each subsection also contains a statement (both qualitative and quantitative) about the extent to
which the training data contains configurations similar to those relevant to the application in that section.
This should inform the reader about the degree of extrapolation inherent in the particular example. In order
to facilitate further scrutiny, we provide a data file for most applications that can be used in conjunction with
the \texttt{chemiscope} tool (at \href{https://chemiscope.org}{\url{chemiscope.org}}) to explore the chemical environments in the training data and
the application example and their relation to one another.
\subsubsection*{Performance summary}
Each subsection also contains a concise statement summarising the performance of the \MLFF{} model in the application.
\tableofcontents
\addtocontents{toc}{\protect\setcounter{tocdepth}{2}}
\clearpage
\section{Further Applications}
\subsection{Self-interstitials in silicon}\label{sec:si_si}
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_diinterstitial/figure.pdf}
\caption{Single- ($I_1$) and di-interstitial ($I_2$) defects in silicon. Left: Nudged elastic band paths between two metastable sites with a subset of images shown (black dots). Right: Diffusion coefficient against inverse temperature and Arrhenius laws with migration energy~$E_\mathrm{m}$. The barrier heights for $I_1\rightarrow I_1$ and $I_2\rightarrow I_2$ agree with the corresponding $E_\mathrm{m}$.}
\label{fig:diinterstitial}
\end{figure}
Di-interstitial silicon ($I_2$) constitutes a test case for the transferability of \MLFF{} to point defects in a periodic lattice. In the following, self-diffusion coefficients $D$ and interstitial migration energies $E_\mathrm{m}$ are obtained from MD simulations without a D3 dispersion correction.
Consistency tests with a $(64+2)$-atom silicon structure from \cite{bartok2018machine} relaxed with the PW91 functional are performed. Relaxing with \MLFF{} showed no change in energy at a force tolerance of \SI{0.05}{eV\per\Angstrom}.
The $I_2$ structure was generated by relaxing a $(216+2)$-atom
diamond structure with lattice constant \SI{5.4}{\Angstrom}.
After running NVT MD, the distance to the closest lattice site for each atom was plotted.
The interstitials propagate consistent with the accepted mechanism~\cite{yoshida2015defects}.
Isolating the trajectories of the point defect shows a characteristic
jump length corresponding to expected jumps between stable sites.
At temperatures below $\SI{700}{\kelvin}$,
the interstitials almost exclusively remain bound in a state corresponding to the ground state with $C_{1h}$ symmetry~\cite{richie2004complexity} and transition between symmetry-equivalent ground states. At higher temperatures, higher-energy states are sampled in which the interstitials separate into tetrahedral interstitial states, lying at a higher energy $\SI{0.25}{\eV}$ (\cref{fig:diinterstitial}).
We calculate $D$ from the mean-square-displacement (MSD) of all atoms in the unit cell~\cite{sahli2005ab}.
The fit to the Arrhenius law (\cref{fig:diinterstitial}) gives a prefactor of $D_0=\SI[separate-uncertainty = true]{1.3(4)e-4}{\centi\meter\squared\per\second}$ and $E_\mathrm{m}=\SI[separate-uncertainty = true]{0.26(2)}{\eV}$, in agreement with~\cite{posselt2005migration} and~\cite{du2005fast}.
The above calculations were repeated for a single interstitial ($I_1$) in a 64-atom cell.
The interstitial predominantly occupies the tetrahedral state and transitions between symmetry-equivalent states via the split $\langle 110\rangle$ state.
While these states are expected, the occupancy of each state shows larger deviations from those reported in~\cite{sahli2005ab}.
\Cref{fig:diinterstitial} shows a nudged elastic band (NEB) between two tetrahedral interstitial sites via a $\langle 110 \rangle$ split interstitial site, from which an energy barrier of $\SI{0.41}{\eV}$ is calculated.
Repeating the MD simulations
at several temperatures (\cref{fig:diinterstitial}) gives
$E_\mathrm{m}=\SI[separate-uncertainty = true]{0.52(5)}{\eV}$, in agreement with LDA~\cite{sahli2005ab}.
While the prefactor is strongly system-dependent~\cite{pichler2012intrinsic}, the migration energy is an intrinsic property of the energy landscape and may be compared with other calculations.
However, the finite system size and the interaction between interstitials results in the difference between the single- and di-interstitial migration energies.
\MLFF{} underestimates the melting point, this is shown by the fact that the $I_1$ structure is unstable at $\SI{1173}{\kelvin}$, whereas \cite{sahli2005ab} observe no melting up to $\SI{1473}{\kelvin}$ (with the LDA functional).
For $I_2$ and a 64-atom diamond structure, the instability is also at about $\SI{1200}{\kelvin}$.
For a 216-atom diamond structure, the phonon instability temperature lies at about $\SI{1500}{\kelvin}$, in agreement with PBE~\cite{dorner2018melting}.
\subsubsection*{Similarity statement}
The MP dataset contains 41 pure silicon structures, including the diamond structure but no self-interstitial defects.
\subsubsection*{Performance summary}
Silicon interstitials display the correct set of local minima, while their relative occupancy at finite temperature is not correct. The predicted activation energies for self-diffusion of the single and di-interstitials are consistent with previous force field and DFT calculations. The melting temperature of silicon too low by about \SI{20}{\%}.
\clearpage
\subsection{Amorphous silicon from melt--quench simulations}\label{sec:si_amo}
\begin{figure}[htbp!]
\centering
\begin{minipage}{0.38\textwidth}
\centering
\begin{subfigure}{1.0\textwidth}
\vspace{0em}
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figures_Si_quench/512-atom-silicon-quench-MACE-GAP-06-lowres.pdf}
\caption{Silicon quench (\SI{e12}{\kelvin\per\second}) protocol}
\label{fig:si-joinedA}
\end{subfigure}
\end{minipage}
\begin{minipage}{0.57\textwidth}
\centering
\begin{subfigure}{0.97\textwidth}
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figures_Si_quench/EXCESS-GAP18vsUMACE-02.pdf}
\caption{Computed excess enthalpies, evaluated with Si-GAP-18 (left) and \MLFF{} (right), as detailed in the text.}
\label{fig:si-joinedB}
\end{subfigure}
\vspace{0em}
\begin{subfigure}{0.485\textwidth}
\centering
\vspace{0.7em}
\includegraphics[width=\linewidth,keepaspectratio]{figures_Si_quench/RINGS-GAP18vsUMACE.pdf}
\caption{Ring count}
\label{fig:si-joinedC}
\end{subfigure}
\begin{subfigure}{0.485\textwidth}
\centering
\vspace{0em}
\includegraphics[width=\linewidth,keepaspectratio]{figures_Si_quench/defects.pdf}
\caption{Defect count}
\label{fig:si-joinedD}
\end{subfigure}
\end{minipage}
\caption{{\bf Amorphous silicon.} We characterise 512-atom structural models of liquid and amorphous silicon simulated using \MLFF{} and, for comparison, using the established Si-GAP-18 model \cite{bartok2018machine}. (a) Evolution of properties during \SI{1e12}{\kelvin\per\second} quench simulations. The top panel visualises the temperature reduction from \SI{1400}{\kelvin} to \SI{300}{\kelvin} over \SI{1100}{\pico\second}. Below, the simulation-cell volume and the average coordination number are presented as a function of simulation time. Inset images in the lower panel visualise the coordination number distribution in simulated liquid and amorphous Si structures. Red (blue) atoms represent coordination numbers higher (lower) than 4, respectively. White atoms are 4-fold coordinated. (b) Excess enthalpies ($\Delta E$) using Si-GAP-18 (left) and \MLFF{} (right) as calculated by relaxing snapshots at seveal points throughout the simulation. (c) Ring-size distribution in relaxed a-Si structures. (d) Coordination defect count in a-Si. Markers indicate values from individual runs and dark blue bars represent a reference 100,000-atom simulation using Si-GAP-18, taken from Ref.~\cite{deringer2021origins}.}
\end{figure}
Amorphous silicon (a-Si) is one of the most widely studied disordered materials. It has served as an example of both the physical and chemical insight afforded by machine-learned potentials \cite{bernstein_quantifying_2019,deringer2021origins} and as a challenging benchmark for the development of new methodology and potential models \cite{bartok2018machine,lysogorskiy2021performant,morrow_indirect_2022}.
In the present example, multiple 512-atom melt--quench simulations (\SI{1e12}{\kelvin\per\second}), describing the change from liquid (\SI{1400}{\kelvin}) to amorphous silicon (\SI{300}{\kelvin}) were performed to assess the qualitative behaviour and quantitative performance of \MLFF{} as compared to an existing, previously validated general-purpose ML potential for silicon~\cite{bartok2018machine}. The latter, known as ``Si-GAP-18'', has been shown to produce a-Si structures in good agreement with experimental observations \cite{deringer2018realistic,deringer2021origins}. Quench simulations were carried out using LAMMPS~\cite{plimpton1995fast} in an NPT ensemble. To test the variability from one run to another, especially given the rather small system size, three independent MD runs were performed using Si-GAP-18 and three with \MLFF{}.
Changes in temperature profiles, volume per atom, and average Si coordination number (determined using a \SI{2.85}{\Angstrom} bond-length cutoff) are presented in \cref{fig:si-joinedA}. When quenched from the liquid state, Si undergoes a vitrification transition that is accompanied by a sudden increase in volume and concomitantly by a decrease in the average atomic coordination number, as the highly-coordinated metallic liquid transforms into the mainly four-fold coordinated semiconducting a-Si. As \cref{fig:si-joinedA} indicates, the Si-GAP-18 and \MLFF{} models predict the occurrence of this transition at markedly different points in the simulation, and therefore predict different vitrification temperatures. That said, all silicon systems before and after the transition described by both models show comparable volumes per atom and an overall mostly fourfold-connected network of atoms, as expected for a-Si.
Further analysis included comparing excess energies ($\Delta E$) that were calculated relative to crystalline diamond-type silicon. These values were computed by taking structural snapshots throughout the quench and relaxing them either with Si-GAP-18 or \MLFF{}, similar to Ref.~\cite{deringer2018realistic}. As \cref{fig:si-joinedB} indicates, both Si-GAP-18 and \MLFF{} show very low energy fluctuations between individual runs for the Si-GAP-18 simulations, while \MLFF{} had some minor variability in each run. For \SI{300}{\kelvin}, Si-GAP-18 predicted enthalpies at around \SI{0.16}{eV/atom}, whereas the relaxation and energy evaluation with \MLFF{} predicts a notably lower value of \SI{0.11}{eV/atom}. Experimentally, the excess enthalpy of a-Si after deposition and annealing is around \SI{0.14}{eV\per atom}~\cite{roorda1991structural}. This enthalpy compares well with results from a previously reported 4,096-atom \SI{1.0e11}{\kelvin\per\second} quench using Si-GAP-18 ~\cite{deringer2018realistic}. Since our quench is one order of magnitude faster, higher enthalpies than \SI{0.14}{eV\per atom} can be expected, and there is likely more variability in the results due to the small system size. Nonetheless, the excess enthalpies predicted by \MLFF{} are qualitatively in the correct ballpark, viz.\ slightly above the corresponding crystalline phase.
As an additional measure of quality probing more subtle features of the structures, the distribution of shortest-path rings and coordination defects were investigated (Fig.~\ref{fig:si-joinedC}). The ring size distribution was assessed using \texttt{matscipy}\cite{grigorev2023matscipy}. Whereas diamond-type Si consists of 6-membered ($m=6$) rings only, a-Si includes some smaller and larger ones. Five- and seven-membered rings are still rather favourable, whereas smaller and larger ring sizes can be considered as topological defects. According to 100k-atom simulations in Ref.~\cite{deringer2021origins}, well-relaxed a-Si was predicted to contain on the order of \numrange{1.5}{2}\unit{\%} of defects, consistent with the 4,096-atom simulations of Ref.~\cite{deringer2018realistic}. Faster quenching leads to higher defect counts, as seen from the Si-GAP-18 results in Fig.~\ref{fig:si-joinedD}, whereas the overall defect count predicted by \MLFF{} is still notably higher than that of Si-GAP-18 at the same quench rate.
\subsubsection*{Similarity statement}
The MP dataset includes 41 different silicon-only structures, however, many of them are very high-density (high coordination number) or crystalline (all 4-fold). There were no cases of wide coordination number distribution, as seen in liquid Si -- however, we found 5 unique a-Si structures (with 100 atoms each) with a mix of slightly higher and lower coordination numbers, providing information about a-Si. Based on a UMAP analysis, the closest structures in the training set are mp-1244971, mp-1245242 and mp-1245041. To help with visualization, we provide \texttt{amorphous-silicon.json} on \url{chemiscope.org}.
\subsubsection*{Performance summary}
The model performs reasonably well for the description of the melt--quench process, leading to good-quality a-Si structures, albeit markedly underestimating the vitrification temperature and the excess enthalpy and overestimating the number of 5-fold coordinated atoms compared to an accurate ML potential.
\clearpage
\subsection{Amorphous carbon}\label{sec:c_amo}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_carbon/IB_ZEM_aC_wide_3-lowres.pdf}
\caption{\textbf{Amorphous carbon.} (a) Count of sp$^{3}$ (fourfold coordinated) carbon atoms in melt-quenched carbon structures as a function of density. The results obtained with the \MLFF{} model are compared to computational and experimental data compiled in Ref.~\cite{DeringerC2017} and references therein, as well as Refs.~\cite{Jana2019} and \cite{Qamar2023}. (b,c) 4,096-atom structures generated using the \MLFF{} potential within ASE. (d) 25 $\times$ 200 atom graphitisation simulations spanning relevant temperature and density ranges, similar to Ref.~\cite{Stenczel2023}. The structures in (b--d) are colour-coded according to coordination numbers as indicated in the legends. (e--f) Shortest-path ring size count for 4,096-atom structures as determined using \texttt{matscipy}\cite{grigorev2023matscipy}. (g--h) Coordination number count for 4,096-atom structures. The results in panels (e--h) are shown for a C-GAP-17-driven simulation (gray) and for a comparable simulation driven by the \MLFF{} potential (red).}
\label{fig:carbon-mp}
\end{figure}
\subsubsection{Melt--quench simulations}\label{sec:C-mq}
Carbon forms many different crystalline and amorphous modifications. The structural diversity of amorphous carbon (a-C), characterised by the simultaneous presence of three-fold coordinated (sp$^2$) and four-fold coordinated carbon atoms (sp$^3$), makes it a challenging system for both classical and ML force fields~\cite{deTomas2019,marchant_exploring_2023}. The correct description of its growth mechanism has been among the early successes of ML-driven materials modeling~\cite{Caro2018}.
We assess the accuracy of the \MLFF{} model in reproducing the structural complexity of amorphous phases by plotting the concentration of four-fold coordinated atoms (sp$^3$) as a function of density in \cref{fig:carbon-mp}. To generate amorphous structures with a given density with the \MLFF{} model, we perform melt-quench simulations. We start by melting diamond structures at a given density by running NVT simulations at \SI{8000}{\kelvin} for \SI{3}{\pico\second}. We then perform a fast quench, reducing the temperature from \SI{8000}{\kelvin} to \SI{300}{\kelvin} at a cooling rate of \SI{1000}{\kelvin\per\pico\second}. Finally, we optimize the geometry with LFBGS to obtain the final structure and determine the count of sp$^{3}$ atoms using a bond-length cutoff of \SI{1.85}{\Angstrom}. We observe in \cref{fig:carbon-mp} that the \MLFF{} model predictions reproduce the trend observed in both the DFT~\cite{Jana2019} and the experimental data extracted from~\cite{DeringerC2017}. We also see good agreement with results of quenches using the carbon ACE reported by Qamar et al.\ in Ref.~\cite{Qamar2023}, and using C-GAP-17 reported in Ref.~\cite{Jana2019}, noting that both potentials had been specifically trained on large carbon datasets.
\subsubsection{Graphitisation}\label{sec:C-gra}
\MLFF{} was used for two graphitisation runs containing 4096 atoms at low density (\SI{1.5}{\gram\per\cubic\centi\meter}) and high density (\SI{3.5}{\gram\per\cubic\centi\meter}). The low-density simulation was run at 2000 K and the high-density simulation was run at 4000 K. Additionally, $25 \times 200$ atom graphitisations spanning from 2000 to 4000~K and \num{1.5} to \SI{3.5}{\gram\per\cubic\centi\meter} were conducted. Both 4,096 atom structures were compared to structures generated using C-GAP-17 using the same protocol, which was also used recently for an ``on-the-fly'' generated GAP potential~\cite{Stenczel2023}. The protocol has two stages. The goal of the stage I is to prepare the starting configuration for the anneal in stage II, and it begins with a random structure with a hard-sphere constraint of $r_{min}$ \SI{\ge 1}{\Angstrom} and equilibrating it at\SI{9000}{\kelvin} for \SI{40}{\pico\second}, followed by cooling to \SI{5500}{\kelvin} over \SI{40}{\pico\second} and subsequent quenching to \SI{300}{\kelvin} over \SI{10}{\pico\second}. The structures are then held at \SI{300}{\kelvin} over \SI{50}{\pico\second} before being rapidly heated up to the annealing temperature over \SI{10}{\pico\second}. This concludes stage I. In stage II, the structure is annealed at \SI{2000}{\kelvin} or \SI{4000}{\kelvin} for \SI{350}{\pico\second} using a time step of \SI{1}{\femto\second}. We used C-GAP-17 to perform stage I because \MLFF{} was found to be unstable at \SI{9000}{K}.
Figures \ref{fig:carbon-mp}b--c show the final structures generated using \MLFF{}. Qualitatively, the low-density structure contains graphitic regions along with large pores of a few nm and sp chains, and the high-density structure contains highly ordered diamond-like regions. This is similar to structures generated with C-GAP-17 where similar features are also observed -- see, for example, a comprehensive survey by de Tomas et al. (Ref.~\cite{deTomas2019}). The same can be observed for a more comprehensive set of smaller structures (Fig.~\ref{fig:carbon-mp}d). Quantitatively, the structures generated by \MLFF{} in annealing simulations agree with the predictions of the established C-GAP-17 model in terms of overall trends. For more detailed insight, Fig.~\ref{fig:carbon-mp}e shows the shortest-path ring distribution for the low-density structure, indicating that \MLFF{} predicts a greater number of 6-membered rings and fewer large rings for the low-density structure compared to C-GAP-17 (thus indicating a higher degree of crystallinity in the \MLFF{} prediction) -- this might be correlated with the higher relative count of graphite versus amorphous-like structures in the training dataset, although further analysis is required. Figure \ref{fig:carbon-mp}f shows close agreement between \MLFF{} and C-GAP-17 in terms of ring count for the high density structure. Finally, Figures~\ref{fig:carbon-mp}g--h shows the coordination number for both 4,096 atom structures. \MLFF{} contains more sp$^{2}$ environments, whereas C-GAP-17 has more sp environments in the low-density structure.
\subsubsection*{Similarity statement}
The MP dataset contains 89 different all-carbon structures, most of which correspond to diamond and related stacking polytypes (lonsdaleite-like and more complex ones, all purely sp$^3$-bonded), as well as graphite in various forms. These structures include a range of mixed configurations with sp$^{2}$/sp$^{3}$ coexistence -- the latter is expected to be critical for a correct description of a-C. The dataset also contains a number of compressed and defective fullerene configurations, with one of those cells containing sp-, sp$^{2}$-, and sp$^{3}$-like environments, and a few hypothetical allotropes (notably ``T-carbon'' and a cubane-motif-based form, representing 3- and 4-membered shortest-path rings, respectively). In essence, the dataset does contain relevant carbon environments but does not contain a significant share of highly disordered carbon configurations. Based on UMAP analysis, we find that the closest structures in the training set are mp-568028 and mp-568806. We provide \verb|amorphous_carbon.json| to help visualize the interactive UMAP on \url{chemiscope.org}.
\subsubsection*{Performance summary}
The \MLFF{} model correctly captures the sp$^{3}$ content as a function of density in melt--quench simulations. Detailed analysis of long annealing simulations shows good agreement with a purpose-trained ML force field for the high-density structures and qualitative agreement for low-density a-C graphitisation.
\clearpage
\subsection{Ceria nanoparticles}\label{sec:ceria}
\begin{figure*}[htbp!]
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figures_CeO2_NPs/ceo2_np_matgpt_3-lowres.pdf}
\caption{Size-dependent formation energies for different shapes and stoichiometries of ceria nanoparticles (NPs). Left panel: Results calculated with the \MLFF{} model, which was not trained on any data from ceria surfaces or NPs. Right panel: Independent validation data from PBE+U calculations. The NP images are just illustrations; the optimized structures from MACE-MP and DFT are compared in more detail in \cref{fig:ceriacloseup}. The formation energy is calculated with respect to stoichiometric bulk \ch{CeO2} (and gas-phase \ch{O2} molecules as needed).
}
\label{fig:UU_ceO2}
\end{figure*}
Cerium oxide (ceria, \ch{CeO2}) is a reducible metal oxide with intriguing chemical and physical properties, and important technological applications especially for nanostructured ceria. Experiments in the literature have for example shown that the oxygen storage capacity (OSC) of ceria at the nanoscale is strongly shape- and size-dependent.
Behind the versatile usage of ceria lies one overriding feature, namely, its exceptional reduction-oxidation (redox) properties enabled by the duality of the cerium ion (\ce{Ce^4+ <-> Ce^3+}). It is generally a formidable task to try to mimic interactions, structure and energetics simultaneously for a compound like ceria without having access to explicit electrons. In earlier work\cite{broqvist2015reaxff}, we constructed a reactive interaction model using the ReaxFF modelling framework \cite{Senftle2016Reaxff} with the aim of handling stoichiometric and partially reduced ceria bulk, surfaces, and nanoparticles (NPs). The model was based on a training set of DFT calculations for a large number of ceria systems in various forms and configurations (bulk, clusters, surface systems; stoichiometric as well as reduced systems). With some exceptions that model performed very well. In the present study, instead, our forcefield is \MLFF{}, where the content of ceria in the training set is only bulk structures, namely exactly 18 bulk polymorphs (stoichiometric or partially reduced).
Here we assess the ability of the \MLFF{} model to describe small ceria nanoparticles of different shapes, sizes and reduction degrees without the training ever including any ceria NPs or surfaces. The optimized NP structures/shapes as well as their formation energies (with respect to stoichiometric bulk \ch{CeO2} and gas-phase \ch{O2} as needed)
will be assessed.
The left panel in \cref{fig:UU_ceO2} shows the \MLFF{} results for optimized particles up to 140 formula units: stoichiometric tetrahedra, stoichiometric truncated octahedra, and perfect octahedra (which are partially reduced by virtue of their shapes). The bottommost curve pertains to ``supercharged'' NPs, i.e. perfect octahedra that are decorated with oxygen molecules. The right panel shows the corresponding results from independent reference calculations at the DFT (PBE+U) level, taken from Refs. \cite{kullgren2013supercharged,broqvist2015reaxff}
The agreement between the energetics in the two panels in \cref{fig:UU_ceO2} is good overall, which is satisfying. However, we note that as far as structures are concerned, the \MLFF{} model (leftmost panel of \cref{fig:ceriacloseup})
is unable to distinguish between \ch{Ce^3+} and \ch{Ce^4+} ions, both of which should in fact be present in a partially reduced perfectly octahedral ceria nanoparticle. This deficiency of the \MLFF{} structure is evident from a comparison with the independent electronic PBE+U calculations in the rightmost panel of \cref{fig:UU_ceO2}, labelled ``DFT+U''. Such calculations involve a Hubbard correction which enforces a stronger and more adequate localization of electrons at \ch{Ce^3+} sites than what is achieved by standard PBE without U, which is the DFT method used in the Materials Project for ceria. The presence of both \ch{Ce^3+} and \ch{Ce^4+} ions in the PBE+U results is seen to lead to local relaxations of the nearest-neighbour oxygen ions around the Ce ions, resulting in symmetry breaking of the NP; see
for example the lack of symmetry with respect to the NP edges in the rightmost panel.
Neither the proper local relaxation nor the symmetry breaking, both seen in the PBE+U results, is captured by the \MLFF{} model. On the other hand, the middle panel of \cref{fig:ceriacloseup} shows our PBE-optimized results for the same NP.
The structural similarity between the \MLFF{} result
and the PBE-optimized nanoparticle is evident.
\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth,keepaspectratio]{figures_CeO2_NPs/avg_Ce-O_dist_blue_2-lowres.pdf}
\caption{Optimized structures of the perfect \ch{Ce44O80} octahedron with three different methods: MACE-MP, PBE, and PBE+U. The purpose of the figure is to highlight the "pattern of distances" rather than quantitative values. The large spheres are the Ce ions (regardless of charge), and the small red spheres are oxygens. The colour scheme indicates the optimized interatomic distances in the following way: for each Ce atom, the distances to its O neighbours in the coordination figure of nearest-neighbours
is measured and the average value is reflected in the colour of the sphere. Light blue indicates a short average Ce--O distance, dark blue indicates a long average Ce--O distance. The distance scale is shown in the colour bar to the right which covers the range from \SI{2.0}{\Angstrom} to \SI{2.5}{\Angstrom}.
}
\label{fig:ceriacloseup}
\end{figure*}
\subsubsection*{Similarity statement:} There are altogether 18 \ch{CeO2} and \ch{CeO_{2-x}} bulk structures present in the MP dataset. No examples of stoichiometric or reduced ceria surface structures or ceria nanoparticles are present in MP.
\subsubsection*{Performance summary}
Broadly correct prediction of the energy of nanoparticles as a function of size, including overoxidised particles, with respect to reference DFT results.
\clearpage
\subsection{Inorganic halide perovskite}\label{sec:pero_ino}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.6\textwidth,keepaspectratio]{figures_perovskites/lattice_constant_plot-lowres.pdf}
\caption{Variation of pseudo-cubic lattice parameters with temperature for \ch{CsPbI3}. The \MLFF{} model is compared to experimental data reported by Even and co-workers \cite{Even2018} and an atomic cluster expansion (ACE) model trained for this material\cite{ACESmall2023}. Inset shows an illustration of the 8640 atom cell used to calculate the lattice constant dependence on temperature.}
\label{fig:perovskite_simulation_cell}
\end{figure}
Halide perovskites have been shown to exhibit subtle phase transitions and long-range structural correlations. The \MLFF{} model has been applied to these systems by predicting phase transitions in the inorganic perovskite \ch{CsPbI3}. This material shows two solid-solid phase transitions between room temperature and \SI{600}{\kelvin}, both of which involve small rotations of the octahedral units and accompanying changes in pseudo-cubic lattice parameters\cite{Even2018}.
To analyze these transitions, we ran constant pressure simulations of an 8640 atom supercell (\cref{fig:perovskite_simulation_cell} inset) with a slowly varying temperature. \Cref{fig:perovskite_simulation_cell} shows the variation in pseudo-cubic lattice parameters with temperature, compared to experimental data. These data were obtained from a \SI{2.5}{ns} simulation during which the temperature was raised from \SI{325}{K} to \SI{525}{K}. The \MLFF{} model correctly predicts the qualitative nature of both phase transitions. There is a shift in both the transition temperatures and the average lattice constant, which has also been observed in other studies of these materials with DFT\cite{ACESmall2023}. It is also known that the choice of exchange-correlation functional has a large effect on transition temperatures for these materials\cite{fransson2023, Jinnouchi2019}.
\subsubsection*{Similarity statement}
There are 57 structures in the MP dataset containing some combination of \ch{Cs}, \ch{Pb} and \ch{I}, without other elements. Of these, 5 structures contain all three of these elements in different compositions spanning several phases of this material. Based on UMAP analysis, these 5 structures are close to the training dataset. In particular, the cubic and orthorhombic phases which are studied in this example are present. Several similar structures with \ch{Br} replacing \ch{I} are also in the training set.
\subsubsection*{Performance summary}
Both structural phase transitions and their transition temperatures well captured, and the 10\% discrepancy in the latter with respect to experimental values is likely due to the PBE functional.
\clearpage
\subsection{Hybrid Organic-Inorganic Perovskites (HOIPs)}\label{sec:pero_organic}
\begin{figure*}[htbp!]
\centering
\includegraphics[width=0.9\textwidth,keepaspectratio]{figures_hoip/hoip_2.pdf}
\caption{(a) Four emblematic HOIPs structures (b) force parity and (c) energy parity plots, with samples taken from MD trajectories and compared directly with their corresponding DFT values. }
\label{fig:hoip}
\end{figure*}
Two-dimensional hybrid organic-inorganic perovskites (HOIPs) have the advantages of enhanced stability and structural tunability over three-dimensional halide perovskites, which makes them candidates for promising applications in photoluminescence (PL), solar cells and light emitting diodes (LEDs) \cite{hoip}. However, HOIPs are difficult to examine using DFT because of their complicated unit cells formed when the organic cations separate the inorganic layers in (100) direction, giving the modified general formula \ch{{A'}_{m}A_{n-1}B_{n}X_{3n+1}}, corresponding to $n$ layers of the 3D-parent \ch{ABX3} structure, separated by a layer of A' organic cations that carry either a single charge ($m$ = 2) or two charges ($m$ = 1); the diversity of these systems can be seen in \cref{fig:hoip}a.
Using \MLFF{}, we have investigated a set of 159 experimentally synthesized 2D HOIPs from the Cambridge Structural Database (none are in the MP) with B = \ch{Pb} and X = \ch{Cl}, \ch{I}, and \ch{Br}. The organic cations (A' and A) are comprised of only the elements \ch{C}, \ch{H}, \ch{N} and \ch{O}, with either a +1 or +2 charge. MD simulations were performed within the NPT ensemble using the \MLFF{} with the D3 correction at a temperature of 300 K and pressure of 1 atmosphere. Out of 55 MD simulations, no bond-breaking or surface cleavage between the organic/inorganic layers occured. Samples were taken every \SI{1}{ps} and the errors in forces (RMSE = \SI{180}{meV/\Angstrom}) and energies (RMSE = \SI{22}{meV/atom}) are calculated relative to PBE+D3 (see \cref{fig:hoip}b and c).
\subsubsection*{Similarity statement}
In the MP training set, there are in total 398 structures with \ch{PbX} (X: \ch{Cl}, \ch{Br}, \ch{I}), and \num{1627} structures with organics made of the elements \ch{C}, \ch{H}, \ch{N} and \ch{O}. Based on UMAP analysis, we observe that some have similarities in environments to the MP training set, but only 22 structures have a mixture of \ch{PbX} and CHNO. From these, there are 14 3D HOIPs in which 12 of them have methylammuniumm (MA) as the organic cation, 3 0D HOIPs, and 4 cases of non-perovskite systems. In the MP, there is one 2D HOIP, which is the most similar to our dataset (mp-1194995), but this structure was not in the our dataset.
\subsubsection*{Performance summary}
Stable NPT MD at ambient conditions for all 159 2D hybrid perovskite materials.
\clearpage
\subsection{Protein Dynamics and Stability}\label{sec:protein_folding}
\begin{figure*}[htbp!]
\centering
\includegraphics[width=0.9\textwidth,keepaspectratio]{figures_protein_folding/mace-mp-chig.png}
\caption{Plot of the radius of gyration (\textup{\AA}) versus the time of simulation (ns) for the three simulations performed on Chignolin (PDB: 1UAO) and Chignolin mutant (PDB: 5AWL). In rectangular boxes initial structures (left boxes) and final structures (right boxes) of the 1 ns simulations are presented with and outline color corresponding to that of the lines of the plot. Orange is the simulation that starts from an unfolded structure of Chignolin, green corresponds to the simulation for the Chignolin mutant and blue corresponds to the simulation that starts from a partially folded structure of Chignolin. The red line corresponds to the computed radius of gyration for the PDB 1UAO structure.}
\label{fig:protein_folding_rg}
\end{figure*}
Understanding the dynamics of proteins is crucial for deciphering their biological function, and remains a core challenge of computational chemistry. Machine learning potentials have the capability of modelling non-covalent interactions, which are key components of secondary and tertiary structures of proteins, due to the quantum mechanical data on which they are trained on.
In this section we perform simulations on the well-known engineered Chignolin peptide (PDB: 1UAO), an artificial prototype for the protein folding phenomenon. This study diverges significantly from the majority of studies discussed in this manuscript as we are employing a machine-learning potential trained mainly on materials chemistry for a biological purpose.
We perform three separate simulations:
\begin{itemize}
\item A simulation starting from a partially folded structure of Chignolin.
\item A simulation starting from an unfolded structure of Chignolin.
\item A simulation of a mutant of Chignolin (PDB: 5AWL) starting from its PDB structure which contains 12 crystallographic waters.
\end{itemize}
Representative starting structures for the folded and unfolded Chignolin were obtained from Ref \cite{Wang2023}, specifically the first (folded) and last (unfolded) geometries from the 9543 conformations sampled by replica exchange molecular dynamics.
We protonate negatively charged residues (1UAO: ASP3, GLU5, GLY10 | 5AWL: ASP3, GLU5, TYR10) and remove the proton on GLY1 for 1UAO and on TYR1 for 5AWL. We are aware that the real stable state of Chignolin is non-neutral and solvated but this test is performed in a neutral environment as \MLFF{} was not trained with a charged based loss and testing stability of the simulation is our primary goal.
Simulations were performed at 300K for 1 ns at 1 fs time-step and a sampling frequency of 1 ps in the NVT ensemble using the Atomic Simulation Environment (ASE) package. The Langevin thermostat was employed with a friction coefficient of 10 ps$^{-1}$. \MLFF{} with the D3 dispersion correction was used for all simulations.
From Figure \ref{fig:protein_folding_rg} we can see that from its unfolded state, Chignolin is compacting throughout the course of the simulation (yellow line), while the folding state is maintained when starting from a partially folded state (blue line). The latter shows a sudden reduction in radius of gyration around 700 ps of simulation. This jump is related to the formation of a hydrogen bond between the OH of a phenolic residue and the oxygen of the backbone of Chignolin effectively making the compacting process converge and the radius of gyration closer to experiment.
Both simulations of Chignolin converge to the radius of gyration computed for the crystal structures 1UAO (red line).
Similarly, the Chignolin mutant shows stability across the simulation and maintains its folded structure (green line).
Figure \ref{fig:protein_folding_proton_transfer} depicts some physical and chemical phenomena happening throughout the simulations. Specifically, we observe continuous proton transfer between the neutralised COOH and NH$_{2}$ functional groups of GLY1 and GLY10 of Chignolin (left) and many proton transfers between various residues of the Chignolin mutant and the crystallographic waters (right). This shows the remarkable capability of MACE-MP-0 in modelling non-covalent interactions but more importantly in modelling reactivity in molecular dynamics simulations. We point out that these observed processes of proton transfer are not possible to observe with classical force-fields as they are generally not parametrised for such effects.
\begin{figure*}[htbp!]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\fbox{\includegraphics[width=\linewidth]{figures_protein_folding/zoom_1.png}}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\fbox{\includegraphics[width=\linewidth]{figures_protein_folding/zoom_2.png}}
\end{subfigure}
\caption{Depiction of representative protons transfers occurring during the 1 ns simulations. Hydrogens that are being transferred are highlighted in purple. The bond connectivity of the original starting structure is kept to highlight the transfer of a proton. The newly formed bonds are shown in dashed black cylinders. (Left) Proton transfer between COOH of GLY 10 and NH$_{2}$ of GLY 1 shown for the last frame of the simulation started from the partially folded Chignolin structure. (Right) Series of proton transfers between residues containing COOH, phenolic groups (-OH) and crystallographic H$_2$O for the last frame of the Chignolin mutant simulation.}
\label{fig:protein_folding_proton_transfer}
\end{figure*}
\subsubsection*{Similarity statement}
The MP dataset encompasses only 99 structures exclusively composed of the elements hydrogen (H), carbon (C), oxygen (O), and nitrogen (N). Based on the UMAP analysis, the atomic environments of Chignolin and Chignolin mutant are clustered similarly to those of the filtered 99 structures (see Figure \ref{fig:umap_protein_folding}), which could mean that the environments of the training set are fairly similar to those of the test structures . However, after manual inspection, we observe that the filtered structures only resemble the protein under study by some functional groups such as carboxylic acid (-COOH), amino groups (-NH\textsubscript{2}), aromatic and amide groups. Some of the most similar structures are \textit{e.g.} mp-998880, mp-1203308, mp-556151, mp-707289 and mp-1203544.
\begin{figure*}[htbp!]
\centering
\includegraphics[width=0.9\textwidth,keepaspectratio]{figures_protein_folding/umap_image.png}
\caption{Comparison of the atomic environments in the 99 filtered structure of the training data (blue dots) and in the Chignolin mutant structure (green dots) in the form of a UMAP plot. Insets of some local environments are shown for a few key functional groups of the protein (green circles) with corresponding similar clustered environments from the training set (blue circles)}
\label{fig:umap_protein_folding}
\end{figure*}
\subsubsection*{Performance summary}
All three simulations performed showed no specific un-physical phenomena. Moreover, for all systems the radius of gyration is maintained or converges to the value computed from the PDB structures. However, we note that residues that were charged in the PDB structures were protonated, thus the experimental folding (i.e. formation of secondary structure) cannot be fully recovered in this context.
\clearpage
\subsection{Hydrogen combustion}\label{sec:h_comb}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.8\textwidth,keepaspectratio]{figures_hydrogen_combustion/combustion_barchart.pdf}
\caption{Comparison of Heats of Reaction of Key Hydrogen Combustion Reactions with ReaxFF and Literature}
\label{fig:hydrogen_combustion_heats}
\end{figure}
Describing the complex reactivity in hydrogen combustion systems is a challenging task, often approached through thermodynamics via quantum mechanical (QM) calculations. However, accurately capturing kinetics in molecular dynamics simulations is hindered by the lack of a suitable transferable empirical force field, a difficulty compounded by the challenge of collecting experimental data in high-pressure, extreme explosion conditions. In this context, we compare our machine learning force field, \MLFF{}, against the reactive classical empirical force field ReaxFF\cite{agrawalla_reaxff}. ReaxFF model was created by fitting to an extensive QM dataset. The creation involved the identification of key reactions and components, and the collection of data on formation heats, reaction heats, and energy barriers, as well as computing bond stretching energies and valence angle distortion energies for all combinations of hydrogen and oxygen. In our study, the performance of \MLFF{} in describing these reactions is evaluated and compared to both ReaxFF and experimental values. It is important to note that \MLFF{} has not been specifically trained for hydrogen combustion reactions as the training data primarily comprises periodic materials and a smaller fraction of molecular crystals; see \cref{sec:hydrogen_combusion_similarity}.
As seen in \cref{fig:hydrogen_combustion_heats}, the ReaxFF and \MLFF{} models agree well with experimental values \cite{baulch_2005}, yielding an RMSE of \numlist{4.82;6.59;4.83}~\unit{kcal\per mol}. In particular, the large \MLFF{} model performs on par with ReaxFF. This is a surprising feat, considering that ReaxFF parameters were tuned using the reaction energies and transition states of relevant hydrogen oxidation reactions.
Packmol\cite{martinez2009packmol} is employed to randomly arrange a 1:1 fuel mixture comprising 128 \ch{H2} and 64 \ch{O2} molecules within cubic cells (side length, a = \numrange{25}{42}~\unit{\Angstrom}), yielding densities ranging from \numrange{0.05}{0.25}~\unit{\kilogram\per\cubic\deci\meter}. Employing NVT simulations for \SI{100}{\pico\second} with the \MLFF{} potential, we tracked the evolution of \ce{H2}/\ce{O2} mixtures. Reactivity analysis focused on water formation, identified via pairwise cutoffs derived from the first minima of the radial distribution function. The relationship between temperature/density variations in the fuel mixture and the water formation rate is depicted in \cref{fig:hydrogen_combustion}a and \cref{fig:hydrogen_combustion}c. We find qualitative agreement with the water formation curves of Ref.~\cite{agrawalla_reaxff}, with a max conversion of approx. \SI{80}{\%}.
\subsubsection*{Similarity statement} \label{sec:hydrogen_combusion_similarity}
We analyze the MPtrj training dataset for the key species in hydrogen combustion (\ce{O2}, \ce{H2}, \ce{H2O}, \ce{H2O2}, \ce{HO2}, and \ce{OH}). These species are present as minority units in other structures, appearing in 2277, 1310, 1342, 232, 21, and 0 structures, respectively. We find 21 molecular crystals composed exclusively of \ce{O2}, 17 for \ce{H2}, 11 for \ce{H2O}, 2 for \ce{H2O2}. There are only eight structures made up of multiple key reaction species. UMAP analysis reveals only 2 MPtrj structures (mp-684678 and mp-1181087) with high similarity to frames within MD simulations.
\subsubsection*{Performance summary}
Heats of reaction close to experimental values (typically within \SI{5}{kcal\per mol}) for 19 reactions. Chemically correct species produced during combustion, with final yield also consistent with reference methods.
\begin{figure}
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figures_hydrogen_combustion/multipanel-lowres.pdf}
\caption{Analysis and visualization of hydrogen combustion in MD simulations. Water formation as a function of elapsed time for a range of (a) densities and (b) temperatures. (c) Representative snapshot during MD simulations, with key species highlighted. Note, although \ce{O3} is not present in the particular simulation frame shown, it is occasionally observed in other simulations.}
\label{fig:hydrogen_combustion}
\end{figure}
\clearpage
\subsection{Sulfur polymerisation}\label{sec:S}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.65\textwidth,keepaspectratio]{figures_sulfur/sulfur-128-atoms-melting-with-pictures-lowres.pdf}
\caption{{\bf Elemental sulfur.} Evolution of the cluster size as a function of temperature in a \SI{20}{\pico\second} heat-up simulation with a linear increase in temperature applied over time. The pictures below are representative snapshots from the simulation, visualised in OVITO \cite{stukowski2009visualization}.}
\label{fig:sulfur}
\end{figure}
We ran a \SI{20}{\pico\second} NPT heat-up of a 128-atom structural model of elemental sulfur from \num{300} to \SI{1400}{\kelvin} at \SI{1}{bar}. At ambient pressure, experiments show that the molecular crystal formed of \ch{S8} rings melts at \SI{392}{\kelvin} and starts polymerizing at \SI{432}{\kelvin}~\cite{steudel2003elemental}, forming large chains that result in a 4-fold increase in viscosity of the liquid. We can qualitatively reproduce this melting and chain formation with \MLFF{}+D3 correction (\cref{fig:sulfur}). The simulated melting temperature does not exactly match the experiment, as expected for a very fast run with only 128 atoms -- however, the simulation does qualitatively reproduce the onset of de-polymerisation with increasing temperature, where large chains break down into smaller ones. For these reasons, we can say that \MLFF{}+D3 is at least qualitatively applicable to simulate the polymerisation of elemental sulfur starting from $\alpha$-rhombohedral crystalline \ch{S8} (with further work being required to test the quantitative agreement).
\subsubsection*{Similarity statement}
The MP dataset contains 31 structures that only contain the element sulfur. Based on UMAP analysis, we see that a large part of the atomic environments in the example system are similar to environments in the training data. The database contains geometry optimizations of sulfur crystals formed of rings with various sizes: \footnotesize6 (mp-7), 7 (mp-557559), 8 (mp-77), 9 (mp-556269), 10 (mp-557031), 11 (mp-561370), 12 (mp-558014), 13 (mp-583072), 14 (mp-561513), 10x6 \normalsize (there exists a sulfur crystal form comprised of \ch{S_{10}} and \ch{S6} rings, \footnotesize 557031), 18 (mp-555915) and 20 (mp-558964)\normalsize. It also contains crystals with planar strands of sulfur \footnotesize(mp-1179643)\normalsize, trigonal polymeric sulfur \footnotesize(mp-555760)\normalsize, and so-called fibrous sulfur (quenched polymeric liquid sulfur, \footnotesize mp-1196831)\normalsize, as well as isolated dimers \footnotesize(mp-1179639)\normalsize, trimers \footnotesize(mp-655141)\normalsize and single atoms \footnotesize(mp-1063988)\normalsize. It does not contain melt or polymeric liquid structures.
Based on the UMAP analysis, the closest (most relevant) structures in the training set are: \footnotesize mp-556269, mp-555915, mp-83, mp-557031, mp-557559, mp-666931\normalsize.
We provide \verb|sulfur.json| to help visualize the interactive UMAP on \url{chemiscope.org}.
\subsubsection*{Performance summary}
Qualitatively correct polymerisation, starting at a temperature that is estimated to be within 20\% of the experimental value.
\clearpage
\subsection{Zeolites}\label{sec:zeo}
\begin{figure*}[htbp!]
\centering\includegraphics[width=\linewidth,keepaspectratio]{figures_zeolites/fig_zeo-lowres.pdf}
\caption{a) MOR-\ch{Al} with acetone, instantaneous bond lengths between \ch{O}@BAS, \ch{H}@BAS and \ch{O}@acetone and \ch{H}@BAS upper panel and angles lower panel - for both \MLFF{} and D3 corrected a hydrogen bond is formed. b) same as a) but for formamide, one can see deprotonation events when \ce{O-H} bond value approaches \SI{1}{\Angstrom}, c) \ch{NH3} case with bonds same as a) \ch{NH3} quickly deprotonates the BAS} to form \ch{NH4^+} d) NEB~\cite{Lindgren2019,Makri2019} path for keto--enol tautomerism of acetone, the path is indicated by the \ch{H} atom traces, only final position of the rest of the atoms is shown
\label{fig:zeo}
\end{figure*}
Zeolites are mesoporous materials with an important role as heterogeneous catalysts in several industrial processes. In this section, we assess the suitability of \MLFF{} to model these materials. We chose two zeolites, Modernite and Zeolite Socony Mobil–5 - ZSM-5 or MOR and MFI by their International Zeolite Association names. We have investigated the dynamic stability of the zeolite frameworks themselves, MOR with \ch{NH3}, acetone and formamide inside one channel and MFI with water, cyclohexane and a mixture of \ch{N2} in one channel and \ch{CO2} in the other. Each zeolite was modified by adding a Br{\o}nstead acide site - BAS, \ch{Al}, and the compensating \ch{H} on the adjacent oxygen, see \cref{fig:zeo}. Another set of simulations was carried out on MOR-\ch{Al}, where in addition to acetone, we introduced 20 water molecules and 32 water molecules, and similarly for MFI-\ch{Al} with cyclohexene instead of acetone.
\MLFF{} correctly identified the adsorption sites, for ammonia, acetone and formamide and the structural motifs in agreement with DFT calculations from, \cite{Trachta2023} in NPT ensemble~\cite{melchionna1993,melchionna2000} simulations carried at \SI{300}{K} and \SI{400}{K} for \SI{125}{ps} each using ASE. Furthermore, in the case of ammonia \MLFF{} correctly predicts the formation of the \ch{NH4^+} and its stabilization around the BAS by the creation of hydrogen bonds with adjacent oxygen atoms, see \cref{fig:zeo} panel c. \MLFF{} also correctly reproduces the DFT findings that acetone does not deprotonate the BAS but forms hydrogen bonds, while formamide predominantly forms hydrogen bonds but deprotonates the BAS occasionally.
Additionally, for the system MOR-\ch{Al} with acetone and 20 water molecules, we have computed the barrier of the keto-acetone to enol-acetone conversion. \MLFF{} gave a barrier of \SI{2.11}{eV} and with D3 correction, \SI{2.20}{eV}, numbers are in good agreement with PBE calculations reported in \cite{Cucinotta2006}. The code used to generate the trajectories is available in the repo \cite{elena2023}.
\subsubsection*{Similarity statement}
For the system MOR-\ch{Al} the training set contains \num{145} structures that have \ch{Si}, \ch{O}, \ch{Al}, and \ch{H} elements on their own or along with other elements. Based on UMAP analysis, we see almost all atomic environments in the example system are similar to environments in the training data. Similar findings hold true for MFI-\ch{Al} with \num{145} structures matches.
One note, on the structures with adsorbates inside, they hold very low structure matches, for examples, acetone in MOR-\ch{Al}, matches only three structures, and formamide only one, and none very close to the studied zeolites. Adsorbants on their own match 1029 structures for formamide, \num{1892} for acetone and \num{3139} for \ch{NH3}. If we consider only \ch{Si}, \ch{O}, \ch{Al}, and \ch{H} elements we have only \num{11} similar structures for both zeolites considered and none is an exact match but they offer good representability of the local environments.
The closest (most relevant) structures in the training set are \ch{CO2} (mp-556034, mp-20066, mp-995224, mp-11725, mp-644607, mp-1102227, mp-1190685, mp-995198, mp-1190699, mp-1077906, mp-1077316, mp-729728). \ch{CO2} alone matches \num{4896} structures with \ch{C}, \ch{O} and alongside other elements.
We provide
\begin{itemize}
\item \verb|MOR-Al_FilterType.exclusive_SiOAlH_chemiscope_input.json|
\item \verb|MFI-Al_FilterType.exclusive_SiOAlH_chemiscope_input.json|
\item \verb|MFI-Al-H2O_FilterType.exclusive_SiAlOH_chemiscope_input.json|
\item \verb|MFI-Al-H2O-cyclohexene_FilterType.exclusive_SiAlOCH_chemiscope_input.json|
\item \verb|MFI-Al-H2O_FilterType.exclusive_SiAlOH_chemiscope_input.json|
\item \verb|MFI-Al-cyclohexene_FilterType.exclusive_SiAlOCH_chemiscope_input.json|
\item \verb|MOR-Al_FilterType.inclusive_SiOAlH_chemiscope_input.json|
\item \verb|MFI-Al_FilterType.inclusive_SiOAlH_chemiscope_input.json|
\end{itemize}
to help visualize the interactive UMAP on \url{chemiscope.org}.
\subsubsection*{Performance summary}
Correct prediction of binding sites, and qualitatively correct reaction behaviour for a range of structures and ligands, including good agreement of predicted reaction barrier with DFT.
\clearpage
\subsection{Open-circuit voltage of lithiated graphite}\label{sec:LiC}
\begin{figure*}[htbp!]
\includegraphics[width=\linewidth,keepaspectratio]{figures_graphite_li_ocv/ocv_li_graphite.pdf}
\caption{(a) The open-circuit voltage profile of lithium in graphite versus lithium metal, computed with \MLFF{} (blue) using a hybrid Grand Canonical Monte Carlo(GCMC)/Molecular Dynamics protocol, contrasted with an experimental reference (black) \cite{stevensdahn2001}. The variance of the simulated voltage is estimated over 100 bootstrapped samples of GCMC/MD trajectories. Representative lithium-graphite configurations are shown at $x = 0.3, 0.5, 1.0$. (b) The contributions to the free energy of formation of sampled lithium-graphite phases. (c) The density plot of individual Li/graphite structures sampled during a GCMC/MD simulation, showing the distribution of potential values over the configurational ensemble.}
\label{fig:OCV}
\end{figure*}
The open-circuit voltage profile of an electrode material is an example of a technologically relevant macroscopic observable that can be accessed through atomistic simulation. We apply \MLFF{} to model the thermodynamics of lithium-ion intercalation in graphite using hybrid grand canonical Monte Carlo/molecular dynamics (GCMC/MD).
Beginning from a \num{720}-atom cell of pristine graphite containing 10 graphene layers, we generate 40 parallel simulation trajectories of 30,000 steps each at a system temperature of \SI{300}{\kelvin}. In our GCMC/MD protocol, at every simulation step, we update the ionic positions according to Verlet dynamics. Every 5 steps, we generate a Monte Carlo proposal on the system volume, followed by a proposal on the system’s composition. For the volume proposal, we sample a perturbation of the unit cell: this is a set of 3 Euclidean vectors sampled component-wise from a normal distribution with a mean of zero and variance of \SI{0.01}{\Angstrom}. We add these random vectors to the existing lattice vectors and rescale the atomic positions to generate the proposed unit cell update. For the composition proposal, we randomly make one of three modifications to the population of lithium ions: insertion, deletion, or swapping. We choose one of these types of modification at random and then generate 5 candidate structures, each with either a single new lithium atom placed in a void in the host lattice (insertion), an existing lithium atom displaced into a void (swapping), or an existing lithium atom removed (deletion). As the lattice in our simulations is \textit{not} fixed, but evolves under molecular dynamics, we use Voronoi triangulation to identify void sites in the atomic lattice, excluding all sites subject to steric overlap according to the atomic radii. Once the set of composition candidates is generated, their energy is evaluated, and the lowest-energy candidate is used as the composition proposal. If a composition proposal is accepted, before proceeding to the next simulation step, we relax the ionic positions and the unit cell for up to ten ionic steps with a force tolerance of \SI{0.05}{eV\per\Angstrom} using the FIRE algorithm.
After sampling configurations with this protocol, we compute the open-circuit voltage as a function of lithium concentration over the sampled ensemble. Following previous work \cite{huang2019anode}, the open-circuit voltage is estimated as the negative of the free energy of formation per atom of the phase with composition \ch{Li_{x}C6} from reference states of graphite (\ch{C6}) and metallic BCC lithium, divided by the lithium concentration: $V(x) = - \Delta G_{f,Li_{x}C_{6}} / x$. The free energy per atom of the metallic lithium reference state is taken as the potential energy predicted by \MLFF{} of BCC lithium after structural optimization, neglecting entropy. To determine the free energies per atom of the \ch{C6} and \ch{Li_{x}C6} phases, we compute the internal energy \( U \) and Gibbs entropy \( S \) as Boltzmann averages over the sample distribution at concentration $x$: $G(x) = U(x) - TS(x)$, $U(x) = \sum_{j}(E_j \cdot p_j)$, $S(x) = - k_B \cdot \sum_{j}(p_j \cdot \ln(p_j))$, with probabilities
$p_j = e^{-\frac{E_j}{k_B T}} / {\sum_{j} e^{-\frac{E_j}{k_B T}}}$, where $E_j$ is the potential energy per atom predicted by \MLFF{} and $j$ indexes the set of all simulation frames with composition \ch{Li_{x}C6}.
\MLFF{} reproduces the experimentally known voltage profile of \ch{Li}/graphite with good quantitative accuracy (\cref{fig:OCV}a). In the regime of $x$ \num{ > 0.04}, the error is \SI{< 0.1}{V}, which reflects the combined error of the model as well as the limitations of the GCMC/MD protocol. This may be compared favorably with a recent purpose-developed model for lithium-graphite energetics \cite{babar2021graphite}, which reported \SI{< 0.1}{V} error for $x$\num{ > 0.0833} in an open-circuit voltage profile produced through GCMC; that model was trained on more than 8,000 system-specific DFT calculations, while \MLFF{} obtains comparable accuracy zero-shot. We note that at very low concentrations, our predicted voltage is higher than the experimental voltage by as much as a factor of 3, indicating overstabilization of dilute lithium. Since the lithium fraction appears in the denominator of the expression for the open-circuit voltage, very slight energetic deviations are magnified in this range; moreover, the free energy at low concentrations is dominated by the entropic contribution (\cref{fig:OCV}b), for which the limited sample size introduces uncertainty. Beyond this lowest-concentration regime, \MLFF{} provides good agreement with experiment.
\subsubsection*{Similarity statement}
There is a skew towards battery materials in MP. Given this, there are several \ce{Li-C} structures that are relevant to this application: mp-1210743 (\ch{Li2C}), mp-976060 (\ch{Li3C}), mp-1223102 (\ch{Li7C120}), mp-1378 (\ch{LiC}), mp-1021323 (\ch{LiC12}), mp-1232339 (\ch{LiC12}), mp-1001581 (\ch{LiC6}). There also exist 62 pure carbon structures including graphite (mp-48).
\subsubsection*{Performance summary}
Correct prediction of voltage as a function of Li concentration with reference to experimental curve.
\clearpage
\subsection{Jahn-Teller Distortions in \ch{LiNiO2}}\label{sec:LNO}
\begin{figure*}[htbp!]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_LNO/LNO_compare_PDFs_new-lowres.pdf}
\caption{a) Ground state structure of LNO. %
b) \ch{Ni-O} pair distribution functions (PDF) as a function of temperature computed by AIMD.\cite{GenreithSchrieverChemrxiv} c) PDFs computeed by \MLFF{}. Inserts indicate shapes of \ch{NiO6} octahedra (blue: distorted, yellow: undistorted).
}
\label{fig:LNO_PDFs}
\end{figure*}
\ch{LiNiO2} (LNO) is an important material in lithium-ion battery research, serving as a model for future high-performance cathode materials with reduced or no cobalt content. Structural and chemical degradation of these materials is a key challenge, so understanding their thermodynamic and dynamic properties is critical. In particular, ordered regions of Jahn-Teller distorted \ch{NiO6} octahedra may influence electron and Li-ion transport. Recent computational and experimental efforts have proposed a phase transition from a low-temperature distorted to a high-temperature undistorted phase \cite{GenreithSchrieverChemrxiv}.
We have investigated the ability of \MLFF{} to reproduce these results.
We used a 256-atom supercell, initialized from the established ground state structure with zigzag long-ranged ordering of the distorted long axes in \ch{NiO6}. We performed NPT-ensemble dynamics using \MLFF{} without a D3 correction, in line with the earlier AIMD simulations at the PBE+U level of theory.
\subsubsection*{Performance}
The \MLFF{} model demonstrates stable dynamics during a heating trajectory from \SI{0}{\kelvin} to \SI{1000}{\kelvin} with temperature increments of \SI{10}{\kelvin\per\pico\second}, time step \SI{1}{\femto\second}. Long simulations (at least \SI{500}{\pico\second}) could be run at \SI{1000}{\kelvin} without significant energy drift or noticeably unphysical behavior.
\Cref{fig:LNO_PDFs} shows that \MLFF{} correctly predicts a phase transition from Jahn-Teller-distorted octahedra (two PDF peaks) at very low temperatures to undistorted (one peak) at higher temperatures. \MLFF{} predicts the onset of the transition at around \SI{150}{\kelvin}, compared with \SI{250}{\kelvin} observed by AIMD simulations and VT-XRD~\cite{GenreithSchrieverChemrxiv}. This discrepancy is due either to deficiencies in the model or to very slight differences in the DFT methods being compared.
\subsubsection*{Similarity statement}
Battery materials are well represented in the MP database. 1393 structures in the training set contain \ch{Li}, \ch{Ni}, or \ch{O} atoms, 143 contain all three elements, and 23 have the exact formula \ch{LiNiO2} representing different crystal structures. However, many of these structures are obtained from geometry optimization and hence neglect the temperature dependence of the equilibrium geometry. \MLFF{} is a useful tool to explore this temperature dependence.
\subsubsection*{Performance summary}
Correctly captures emergence of Jahn-Teller distortions as a function of temperature, with an underestimation of the transition temperature compared with DFT.
\clearpage
\subsection{Point and extended defects in BCC metals}\label{sec:bcc_defects}
\begin{table}[h!]
\centering
\begin{tabular}{lrr}
\hline
& DFT & \MLFF{} \\
\hline
W & 3.185 & 3.188 \\
Mo & 3.163 & 3.170 \\
Nb & 3.322 & 3.313 \\
\hline
\end{tabular}
\caption{Lattice constants (in units of \AA{}) for W, Mo and Nb computed by variable-cell minimisation. DFT reference data is from Ref.~\cite{Hiremath2022} for W, Ref.~\cite{Smirnova2020} for Mo and Ref.~\cite{Yang2019_Nb} for Nb.}
\label{tab:BCC-lattice}
\end{table}
\begin{figure}[h!]
\centering
\includegraphics[width=1.0\textwidth,keepaspectratio]{figures_bcc_defects/elastic_constants.pdf}
\caption{Cubic elastic constants for W, Mo and Nb computed with linear stress-strain fits for DFT (orange; from Ref.~\cite{Hiremath2022} for W, Ref.~\cite{Smirnova2020} for Mo and Ref.~\cite{Yang2019_Nb} for Nb) and the \MLFF{} (blue).}
\label{fig:BCC-elastic}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=1.0\textwidth,keepaspectratio]{figures_bcc_defects/eos_W_Mo_Nb_medium.pdf}
\caption{Energy vs volume curves for W, Mo and Nb computed with the medium \MLFF{} model for FCC (purple) and BCC (green) crystals. The DFT reference data shown with the circular (BCC) and square (FCC) points is from Ref.~\cite{Cak2014}.}
\label{fig:BCC-eos}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=1.0\textwidth,keepaspectratio]{figures_bcc_defects/stacking_fault_energy_W_Mo_Nb_medium.pdf}
\caption{Generalised stacking fault profiles for $(112)[111]$ and $(110)[111]$ $\Gamma$-surfaces predicted by the \MLFF{} model, shown in blue. The DFT reference data, shown in orange, is from Ref.~\cite{Starikov2024}.}
\label{fig:BCC-gamma}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.85\textwidth,keepaspectratio]{figures_bcc_defects/vac_SIA_W_Mo_Nb_medium.pdf}
\caption{Vacancy and SIA formation energies for W, Mo and Nb computed with the MACE-MP-0 model (blue), and DFT reference data from Ref.~\cite{Ma2019} (orange).}
\label{fig:BCC-point-defects}
\end{figure}
This test explores bulk and extended defect properties of three prototypical BCC metals: W, Mo and Nb. An accurate description of these properties is essential to enable predictive modelling of mechanical responses to applied loads such as dislocation glide\cite{rodney2017ab}, dislocation climb through interaction with point defects, grain boundary motion, and the competition between cleavage and dislocation emission that underpins the brittle to ductile transition in fracture.
Across-the-board accuracy for bulk and defect properties in these systems is challenging even for bespoke machine learning potentials fit to carefully curated datasets~\cite{Goryaeva2021}.
Lattice and elastic constants are shown in Table~\ref{tab:BCC-lattice} and Fig.~\ref{fig:BCC-elastic}, respectively. Lattice constants are generally well reproduced, but there is a general softening of the elastic response.
In Fig.~\ref{fig:BCC-eos} we compare the energy-volume (E-V) curves predicted by the \MLFF{} model for BCC and FCC phases of the three metals with reference DFT data from Ref.~\cite{Cak2014}. The BCC cases show generally good agreement, consistent with the inclusion of some BCC data in the training set. There is room for improvement in the curvature of the E-V curves (critical for the elastic properties) for W and Mo, while Nb is well described. FCC energies are underestimated, which is unsurprising as no FCC structures for any of these elements are present in the training set. The curvature is approximately correct, also for FCC, giving reasonable predictions of the elastic response.
We next investigated point defect formation energies, including vacancies and self-interstitial atoms (SIAs). Calculations were performed in a $5\times5\times5$ supercell and were relaxed to a force tolerance of \SI{1.e-5}{eV\per\Angstrom}. For the SIAs, a short MD run was performed to escape the initial metastable configuration. The results, illustrated in Fig.~\ref{fig:BCC-point-defects} show good agreement with reference DFT data is from~\cite{Ma2019}: vacancy energies are predicted within ca. 20\% of the DFT value and SIA energies within 50\%. For all three elements the \MLFF{} predicts that the $\langle111\rangle$ dumbbell is the most stable SIA configuration, in agreement with DFT.
Subsequently, we looked at generalised stacking fault energy profiles for the $(112)$ and $(110)$ $\Gamma$-surfaces along the $[111]$ direction as shown in Fig.~\ref{fig:BCC-gamma}. These results were obtained with constrained minimisation where atoms were allowed to move only in the direction perpendicular to the cut surface and with a force tolerance of \SI{1.e-3}{eV\per\Angstrom}. The details of the method are explained in Ref.~\cite{Ventelon2010}. Nb is very well described by the MACE model, while there is an underestimate in the stacking fault energies in W and Mo by around a factor of two that explains the underestimates in the screw dislocation glide barriers discussed below.
Dislocations in BCC materials lie predominantly in the
$\langle111\rangle\{110\}$ and $\langle100\rangle\{010\}$ slip systems.
We investigated the characteristics of $\langle111\rangle$ screw and $\langle100\rangle$ edge dislocations by calculating the transition pathways and Peierls barriers using the nudged elastic band (NEB) method and the \MLFF{} potential
comparing against DFT~\cite{Dezerald2014} and hybrid QM/MM calculations~\cite{grigorev2023calculation,Swinburne2017} where the data is available.
The cells contained $\approx$1400 atoms for the [111] screw dislocation and $\approx$2200 for the [100] edge dislocation. Geometry optimisations to obtain starting configurations used the FIRE algorithm~\cite{bitzek2006structural} with a force tolerance of \SI{1e-6}{eV\per\Angstrom}. To create and analyze atomistic dislocation configurations we employed the \texttt{matscipy.dislocation} module~\cite{grigorev2023matscipy}.
The transition path calculation is performed with an adaptive ODE solver~\cite{Makri2019} following the approach of Refs.~\cite{grigorev2020hybrid, grigorev2023calculation}, using fifteen intermediate images with stopping force tolerance of \SI{0.025}{eV\per\Angstrom}.
Starting positions for the NEB relaxation were obtained by linear interpolation between initial and final configurations.
Figs.~\ref{fig:BCC-screw-glide} and \ref{fig:BCC-edge-glide} illustrate the NEB minimum energy path depicting the Peierls barriers for $\langle111\rangle$ screw and $\langle100\rangle$ edge dislocations in W, Mo and Nb. We compare to DFT results where they are available.
Insets within Fig.~\ref{fig:BCC-screw-glide} illustrate the dislocation core structures at the initial, intermediate, and final positions along the \MLFF{} minimum energy path.
Screw dislocations are known to be a sensitive probe of potentials, since the accuracy required is on the meV/atom level.
For Nb the dislocation core structure is correctly predicted as the easy core seen in DFT~\cite{Dezerald2014}, while for W and Mo the degenerate core is incorrectly predicted. Moreover, for W the \MLFF{} predicts the hard-core structure at the saddle point rather than the split core seen in DFT. The observed agreement with the DFT glide barrier is therefore somewhat fortuitous.
The barrier height is underestimated for Mo and Nb and overestimated for W compared to reference DFT results~\cite{Dezerald2014}. For the edge dislocation, where energy differences are larger, we find that the barrier height aligns well with QM/ML results in W~\cite{grigorev2023calculation}, while it is an overestimate in comparison with QM/MM results for Mo~\cite{Swinburne2017}. However, the presence of a minimum along the transition path results in a spurious stable dislocation configuration. In Mo the barrier is overestimated.
We anticipate the performance of \MLFF{} for all properties considered here would be substantially improved by enhanced accuracy in stress and a more precise agreement on the elastic constants, followed by fine-tuning on defect configurations where necessary.
\begin{figure}[h!]
\centering
\includegraphics[width=1.0\textwidth,keepaspectratio]{figures_bcc_defects/screw_dislocation_glide_W_Mo_Nb_medium.pdf}
\caption{Screw dislocation glide barriers for W, Mo and Nb. DFT data from~\cite{Dezerald2014}}
\label{fig:BCC-screw-glide}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=1.0\textwidth,keepaspectratio]{figures_bcc_defects/junction_dislocation_glide_W_Mo_Nb_medium.pdf}
\caption{Edge $\langle100\rangle$ dislocation glide barriers for W, Mo and Nb. QM/ML data for W from Ref.~\cite{grigorev2023calculation}, QM/MM data for Mo from Ref.~\cite{Swinburne2017} }
\label{fig:BCC-edge-glide}
\end{figure}
\subsubsection*{Similarity statement}
The MP dataset includes 3 elemental tungsten, 2 elemental molybdenum, and 2 elemental niobium structures. They are all crystalline without any defects. Based on UMAP analysis, we find that the closest structures in the training set are mp-8641 for tungsten, mp-8637 for molybdenum and mp-8636 for niobium. We provide \verb|W_input.json|, \verb|Mo_input.json| and \verb|Nb_input.json| to help visualize the interactive UMAP on \url{chemiscope.org}.
\subsubsection*{Performance summary}
Energy-volume curves for BCC are well reproduced, while for FCC structures they show a ca.~0.5 eV/atom shift in energy. Stacking fault profile energies for Nb are well reproduced, while for W and Mo they are underestimated by a factor of around two with respect to DFT. Relaxed point defect structures are reasonable in all cases, with formation energies within 50\% of reference DFT values. Peierls energy barrier profiles for dislocation glide are qualitatively correct, with the barrier correct for the edge- underestimated for the screw-dislocation. There is a small spurious local minimum near the top of the barrier for the edge dislocation.
\clearpage
\subsection{Alumina defects and bulk diffusion}\label{sec:alumina}
\begin{figure*}[htbp!]
\centering
\includegraphics[width=0.9\linewidth,keepaspectratio]{figures_tks/Al2O3_all-in-one_larger-font.pdf}
\caption{
\textbf{a-b}: Arrhenius plot of elemental diffusion in \ch{Al2O3} compared with experimental results from \cite{diffusion_in_alumina_2006};
\textbf{c:} Comparison of \MLFF{} NEB barrier paths (blue) and PBE single point evaluation of the \MLFF{} transition state (red) for elements in \ch{Al} or \ch{O} sites in \ch{Al2O3} moving to neighboring sites (dashed lines are just guide to the eye);
\textbf{d--e}: lowest energy NEB path for \ch{Y} (where \MLFF{} is accurate) and \ch{Co} (where there is a substantial discrepancy), with single point PBE evaluations indicated by red circles.
}
\label{fig:al2o3}
\end{figure*}
\subsubsection{Bulk diffusion}
One \ch{Al} and one \ch{O} vacancy were introduced into a 270-atom alumina supercell, and over \SI{2}{ns} (at \SI{2.5}{fs}) MD was used at temperatures between \numrange{2800}{3200}~\unit{\kelvin} to measure diffusivities of the two elements.
Diffusivities agree within one order of magnitude for \ch{Al}, and activation energies are underestimated for both compared to extrapolated experimental values~\cite{diffusion_in_alumina_2006}.
This demonstrates the long-timescale stability of the model, even at high temperatures and for long MD trajectories, but highlighting a shortcoming of the model for quantitative prediction of macroscopic observables.
\subsubsection{Elemental defects}
Elemental defects in \ch{Al2O3} were investigated by substituting \ch{Si}, \ch{S}, \ch{Ti}, \ch{V}, \ch{Cr}, \ch{Fe}, \ch{Co}, \ch{Ni}, \ch{Cu}, \ch{Y}, \ch{Ag}, and \ch{Pt} into lattice sites in a $2 \times 2 \times 1$ (120 atom) supercell and minimal energy paths to neighboring sites were obtained using NEB\cite{neb_aseneb, neb_jk_optimiser} starting from a linear interpolation. Paths were converged (max 50 steps, \SI{0.3}{eV\per\Angstrom} tolerance on projected forces) and the lowest energy one was tested with PBE single point evaluations using CASTEP \cite{clark2005first}.
Comparing \MLFF{} and PBE on \cref{fig:al2o3}c--e there are large discrepancies, with a total force component RMSE of \SI{0.33}{eV\per\Angstrom} across 139 structures evaluated.
A few NEB calculations failed catastrophically, this was found to be due to lack of sufficient repulsion between \ch{Al}-\ch{Ag} for distances under \SI{0.8}{\Angstrom}, and could be prevented by using an initial guess path with atoms not so close to each other.
\subsubsection*{Similarity statement}
There are 109 structures in the MP dataset containing exclusively \ch{Al} \& \ch{O}, pure \ch{Al2O3} appears as \verb#mp-1143# (used to generate supercells).
There is a total of 243 structures in the training set with \ch{Al} \& \ch{O} and exactly one of \ch{Si}, \ch{S}, \ch{Ti}, \ch{V}, \ch{Cr}, \ch{Fe}, \ch{Co}, \ch{Ni}, \ch{Cu}, \ch{Y}, \ch{Ag} or \ch{Pt}.
\subsubsection*{Performance summary}
Activation energies for self-diffusivity are underestimated compared to experimental values (extrapolated from lower temperatures). Dopant atom migration minimum energy paths are all stable, and are sometimes accurate (e.g. \ch{Yb}) and sometimes only qualitative (e.g. \ch{Co}) with respect to DFT single point reevaluations.
\clearpage
\subsection{Random structure search: Arsenic}\label{sec:rss}
\begin{figure*}[ht!]
\centering
\includegraphics[width=0.9\textwidth,keepaspectratio]{figures_arsenic_RSS/mace_mp_medium_As_RSS.pdf}
\caption{Densities of states of random structure search (RSS) minima for As at \num{0.1}, \num{10} and \SI{50}{\giga\pascal} obtained using \MLFF{} (black, squares) and DFT (brown, circles), with known As structures outlined in various colors (see legend). The bottom right panel shows how the binding energy of 7 (unrelaxed) randomly generated structures (colored lines), and one pathological structure found during RSS (black line and inset) vary as the structures are compressed uniformly with fixed fractional atomic coordinates. The \ce{As-As} dimer energy and the RDF for the \SI{0.1}{\giga Pa} \ch{As} RSS structures are shown for comparison. }
\label{fig:As_RSS}
\end{figure*}
\emph{Ab initio} Random Structure Searching (AIRSS) \cite {pickard2011ab} is a simple, yet highly successful, approach for discovering new materials computationally. Multiple candidate structures are generated randomly, subject to physically motivated constraints, and then relaxed to local enthalpy minima using ab-inito methods such as DFT. There is great interest in accelerating structure prediction by using surrogate models \cite{bernstein2019novo, podryabinkin2019accelerating, pickard2022ephemeral, merchant_scaling_2023}, such as ML potentials, to perform the initial structural relaxations. Here we test the suitability of \MLFF{} (without the D3 correction) for this task by searching for structures of Arsenic at \num{0.1}, \num{10} and \SI{50}{\giga\pascal}. The exceptional structural variety encountered during RSS probes the robustness of the model in an extremely extrapolative regime; there are only six \ch{As} structures and no high-pressure data in the training set.
At each pressure, 2000 ($\times 100\, n$) random structures were generated using $n=2-6$ atoms per primitive unit cell, 2-4 randomly chosen symmetry operations, minimum distance constraints of \SI{2}{\Angstrom} and a volume per atom of \num{15}--\SI{40}{\cubic\Angstrom}. The structures were then relaxed with \MLFF{} (ASE, force tolerance of \SI{1e-3}{eV\per\Angstrom}) and CASTEP \cite{clark2005first}; PBE exchange-correlation functional \cite{perdew1996generalized}, \SI{400}{eV} cutoff energy, k-point spacing of $2\pi\times$ \SI{0.05}{\per\Angstrom} and Vanderbilt ultrasoft pseudopotentials \cite{vanderbilt1990soft} with a force tolerance of \SI{0.05}{eV\per\Angstrom} and stress tolerance of \SI{0.1}{\giga\pascal}. With these settings 96\% and 99\% (at \num{0.1} and \SI{10}{\giga\pascal}) of structural relaxations were successful with DFT and \MLFF{} respectively. The distributions of relaxed structures are depicted in \cref{fig:As_RSS} and the known structures listed in \cref{tab:As_RSS} are highlighted with colored symbols. At \num{0.1} and \SI{10}{\giga\pascal} the energy and volume distributions are visually similar (note the systematic shift in volume distribution) with the relative energy differences between the highlighted structures, particularly the low-energy ones, generally being small compared to the overall range. Inspection of the structures at \SI{0.1}{\giga\pascal} reveals that similar 3-fold coordinated 3D, layered, and 1D structures are found with both \MLFF{} and DFT. Furthermore, we note that the known structures listed in \cref{tab:As_RSS} were all found using \MLFF{}, including a simplified packing of the \ch{As4} tetrahedra found in yellow As. The lowest energy structures found at \SI{0.1}{\giga\pascal} contain \ch{As8} cubes. The cluster of dense structures found about \SI{0.3}{eV/atom} above the minimum at \SI{10}{\giga\pascal} contain 6-fold coordinated As in a 3D kagome-like lattice. Relaxing these structures with DFT preserves this connectivity but causes the the volume to increase to about \SI{19}{\cubic\Angstrom\per atom}.
At \SI{50}{\giga\pascal}, the similarity between the DFT and \MLFF{} results decreases markedly, with 7\% of relaxations failing, compared to 1\% at \num{0.1} and \SI{10}{\giga\pascal}. These failures are caused by ``holes'' in the \MLFF{} potential energy surface, where exceptionally dense, highly coordinated structures are predicted to be overly stable. An example of such a structure is shown in the inset of the bottom right of \cref{fig:As_RSS}. The repulsion seen in the As--As dimer curve indicates that these holes are caused by higher body-order terms in an extrapolative regime - holes occurs at shorter \ce{As-As} distances than occur in the radial distribution function of the \SI{0.1}{\giga\pascal} RSS results. We note that such holes are not typically an issue during ambient pressure MD, due to the large energy barriers seen in \cref{fig:As_RSS}, and can likely be fixed by including some high pressure data in the training set.
\begin{table}[htbp!]
\centering
\caption{Summary of known Arsenic structures. The white \ch{P} structure type is used as a proxy for yellow \ch{As} as the structure is unknown \cite{hart2018one}.}
\resizebox{\textwidth}{!}{
\begin{tabular}{cccccc}
\\ \hline
\hline
\textbf{} Structure & Pressure & In training set? & Space Group & Z & Found with \SI{0.1}{\giga\pascal} \MLFF{}? \\
\hline
A7, grey As \cite{schiferl1969crystal} & ambient & yes & $R\bar{3}m$ & 2 & yes \\
black P \cite{smith1975structures} & ambient & yes & $Cmce$ & 4 & yes \\
white P & ambient & no & $P\bar{1}$ & 24 & \ch{As4} tetrahedra found \\
simple cubic \cite{silas2008density} & \num{27}--\SI{57}{\giga\pascal} & yes & $Pm\bar{3}m$ & 1 & yes \\
bcc \cite{silas2008density} & \SI{\geq 110}{\giga\pascal} & no & $Im\bar{3}m$ & 1 & yes \\ \hline
\end{tabular}}
\label{tab:As_RSS}
\end{table}
\subsubsection*{Similarity statement}
The MP dataset set contains only six pure As structures. Of these, grey arsenic ($R\bar{3}m$) \cite{schiferl1969crystal} and the orthorhombic allotrope ($Cmce$, isostructural with black phosphorus) \cite{smith1975structures} have both been observed at ambient conditions whilst the simple cubic structure (Pm$\bar{3}$m) \cite{silas2008density} is stable at moderate pressure between \num{27}-\SI{57}{\giga\pascal}. The remaining three structures are \SI{> 0.4}{eV/atom} above grey arsenic. There are an additional 3857 unique structures that contain As and other elements. Within these optimised structures there are a total of 22047 As environments of which 1606, 1537, 534 and 12 are 1, 2, 3 and 4-fold coordinated by neighbouring As respectively (\SI{2.7}{\Angstrom} cutoff). Many of the 3-fold As environments are found in \ch{AsX} compounds where X is a group I or II element and the As atoms are arranged in local clusters. There is one As atom which is bonded to 4 neighboring As atoms (\ch{Cs7(InAs2)3}, mp-1203378), one structure containing isolated \ch{As4} tetrahedra (\ch{AsO3}, mp-1215144) and two structures containing connected \ch{As4} tetrahedra (\ch{Re4As5S4} mp-1209063 and \ch{Re4As6S3} mp-1219545).
\subsubsection*{Performance summary}
All expected low enthalpy stable structures found. Unphysically low energy structures uncovered at high pressures and small internuclear distances.
\clearpage
\subsection{Properties of bulk and nanoconfined water}\label{sec:water-bulk-nano}
See main text \cref{sec:water_results} for results and discussion.
\subsubsection*{Similarity statement}
The MP dataset contains 21 structures composed of \ch{O} and \ch{H} elements and 7769 structures that have \ch{O} and \ch{H} elements alone or together with other elements. Based on UMAP analysis, we see that some atomic environments in the example system are similar to environments in the training data. For instance, bulk water and ice comprise typical molecular environments (\textit{e.g.}, the environment of atom 20 in structure 13 of \verb|water_exclusive_OH_chemiscope_input.csv|) but also environments of hydrogen peroxide (\textit{e.g.}, the environment of atom 1 in structure 14 of \verb|ice_exclusive_OH_chemiscope_input.csv|). Despite being two dimensional, the superionic phase also comprises distinct environments mimicking those of water molecules (\textit{e.g.}, the environment of atom 3 in structure 20 of\\ \verb|superionic_exclusive_OH_chemiscope_input.csv|) and dissociated environments mimicking those of hydrogen peroxide (the environment of atom 12 structure 12 of\\ \verb|superionic_exclusive_OH_chemiscope_input.csv|). The environments farthest from the MP dataset are the monolayer oxygen environments surrounded by a (flat) hexagon of 6 other oxygen atoms.
\subsubsection*{Performance summary}
The \MLFF{} model demonstrates stability and reliable performance in conducting simulations across diverse conditions for both bulk and confined water. It maintains stability in NVT simulations at experimental densities and temperatures for bulk water, ice Ih, and reactive proton defects (\ch{OH^-} and \ch{H3O^+}). The model describes extensive proton transfer in nanoconfined water at \SI{4}{\giga\pascal} and \SI{600}{K}, in good agreement with reference methods.
\clearpage
\subsection{Ethanol-water density-composition curves}\label{sec:ethwat}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.6\linewidth,keepaspectratio]{figures_ethanol_water/ethanol_water_dens_curve.pdf}
\caption{Ethanol--water density curves obtained by NPT MD using \MLFF{} in double precision, compared to experimental data taken from Ref.~\cite{southard2018perry}. }
\label{fig:ethanol_water_density}
\end{figure}
In this section, we investigate the ability of the \MLFF{} model to describe mixtures of molecular liquids. In particular, we study the density-composition curve for a range of volume fractions of ethanol in water. Initial configurations were generated with Packmol~\cite{martinez2009packmol}, with 120 molecules per box, and the initial box vectors were set to be slightly below the experimental density for each composition. Initial structures were minimised to a tolerance of 0.1 eV/\AA{} with the L-BFGS algorithm. Trajectories were generated in the NPT ensemble using ASE, including a D3 dispersion correction with the Becke-Johnson damping function. Final densities were computed as the averaged of the final 1000 snapshots from the simulation, once the density had converged.
\subsubsection*{Similarity statement}
The MP dataset contains 37 structures that contain only the elements \ch{C}, \ch{H}, and \ch{O}. Based on UMAP analysis, we observe that almost all atomic environments in the example system are similar to environments in the training set. On closer inspection, we find that the most similar environments to the majority of the example configurations are clusters primarily containing water, hydroxide and atomic hydrogen and oxygen, with a few examples containing small hydrocarbon-type fragments.
\subsubsection*{Performance summary}
For low ethanol volume fractions, \MLFF{} predicts the density deviation from the linear behaviour (with a 4\% error in the absolute density, not uncommon for a GGA DFT functional for molecular fluids), however simulations of pure ethanol result in the boiling off of the liquid at ambient conditions, resulting from the underestimation of the boiling point.
\clearpage
\subsection{Solvent mixtures}\label{sec:mixtures}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_solvent_mixtures/solventLine1ns.png}
\put(-455,105){Water - Benzene}
\put(-330,105){Water - Heptane}
\put(-210.5,105){Water - Ethanol}
\put(-90.7,105){Ethanol - Benzene}
\caption{Snapshots of solvent mixtures after \SI{1}{\nano\second} of NVT MD. The solvents shown are benzene (blue), heptane (green), ethanol (red), and water (grey). Axis orientations in the figures were chosen to highlight the phase separation in these systems.}
\label{fig:mixtures}
\end{figure}
Modeling solvent mixtures requires an accurate description of intermolecular forces within highly disordered systems. To investigate the performance of \MLFF{} in this setting, MD simulations were performed for four mixtures of solvents of varying polarity. The investigated systems are water-benzene, water-heptane, water-ethanol, and benzene-ethanol. Simulations were performed at \SI{300}{\kelvin} in the NVT ensemble via the ASE interface. A time step of \SI{1}{\femto\second} and a friction constant of \SI{0.001}{\per\femto\second} were used. In the case of immiscible solvents, a mixture of equal volumes of both solvents with their corresponding densities was assumed. In the case of miscible solvents, the experimental density of the mixtures was used. All systems were initialized with a uniform random mixture of both solvents using the \texttt{packmol} code. \cite{martinez2009packmol}
\Cref{fig:mixtures} shows the states of all systems after \SI{1}{\nano\second}. Notably, mixtures of water with apolar solvents (heptane and benzene) quickly form separate phases, whereas the ethanol-water system remains mixed on the timescale of the simulation. This is in good agreement with experiment. However, the mixture of ethanol and benzene (which are fully miscible at \SI{300}{\kelvin}) also shows phase separation.
\subsubsection*{Similarity statement}
The MP dataset contains 37 structures composed exclusively of \ch{C}, \ch{H}, and \ch{O}, and 1902 structures that contain \ch{C}, \ch{H}, and \ch{O} along with other elements. Regarding the specific molecules, several ice structures but none of the other molecules are included as pure compounds. The closest to benzene (with the ratio of \ch{C}:\ch{H} 1:1) is mp-995197 containing chains of dimethylbenzenes with methyl-methyl bridges. The UMAP analysis shows that many atomic environments from our structures have similar environments in the training data. However, no liquid configurations are included in the MP. We provide two files to visualize the interactive UMAP on \url{chemiscope.org}. \verb|solvents_mixtures_CHO.json| contains structures exclusively containing \ch{C},\ch{H}, and \ch{O}. \verb|solvents_mixtures_CHOplus.json| includes structures containing \ch{C}, \ch{H}, and \ch{O} along with other elements.
\subsubsection*{Performance summary}
Miscibility of three out of four mixtures correctly predicted, ethanol and benzene incorrectly showed phase separation.
\clearpage
\subsection{Aqueous interfaces}\label{sec:aqueous_interfaces}
\begin{figure}[ht!]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_aqueous/waterdensity-lowres.pdf}
\caption{\textbf{Water structure and density at various interfaces}. Representative sections of the simulation cells with the equilibrium water density profile below them. The system depicted are water on silicon dioxide (\ch{SiO2}), titanium dioxide (\ch{TiO2}), copper (\ch{Cu}), between two layers of molybdenum disulfide (\ch{MoS2}), and on sodium chloride (\ch{NaCl}).}
\label{fig: aqueous_density}
\end{figure}
Simulating complex systems, such as solid-liquid interfaces, is a difficult endeavor, as the potential must simultaneously describe the two materials and their interface. We tested the effectiveness of \MLFF{} on a wide range of aqueous interfaces, from oxides and metals to confinement.
NVT MD simulations were performed on a variety of surfaces at a temperature of \SI{330}{K}. The average density of water above the surface is shown in \cref{fig: aqueous_density}.
\ch{SiO2} and \ch{TiO2} were two notable oxide systems in which dissociative and molecular adsorption was observed, respectively. Deprotonation of water was expected on the surface of silicon dioxide, which is evidenced by the shoulder in the water density plot. These figures show that the interfacial water property is accurately reproduced; however, the liquid phase is overstructured, which is a common characteristic of the PBE functional \cite{gillan2016perspective} used in the Materials Project.
Water in confinement was also investigated within \ch{MoS2} slit pores. The simulation captures the pronounced stratification characteristic of the aqueous phase perpendicular to the two-dimensional layers. This was also observed for water confined between graphene sheets and boron nitride nanotube. In particular, in \cref{fig: aqueous_density} we show sharply defined interfacial water layers between the \ch{MoS2} sheets. Upon the addition of interlayer spacing (not shown), we also capture additionally smoother intermediate layers, noting that with more layers, we lose the sharp peaks at the surfaces.
Finally, the \ch{NaCl} (001) surface in contact with water was simulated. The system comprised a ($3\times4$) \ch{NaCl}(001) supercell containing \num{24} atoms, with 3 \ch{NaCl} layers and a unit cell lattice constant of \SI{5.72}{\Angstrom} on top of which were \num{89} water molecules. A subsequent \SI{25}{\Angstrom} of vacuum was added between the adsorbed water layer and the lower layer of the surface.
The layered structure of the water as previously observed in \textit{ab initio} PBE simulations in Ref. \cite{liu2008density} is captured by the \MLFF{} model, with the positions of the density minima and maxima qualitatively agreeing with the PBE simulations.
\begin{figure}[ht!]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_aqueous/nacl-lowres.pdf}
\caption{\textbf{Dissolution of NaCl in water}. (a) Density profile of water in contact with \ch{NaCl}(001) surface, with a representative snapshot from the simulation showing no dissolution events from the pristine surface. (b) Evolution of crystal size of $4\times4\times4$ \ch{NaCl} nanocrystal in water over time, comparing the \MLFF{} (blue line) with an ML model explicitly trained to capture \ch{NaCl} dissolution (black dashes) ~\cite{oneill2022crumbling}. Representative snapshots showing the dissolution progress of the crystal are shown above the plot.}
\label{fig: NaCl_dissolution}
\end{figure}
Simulation of dissolution processes is another challenge for the \MLFF{} model. It must be able to describe the very different chemical environments of the bulk crystal surrounded by water going through the stages of ions detaching from the crystal to fully solvated ions in solution. In \cref{fig: NaCl_dissolution}, we compare the \MLFF{} model in NVT simulations of the pristine \ch{NaCl} (001) interface in contact with water and a \ch{NaCl} nanocrystal surrounded by water at \SI{400}{\kelvin}.
The nanocrystal system simulated comprised a $4\times4\times4$ \ch{NaCl} nanocrystal comprising 32 ions, with lattice constant \SI{5.72}{\Angstrom} surrounded by 625 water molecules, giving a final concentration when dissolved of \SI{2.84}{mol/kg}.
As expected, for a pristine \ch{NaCl} surface, the model predicts no dissolution events on the time scale of the simulation.
Meanwhile, for the nanocrystal surrounded by water, the model captures a dissolution mechanism similar to that reported by Ref. ~\cite{oneill2022crumbling} with an ML model trained specifically to capture \ch{NaCl} dissolution at revPBE-D3 level of theory. The dissolution proceeds via a crumbling mechanism, where an initial steady loss of ions precedes rapid disintegration of the crystal. The resulting solution of ions in water also displays correct expected orientation of the water molecules with respect to the ions.
\subsubsection*{Similarity statement}
The MP dataset contains 460, 100, 112, 13 and 29 structures composed exclusively of [H, O, Si], [H, O, Ti], [H, O, Cu], [H, O, Mo, S] and [H, O, Na, Cl], respectively. The corresponding number of structures inclusive of the given atoms along with other elements is 477, 215, 435, 260 and 190.
Based on UMAP analysis, the closest atomic environments for each of these systems are mp-626085, mp-626550, mp-697660, mp-990086 and mp-504600.
Two files are provided for each of the systems for visualising using \verb|chemiscope|, one for inclusive and one for exclusive matches in the training set.
\subsubsection*{Performance summary}
All interface structures correctly predicted, including dissociative adsorption on \ch{SiO2} and molecular adsorption on \ch{TiO2}. At the salt/water interface correctly predicted dissolution from nanocrystal and no dissolution from flat surface on nanosecond time scale.
\clearpage
\subsection{Molten salts}\label{sec:molten_salts}
\begin{figure*}[htbp!]
\centering
\begin{subfigure}[b]{0.6\linewidth}
\includegraphics[width=\linewidth,keepaspectratio]{figures_molten_salts/NaCl-UCl3.pdf}
\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.39\linewidth}
\includegraphics[width=\linewidth,keepaspectratio]{figures_molten_salts/polyhedra.png}
\caption{}
\end{subfigure}
\caption{\ch{NaCl-UCl3} molten salt mixtures at 1100K. (a) Pair correlation function of \ch{NaCl-UCl3} mixtures at \SI{1100}{\kelvin}. (b) Example \ch{U-Cl} oligomers forming vertex sharing coordination polyhedra (\ch{U}: yellow, \ch{Cl}: green, \ch{Na}: purple).}
\label{fig:NaCl-UCl3}
\end{figure*}
With increasing interests in molten-salt energy technologies, we have simulated binary \ch{NaCl-UCl_3} salt mixtures \ch{(NaCl)_{(1-x)}(UCl_3)_x} at different compositions using \MLFF{}. The initial structures were randomized using Packmol in a cubic cell at the density estimated by the linear interpolation of the densities of constituent solid-state salt at \SI{0}{\kelvin} divided by a constant factor of \num{1.05}. \cite{martinez2009packmol,Chiang_muse_2023}. We then implemented geometric optimization using Lennard-Jones potential and further relaxed structures using \MLFF{} with a three-step process: geometric optimization, NVT relaxation through annealing at $1.2\times$ target temperature, and NPT relaxation at \SI{1100}{\kelvin} and ambient pressure. \Cref{fig:NaCl-UCl3}a presents the pair correlation functions between \ce{Cl-U}, \ce{U-U}, \ce{Cl-Cl} in salt mixtures. The characteristic peaks and transitions are consistent with previous polarizable ion models \cite{van2021coupled} and AIMD simulations \cite{andersson2022ab}, except for a noticeable shift of U-U peak from \num{4.5} to \SI{4}{\Angstrom}. The shift could be explained by the lack of Hubbard $U$ correction for rare earth elements in MP, leading to unrealistic ionic radii and solvation shell in the mixture. We also note that there is a small U-U peak around \SI{2.5}{\Angstrom}. This peak is absent in previous molten salt studies at high temperature \cite{andersson2022ab}, but as it is close to the equilibrium distance between \ch{U} as demonstrated by the homonuclear diatomic curve (\cref{fig:homonuclear-2b}), its appearance indicates the formation of a few \ce{U-U} bonds at a high fraction of molten \ch{UCl3} salt.
\subsubsection*{Similarity statement}
The MP dataset contains 573 structures composed of at least one Na, U, or Cl atom, 14 elemental Na crystals, 14 elemental U crystals, and 3 \ch{Cl2} molecular crystal structures. Based on UMAP analysis, we see that all atomic environments in the example system are similar to environments in the training data. We found that \ch{Cl2} molecular crystals are close to the molten salts but most of the pure \ch{U} metals are found separated from the molten salt in terms of MACE descriptors. We provide
\begin{itemize}
\item \verb|T_1100-P_0-seed_3-npt-5_chemiscope_input.json|
\end{itemize}
to help visualize the interactive UMAP of molten \ch{Cl64Na28U12} on \url{chemiscope.org}.
\subsubsection*{Performance summary}
Correct pair distribution peaks and variation of peak positions as a function of concentration, with a notable shift in the first U-U peak position, due to absence of Hubbard-$U$ correction.
\clearpage
\subsection{Room temperature ionic liquids}\label{sec:ionic_liquids}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\linewidth]{figures_RTIL/bmim_bf4_full-lowres.pdf}
\caption{MD simulation of the BMIM \ch{BF4} room temperature ionic liquid. The left panel shows energy
as a function of the trajectory, starting from an energy minimization, followed by MD simulation in an NVT ensemble with a step-wise increase in temperature (dashed lines, right axis). The middle column of panels
shows energy and force parity plots for configurations from the trajectory (red markers). The right top panel compares RDF to AIMD, and the right bottom panel shows a rigid-molecule volume-scan test \cite{magduau2023machine}.
}
\label{fig:rtil_bmim_bf4}
\end{figure}
Room temperature ionic liquids provide a class of organic solvents with desirable properties such as low melting and high boiling points, chemical inertness, and good ionic conductivity, making them applicable to different chemical and physical applications. Furthermore, these properties can be tuned by changing substituents on the anion or cations. The vast availability of substituents makes simulations at quantum mechanical accuracy to optimize these \textit{in silico} a very interesting approach.
For the given example, the class of imidazolium-based ionic liquids was chosen. The introduced MACE model struggles with running simulations that are only composed of a single chlorine anion, resulting in a \ch{Cl} bonded to the aromatic ring.
Therefore, simulations using the more commonly used \ch{BF4} anion with the 1-butyl-3-methylimidazolium (BMIM) cation were conducted. A single MD simulation was performed starting at the experimental density~\cite{rebeloDetailedThermodynamicAnalysis2004} of BMIM \ch{BF4} at \SI{273}{\kelvin}. The temperature was stepwise increased from \SI{273}{\kelvin} to \SI{323}{\kelvin}. Between each increase, the cell was adjusted to the new density and equilibrated over \SI{500}{\femto\second}. All simulations are conducted using the \MLFF{} with additional D3(BJ) corrections.
At each temperature, an NVT simulation using a Langevin thermostat was conducted for \SI{50}{\pico\second} with a time step of \SI{0.5}{\femto\second}. From the final trajectory spanning \SI{250}{\pico\second} over 5 different temperatures, data points were uniformly selected, and energies and forces were compared to DFT~\cite{perltFindingBestDensity2018}. Additionally, the radial distribution function was compared to an AIMD simulation, indicating a shift in the hydrogen positions compared to DFT. Furthermore, the interatomic interactions are probed using a volume scan, showcasing the importance of the additional D3(BJ) correction to stabilize the correct volume (see Figure~\ref{fig:rtil_bmim_bf4}).
Finally, a MD simulation in an NPT ensemble at \SI{1}{atm} and \SI{300}{\kelvin} showed that the model reproduces the experimental density within 5\%.
\subsubsection*{Similarity statement}
There are 52 structures in the MP dataset that explicitly include the \ch{BF4} anion.
Although there are organic nitrogen-containing molecules as well as heterocyclic systems, there are no alkyl-substituted imidazolium derivatives like BMIM in the training dataset.
We provide \verb|BMIM_BF4.json| for a comparison of snapshots from the MD trajectory to the training dataset on \url{chemiscope.org}.
\subsubsection*{Performance summary}
Stable MD for \ch{BF4} anion, but \ch{Cl} anion bonded to imidazolium. Intermolecular distance distribution shows small peak shift, and intermolecular attraction underestimated.
\clearpage
\subsection{High-pressure hydrogen}\label{sec:h2}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figures_highPressure_H/Hydrogen-lowres.pdf}
\caption{(A) Illustrative phase diagram of high-pressure hydrogen \cite{gregoryanz2020everything, magduau2017simple}. Horizontal lines show the NPT-MD simulations: \MLFF{} (orange), MACE/stress (blue) (see text) and AIMD/PBE (green)~\cite{magduau2017theory}. (B) Temperature and pressure during MACE-MD, and those externally applied by the thermostat/barostat. The bottom panel shows the pressure/density dependence in MACE-MD compared to AIMD. (C) MACE properties (energy, forces, pressure) reevaluated on the AIMD trajectory and compared to the original PBE result. (D) At low pressure MACE reproduces solid hydrogen phase I as a hexagonal-close-packing (HCP) lattice of free rotors. (E) At higher pressure (\SI{60}{\giga\pascal}) the potential fails and forms unphysical hydrogen chains before exploding. (F) MACE energy curves computed for different orientation of the \ch{H2}-\ch{H2} dimers. The distance is measured between the centers of mass of the two molecules.}
\label{fig:HPH}
\end{figure}
Condensed-phase hydrogen is an exotic state of matter that forms at extreme conditions in the core of larger planets and in specially-designed laboratory diamond anvil cells. Despite the simplicity of the hydrogen atom and molecule, the condensed phase exhibits fascinating phenomena such as entropy-driven phase transitions \cite{pickard2012density, magduau2013identification}, phonon localization \cite{howie2014phonon, magduau2017infrared}, quantum rotor solid phases \cite{cooke2020raman}, and an insulator-to-metal transition \cite{cheng2020evidence, zong2020understanding}. AIMD has been used extensively to study the molecular mechanisms underlying these phenomena, however, simulations are often affected by finite-size effects. Bespoke ML potentials, fitted to reproduce the interaction of hydrogen molecules with ab initio accuracy, have demonstrated simulations at an unprecedented level of detail, unlocking new scientific observations \cite{cheng2020evidence}. Being so different from other materials in the MP database, solid hydrogen is a uniquely challenging test for \MLFF{}.
The stability of the potential on this system was tested by running MD simulations at high pressure and by investigating the H-H and \ch{H2}-\ch{H2} dimer curves for molecules in different orientations. The MD simulation started from a thermalized crystal structure with P6$_3$/m symmetry and pressure was slowly ramped up at constant temperature. \MLFF{} becomes unstable and explodes at pressures around \SI{10}{\giga\pascal}. Attempting a simple remedy, we fitted a new model called ``MACE/stress'' which differs from \MLFF{} by having a larger weight on the stress error (by a factor of 10) in the loss function. This potential is more robust and remains stable up to \SI{60}{\giga\pascal}. At low pressures, both potentials reproduce the correct behaviour of hydrogen phase I: HCP crystal of freely rotating hydrogen molecules. The accuracy of the potential was also quantified by re-evaluating the energy, forces and pressure on an existing AIMD trajectory \cite{magduau2017theory}. \MLFF{} reproduces the PBE energy on the AIMD trajectory up to surprisingly high pressures (\SI{100}{\giga\pascal}), however, during dynamics it forms unphysical hydrogen chains well below that pressure (\SI{60}{\giga\pascal}). The energy--separation curves for \ch{H2}-\ch{H2} dimers are generally smooth, with a few exceptions in MACE/stress, where the lower repulsion between \ch{H2} molecules results in chain formation, but also explains why this simulation is stable up to higher pressures.
\subsubsection*{Similarity statement}
The MP dataset contains 17 structures crystal structure composed exclusively of H. From these, only 2 structures are above the \SI{0.2}{\gram\per\cubic\centi\meter} density value: mp-1096977 (hexagonal P4/mmm \SI{0.24}{\gram\per\cubic\centi\meter}) and mp-754417 (hexagonal P6/mmm \SI{0.24}{\gram\per\cubic\centi\meter}), yet \MLFF{} was found to extrapolate well up to around \num{0.30}--\SI{0.35}{\gram\per\cubic\centi\meter} where the potential starts to break down. We provide \verb|hydrogen_exclusive.json| for visualization on \url{chemiscope.org}.
\subsubsection*{Performance summary}
Correct solid hydrogen structure reproduced at low to moderate pressures. At pressures above \SI{10}{\giga\pascal}, the potential failed and resulted in unphysical structure. A refitted potential with higher weight on stress errors failed at \SI{60}{\giga\pascal}.
\clearpage
\subsection{Ammonia and borane thermal decomposition}\label{sec:amonia_borane}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_BN/bn-figure-lowres.pdf}
\caption{Decomposition of ammonia and borane at \SI{1600}{K}. The plot shows the time evolution of cluster size in terms of the fraction of heavier atoms (\ch{B} and \ch{N}) found in each cluster size group. Snapshots of the system at different times show the growth of \ch{BN} clusters and evolution of \ch{H2} molecules.}
\label{fig:borane-ammonia}
\end{figure}
Ammonia and borane form an adduct \ch{NH3BH3}. At high temperatures, these molecules lose hydrogen gas to give increasingly heavier \ch{B}- and \ch{N}-containing molecules, ultimately resulting in the growth of hexagonal boron nitride \cite{Frueh2011}. We simulated this process with \MLFF{} by running NVT molecular dynamics simulation of 32 ammonia and 32 borane molecules in a cubic box of length \SI{16}{\Angstrom}, at a temperature of \SI{1600}{\kelvin} for \SI{1}{\nano\second} with \SI{0.5}{\femto\second} time step. We analysed the time evolution of the system in terms of the sizes of heavier atom clusters (excluding hydrogen), illustrated in \cref{fig:borane-ammonia}. Initially, the formation of ammonia borane adduct and small borane clusters is seen, while in 100 ps timescales \ch{B} and \ch{N} atoms are increasingly more clustered, exhibiting \ch{B}-\ch{N} bonds in chains and six-membered rings as found in borazine and hexagonal boron nitride.
\subsubsection*{Similarity statement}
The training set contains 67 structures composed of \ch{H}, \ch{B}, \ch{N} elements.%
The training set contains various structures encountered during the simulation including ammonia, borane, and \ch{HBN} compounds of various stoichiometries, for example borazine (\ch{B3N3H6}) and \ch{(BNH2)_n} chains. We performed UMAP analysis for 100 frames taken from the 1 nanosecond simulation against training data containing at least one of the \ch{HBC} elements and any other elements. Based on the UMAP values, most of the simulation environments are clustered near the training data, with exceptions being species with unusual valency (e.g. \ch{BH2}). The closest structures in the training set are mp-1197795, mp-1203334 (both containing B and N, among other elements) and mp-1214811 (\ch{B6N6H10} bicyclic aromatic compound). We provide \verb|ammonia-borane.json| for visualization on \url{chemiscope.org}.
\subsubsection*{Performance summary}
Model correctly predicts hydrogen production and \ch{BN} cluster formation from thermal decomposition of \ch{NH3BH3}.
\clearpage
\subsection{Heterogeneous Catalysis}
Computational heterogeneous catalysis evolves around the exploration of \textit{operando} catalyst stability and catalytic reaction mechanisms to provide information about the nature of the active site that defines a catalyst's performance. This information provides a basis for screening applications to find efficient and ideally non-precious and non-toxic catalysts. To this end, a variety of atom-scale properties are investigated, including bulk and surface energies to evaluate catalyst stability in (surface) phase- or Pourbaix diagrams, as well as adsorption energies, reaction thermodynamics, and reaction barriers that are key to elucidating mechanisms and catalytic activity \cite{Norskov2014}. Local geometry optimizations and transition state searches via \textit{e.g.} NEB calculations \cite{neb_aseneb, neb_jk_optimiser} that yield target properties are usually conducted on slab models that exemplify the catalyst surface. By expanding the usual surface science approaches via thermodynamic referencing of protons and electrons to pH and applied potential on basis of the computational hydrogen electrode (CHE) \cite{Norskov2004-ow}, concepts in thermal catalysis can be extended to electrocatalysis. This approach provides fairly robust results even though the simulation of the electrolyte, charged species or an applied potential and thus the direct influence of the electrified solid/liquid interface is omitted. The computationally involved methodology is fully transferable to \MLFF{} and we included all mentioned aspects in the examples presented in the main text and below.
\subsubsection{Pourbaix diagrams}\label{sec:pour}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\textwidth,keepaspectratio]{figures_pourbaix/mace_pourbaix-lowres.pdf}
\caption{Pourbaix diagrams of \ch{CuO} bulk systems with energies of relevant solid compounds taken from (a) the {\MLFF{}} calculations and (b) the MP reference. (c) and (d) shows the {\MLFF{}}-calculated Pt(111) surface Pourbaix diagrams at pH = 0 and at various pH, respectively, which are in good agreement with \cite{Hansen2008-zz}. Different stable surface structures are represented in different colors. (e) shows the stable Pt(111) surface structures from low to high applied potentials. The red dashed lines indicate the stable window of water ranging from $U_{\mathrm{RHE}}$ = \SI{0}{V} to $U_{\mathrm{RHE}}$ = \SI{1.23}{V} ($U_{\mathrm{RHE}} \approx U_{\mathrm{SHE}} + 0.059 \cdot \mathrm{pH}$).}
\label{fig:pourbaix}
\end{figure}
In \cref{fig:pourbaix} we show the Pourbaix diagrams, calculated by the {\MLFF{}} with D3 corrections, which illustrate the aqueous stability for a \ch{CuO} bulk and a Pt(111) surface in dependence of applied potential and pH as referenced by the CHE. Structures for bulk \ch{CuO} and all other related oxide and peroxide compounds are taken from MP and are subsequently optimized (both atomic positions and cell parameters) using {\MLFF{}}. The energy corrections for oxides and peroxides, as well as the free energies for aqueous ions, are consistent with the values used in MP. As shown in \cref{fig:pourbaix}a and b, the overall trend of the \ch{CuO} stability predicted by {\MLFF{}} is well-aligned with the result given by MP, except for the narrow region of the \ch{Cu2}O phase that is not reproduced by \MLFF{}. For the \ch{Pt}(111) surface, {\MLFF{}} is used to optimize the surface geometry with different coverages of O*/OH* adsorbates together with an ice-like water layer.
As depicted in \cref{fig:pourbaix}c--e, {\MLFF{}} predicts that the \ch{Pt}(111) surface starts to oxidize at $U_{\mathrm{SHE}}$ = \SI{0.77}{V} (pH=0), followed by a step-wise increasing O* surface coverage with more positive electrode potential. This is in good agreement with the Pt(111) surface Pourbaix diagram reported previously\cite{Hansen2008-zz}, except that in our study the 0.5 ML O* coverage is not found to be stable at any electrode potential.
\subsubsection{Linear Scaling Relationships (LSR)}\label{sec:lsr}
\begin{figure}[ht!]
\centering\includegraphics[width=0.8\textwidth,keepaspectratio]{figures_catalysis_LSR/LSR_SI_figure-lowres.pdf}
\caption{Correlation plots (a) between the adsorption energies of two intermediates at the same (hollow site) of tightly packed metal surfaces (b). The correlation between *O and *C (c) is not linear (in agreement with the literature).}
\label{fig:catalysis_lsr}
\end{figure}
Adsorption energies of molecules and intermediaries are indicative of catalyst reactivity and often used as descriptors in screening studies for catalyst materials. The adsorption energies are governed by electronic and geometric factors. Provided a consistent geometric environment (\textit{e.g.} a hollow site of a tightly packed metallic lattice) across different metal surfaces, the trend in adsorbate binding energies resulting primarily due to electronic effects can be observed. A well-known property of catalytic surfaces (\textit{e.g.} transition metals) is that binding energies of individual intermediates are not independent of each other, as a consequence of the varying degree of occupation of the metallic d-band~\cite{norskov2007linear,norskov2022nonlinear}. Linear scaling relationships (LSR) were found for a range of metallic surfaces and molecules that bind to this surface through the same atom (\textit{e.g.} E scales with EH$_\mathrm{x}$, where E = C, O, N, S and x = 1,2,3), however, this scaling is not linear when comparing adsorbates that bind through different atoms (\textit{e.g.} C versus O).
In \cref{fig:catalysis_lsr} we show the correlation between the adsorption energies of EH$_\mathrm{x}$, where E = C, O, N, S, and x = 0, 1, 2, 3. The structures (\cref{fig:catalysis_lsr}b) were relaxed with the {\MLFF{}} model with D3 correction (cutoff = 4 nm), and the adsorption energy was computed as $\Delta E_{\textrm{ads}} = E(a*) - E(*) - E(a)$, with $a$ as the adsorbate and * as the surface slab. The observed correlations are linear in all cases except for the correlation between \ch{O} and \ch{C}. In \cref{fig:catalysis_lsr}, the mace computed adsorption energies (blue/green/orange circles) are compared to the corresponding DFT values (connected with faint gray lines to faint gray circles) as reported by Norsk{\o}v~\cite{norskov2022nonlinear}. In this plot, although the absolute error of the obtained adsorption energies in comparison to the DFT values is high (which is not surprising as the model is extrapolating in this example), the trend of grouping metals into passive (noble, \textit{e.g.} Au) catalytic (so-called Pt group) and non-reducible (\textit{e.g.} Zr) is correctly captured and the essence of the LSR relationships was reproduced with the {\MLFF{}}+D3 model.
\subsubsection{\ch{CO} (electro-)oxidation on \ch{Cu}}\label{sec:oxi}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.9\textwidth,keepaspectratio]{figures_COoxCu/COoxCu_barplot_revised.pdf}
\caption{NEB profiles (top) and extracted barriers (bottom) of reactions (i-iv) computed for the non-potential dependent steps in the multistep reaction mechanism of \ch{CO} electro-oxidation on the low index Cu facets (110), (111), and (211) (for profile of Cu(100) see main text). The reaction mechanism and the original BEEF-vdW NEB data without implicit solvent (light-blue) is adapted from \cite{Tiwari2020}. The NEB calculations were repeated via \MLFF{} (orange) with an additional D3 correction and single point PBE+D3 calculations (dark-red) were performed for the \MLFF{}+D3 NEB-path.}
\label{fig:catalysis_eCOoxCu}
\end{figure}
We test the ability of \MLFF{} to predict the catalytic reaction mechanisms for the oxidation of CO on different facets of Cu which was previously explored via DFT\cite{Tiwari2020}. The evaluated reaction barriers present the potential-independent reaction steps of the CO oxidation via electrochemically adsorbed OH* groups. The electrochemical OH* adsorption (not included here) can be described via the CHE \cite{Norskov2004-ow}. We recompute the reaction pathways using \MLFF{}+D3 via NEB calculations in lattice-parameter-adjusted simulation cells.
In \cref{fig:catalysis_eCOoxCu} we show the \MLFF{}+D3 reaction pathways and barriers together with corresponding PBE+D3 single points and the original BEEF-vdW NEB calculations for two Cu terraces (111) and (100) and two step-sites (110) and (211). \MLFF{}+D3 displays a mixed performance for the various reaction steps i-iv.
The coupling (i) and dehydrogenation (iii) barriers are reproduced within 10 \% of the original BEEF-vdW data\cite{Tiwari2020}. An exception is the coupling barrier on Cu(111) which is overestimated by 100 \%. For this latter transition state we, however, find good agreement with the single point PBE-D3 data which indicates divergence due to the DFT functional (PBE+D3 vs. BEEF-vdW). A throughout gross and moderate underestimation of the torsional rearrangement of \ch{COOH} (ii) and desorption of \ch{CO2} (iv) is found. We believe that this shortcoming is likely due to missing training data which is suggested by large deviations from the PBE+D3 single point calculations. Despite the quantitative differences in reaction barriers, we find \MLFF{}+D3 to mostly preserve the qualitative trends between the reaction steps i, iii, and vi (ii is inaccurately described) and facets as well as the overall reaction mechanism.%
\subsubsection{\ch{In2O3}}\label{sec;in2o3}
As a final test system we investigate a key step (\ch{CH2O2 -> CH2 + O}) in carbon dioxide hydrogenation to methanol over indium oxide via an NEB transition state search. This reaction has been extensively studied with \textit{ab initio} methods due to indium oxide's promising selectivity compared to conventional modified copper catalysts~\cite{dang2020rationally, schaaf2023accurate}. First, we perform a global geometry optimization of the reactant near an oxygen vacancy. \MLFF{} correctly identifies the three-oxygen-coordinated indium as the active site~\cite{schaaf2023accurate}. Following a NEB calculation \MLFF{} predicts the reaction barrier within 10\% of that investigated with DFT (\SI{1.30}{eV} vs.\ \SI{1.16}{eV}), as visible in \cref{fig:cat_main}.%
\subsubsection*{Similarity statement}
With the exception of the \ch{CuO} bulk Pourbaix diagram, which is based on structures from the Materials Project, all presented examples treat surface slab models (with and without reacting adsorbates) which are a strong extrapolation of \MLFF{}'s dataset. The dataset does not include any such slab models but only related bulk structures. These bulk structures include 6 bulk structures that contain Pt, O, H and 15 bulk structures that contain Pt and O for the Pt Pourbaix diagram and the corresponding LSR example (similar number for other metals in the LSR), 8 different Cu-bulk phases and 111 structures composed of Cu, O, Cu, and H along with other elements for the example of CO oxidation on different Cu facets, and 9 structures that contain In, O, H and C and a 825 bulk structures that contain indium and oxygen for the \ch{In2O3} example. The most similar configurations for the CO oxidation example are \ch{Cu2H4C4N3}O with the Materials Project ID mp-686268 and for the \ch{In2O3} example \ch{(NH4)In(OH)PO4} with the ID mp-764968. We provide \verb|Pt_LSR.json| and \verb|COoxCu_closest_training_points.csv| for the LSR and CO oxidation example to help visualize the interactive UMAP on \url{chemiscope.org}.
\subsubsection*{Performance summary}
Solid energies accurately predicted leading to correct Pourbaix diagrams. Adsortion energies overestimated, but linear scaling relationships between different surface/adsorbate pairs preserved. Minimum energy paths for reactive steps qualitatively correct, in some cases small (\SI{0.2}{eV}) in other larger (\SI{0.5}{eV}) energy errors.
\clearpage
\subsection{Carborane rearrangement}\label{sec:carborane}
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_boranes/carborane-o-m-lowres.pdf}
\caption{DFT NEB pathways for the isomerization reaction of \textit{ortho}-carborane to \textit{meta}-carborane. \MLFF{} energies, evaluated on the same pathways, are shown. All energies are shown relative to the \textit{ortho} isomer energy. On the right, end point and transition state structures are illustrated.}
\label{fig:carborane}
\end{figure}
Carborane (\ch{C2H12B10}) is an organoboron compound with uses in drug discovery \cite{Marfavi2022} and organometallic chemistry \cite{Sivaev2000}. It adopts icosahedral cluster structures, with three isomers based on different relative positions of carbon atoms: \textit{ortho} (\textit{o}), \textit{meta} (\textit{m}), and \textit{para} (\textit{p}). The thermally activated rearrangements between these isomers have been thoroughly studied \cite{Brown2006}, with several mechanisms proposed involving triangular face rotation (TFR). We used \MLFF{} to study two pathways from the \textit{ortho} isomer to the \textit{meta} isomer: one involving an anticuboctahedral transition state (by mutual rotation of two opposite faces), the other involving the rotation of a single triangular face.
We used \MLFF{} to provide preliminary rearrangement pathways for each mechanism to be refined with DFT using the NEB method. For each mechanism, a pair of endpoint structures was built, with the atom labels consistent with the mechanism. The endpoint structures were relaxed with force tolerance of \SI{0.01}{eV\per\Angstrom} at PBE/def2-TZVPPD level of theory using ORCA 5.0.1 \cite{Neese2022}. The initial NEB pathways with 20 images and a spring constant \SI{2}{eV\per\square\Angstrom}, were relaxed using \MLFF{} and then further relaxed at PBE/def2-TZVP level of theory. \Cref{fig:carborane} shows the rearrangement energy profiles with their endpoint and transition state structures. \MLFF{} qualitatively captures the structural details of each rearrangement while underestimating the barrier height of the rearrangement.
\subsubsection*{Similarity statement}
The training set contains 5 structures composed of H, B, C elements, and 21837 structures that have H or B or C along with any other elements. The dataset contains 2 structures or containing icosahedral \ch{C2H11B10} clusters linked by a C-C bond to form a dimer, 1 icosahedral borane \ch{B12H12} cluster and 18 other borane clusters containing trigonal \ch{B3} faces. Along these, the dataset contains several hundred structures involving derivatives of borane and carborane clusters such as salts, metal complexes, and halogenated species. The closest structures in the training set are mp-1194548 (\ch{C4H22B20} containing two icosahedral carborane clusters joined with a C-C bond), and metal complexes containing carborane ligands: mp-759303, 705569, 1199795, 1198024.
\subsubsection*{Performance summary}
Carborane rearrangement minimum energy paths geometrically correct, with the energy barrier underestimated.
\clearpage
\subsection{Transition Metal Dichalcogenides}\label{sec:tmd}
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figures_TMDs/TMD_MACE_Mp0.pdf}
\caption{(a) shows four \ch{MoS2} edge models of different type and sulfur coverage ($\theta_S$). DFT-predicted structures are obtained from \cite{rosen2018comprehensive}. \MLFF{} structures are the result of \SI{100}{ps} of MD at \SI{300}{\kelvin} followed by a geometry optimization. (b) compares the defect formation energies predicted by \MLFF{} and DFT-PBE results from \cite{kieczka2023defects}. Geometries of defects considered here as relaxed using \MLFF{} are also shown in (b).}
\label{fig:TMDs}
\end{figure}
Edges in \ch{MoS2} and TMDs more broadly are known to be sites of high reactivity with much relevance to TMD-catalyzed reactions and material aging studies. Studies have recently addressed the structure of \ch{MoS2} edges under various conditions using DFT \cite{rosen2018comprehensive}. Here, we examine stable Mo-edge and S-edge configurations with sulfur coverage $\theta_S = 0.5$ and $\theta_S = 1$ from \cite{rosen2018comprehensive}. The edge models are multi-layer models, i.e stacked infinite stripes. We assess the MD stability of \MLFF{} + D3 dispersion on these structures by running \SI{100}{ps} of MD (NVT ensemble) at \SI{300}{K}. Final structures are geometry optimized and taken as equilibrium geometries. \Cref{fig:TMDs}a shows the \ch{MoS2} edge structures considered. All configurations exhibited stable MD for \SI{100}{ps}. The Mo-edge and S-edge configurations largely retain the structural features predicted by DFT. The $\theta_S = 1$ S-edge is seen to have a sulfur dimer forming which is a favorable reconstruction in monolayer and not multi-layer models \cite{rosen2018comprehensive}.
Defects have a significant impact on the optical and electronic properties of 2D TMDs and come in various types including vacancies. We assess the ability of \MLFF{} to describe defect formation energies ($E_f$) of various defects in \ch{WS2} as compared to PBE from \cite{kieczka2023defects}. The formation energy is calculated using the formula:
\begin{equation}
E_{\text{f}} = E_{\text{defect}} - E_{\text{pristine}} - \sum \Delta n_i \mu_i
\end{equation}
where \(E_{\text{defect}}\) and \(E_{\text{pristine}}\) are energies of the \ch{WS2} with and without defects, \(n_i\) and \(\mu_i\) are the number of atoms and chemical potential of element \(i\). The chemical potential of \ch{S} obeys the equilibrium condition $\mu_{\ch{WS2}} = \mu_{\ch{W}} + 2 \mu_{\ch{S}}$, and is bounded by predefined S-poor and S-rich conditions: $\mu_S^{\text{bulk}_{\ch{W}}} \leq \mu_S \leq \frac{1}{2}\mu_S^{\ch{S2}}$. It is calculated with respect to the \(\alpha\)-S as the reference state\cite{kieczka2023defects}. \Cref{fig:TMDs}b shows $E_F$ values predicted by \MLFF{} and DFT are in qualitative agreement. Large deviations can be observed in some of the high-energy defects such as $V_{WS_3}$ and $V_W$ with a consistent underestimation of energies and forces by \MLFF{}.
\subsubsection*{Similarity statement}
Majority of the \ce{Mo-S} or \ce{W-S} containing structures in the database include primitive units of layers of \ch{MoS2} or \ch{WS2}. However, clusters of \ce{Mo-S} and \ce{W-S} were found as well. Edge models for \ch{MoS2} are rare, with three structures identified (mp-990083, mp-989179 and mp-990086). These include a variant of the $\theta_S = 1$ S-edge and the $\theta_S = 0$ Mo-edge. No defect models of \ch{WS2} were found in the dataset.
\subsubsection*{Performance summary}
Geometric reconstruction of nanoribbon edges mostly correct, apart from small deviations. Ordering of defect formation energies qualitatively correct with an overall tendency to underestimate precise values.
\clearpage
\subsection{Electrode-electrolyte interface / Battery system}\label{sec:battery}
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_electrode-electrolyte/Battery-lowres.pdf}
\caption{(A) Flowing EC/EMC \ch{LiPF6} electrolyte with stable density and intact molecules at \SI{500}{\kelvin} \MLFF{}+D3 NPT MD. Comparison with a bespoke MACE model trained and tested exclusively on PBE/D2 EC/EMC solvent mixture, which will be published elsewhere. Final simulation snapshot, repeated across PBC (unwrapped) highlighting the diffusion of the simulated molecules `outside' the simulation box. (B) Intra-/Inter-molecular energy / forces / virial evaluated on independent PBE test set spanning all possible compositions of EC/EMC LiPF$_6$ electrolyte at densities between \num{0.1}--\SI{2.5} {\gram\per\centi\meter}. (C) Rigid-molecule volume-scan test \cite{magduau2023machine} for both charge-neutral solvent and full electrolyte, compared to PBE. (D) Full battery simulation (Cu | H-capped graphite+Li | EC/EMC+\ch{LiPF6} | NMC+Li \cite{fang2021formation}), final snapshot of \SI{500}{\kelvin} \MLFF{}+D3 NVT-MD, showing degraded solvent (grey iso-surface molecules), new \ch{CO2} and \ch{H2O}, oxygen atoms originating from the cathode floating in the electrolyte. (E) Close-up of cathode-electrolyte interface showing solvent molecules chemisorbed to surface. (F) Time progression of the predominant molecular species in the electrolyte for the two NVT simulation settings (1. neat NMC, 2. H-capped NMC (\ch{H} on exposed \ch{O} atoms)).}
\end{figure}
Atomic-level interactions between the electrodes and electrolyte play a crucial role in determining the performance of electrochemical devices, including batteries, fuel cells, and electrocatalysts. Understanding these interactions is essential for optimizing the energy storage, conversion, and catalytic properties of these devices and to this end molecular modelling plays a crucial role. The remaining challenge is that processes underpinning transport and degradation in these devices take place on a long time scale, inaccessible to ab initio simulations. MLIPs are ideally suited to bridge this gap, bearing in mind that these complex materials and heterogeneous interfaces cover an extensive chemical space which poses a big challenge to ML models. Here we test the performance of \MLFF{} on three separate systems -- pure (EC/EMC \ch{LiPF6}) electrolyte, the electrolyte-graphite anode interface, and the complete battery including the copper interface, anode, electrolyte and NMC cathode (totalling 9 chemical elements). We performed MD simulations at 500 K using \MLFF{}+D3 to stress-test the qualitative robustness of the potential. Further, we quantitatively assessed the potential on a separate PBE test set in order to establish the accuracy of describing intra- and intermolecular interactions.
Previous work~\cite{magduau2023machine} how shown that modelling even the neat solvent is a challenge to MLIPs owing to the weak, but crucially important, inter-molecular interactions. Here we find that \MLFF{} is stable in the NPT ensemble at \SI{500}{\kelvin}, the density is preserved while the electrolyte (solvent+salt) remains liquid and all molecules remain intact for the entire duration of the simulation (approx.\ \SI{150}{\pico\second}). A similar NPT simulation was performed for the pure solvent with a bespoke MACE model trained exclusively for EC/EMC solvent mixtures at PBE/D2 level (will be published elsewhere). Both the densities and diffusivities of the two simulations closely agree with each other, demonstrating that the out-of-the-box \MLFF{} can reproduce the performance of the custom trained model. The accuracy of the model was also tested on a diverse PBE test set of electrolytes spanning all physical compositions and densities (will be published elsewhere). \MLFF{} reproduces both intra- and inter-molecular interactions and performs surprisingly well on rigid-molecule Volume scans \cite{magduau2023machine} which test the inter-molecular PES. The potential capture the $1/r$ scaling of the energy in the charged system to surprisingly large volume (low density) values.
The electrolyte-anode interface, as well as the full battery were simulated in the NVT ensemble, since the volume of the entire system was not stable in NPT simulations (possibly the result of large compressibility differences along the x-direction normal to the liquid electrolyte layer vs the y/z-directions along the solid electrode slabs). All NVT simulation were stable at \SI{500}{\kelvin} for the entire simulation time (\num{100}-\SI{200}{\pico\second}). The electrolyte \ch{Li}-ions were found to deintercalate from both the graphite anode and NMC cathode and the electrolyte was mobile. The H-capped graphite was found to be inert, whereas the cathode-electrolyte interface exhibited pronounced reactivity. Evident from the start of the simulation was the extensive proton transfer from the carbonate solvent (EMC in particular) to the oxygen atoms in NMC. This in turn led to continuous breakdown of solvent molecule (which became a radical) and chemisorption onto the cathode surface, possibly demonstrating the initial steps of SEI formation. Notably, substantial amounts of \ch{CO2} and \ch{H2O} were generated in the process, and oxygen atoms were easily extracted from the cathode leaving behind binding sites for the oxygen-rich carbonate molecules. A separate simulation setting was tested where the exposed oxygen atoms of the cathode were hydrogenated before the simulation. Similar reactivity was observed albeit with different outcomes, notably more water molecules and less carbon dioxide was generated in the process. These early simulations demonstrate \MLFF{} is robust for battery interfaces and showcase the initial steps in modelling the SEI formation with ab initio accuracy -- which has been a long-held dream of the scientific community.
\subsubsection*{Similarity statement}
To perform the similarity analysis, 100 representative (decorrelated) structures were taken from the previously described simulations. While the MP dataset does not contain liquids, for all three simulations the UMAP analysis showed that all atomic environments were well represented in the MP dataset. More specifically, the pure electrolyte was found to contain environments close to mp-995234 and mp-995218 which correspond to \ch{HCO} and \ch{H_4C_5O_2}, respectively. These configurations were also found to be similar for the snapshots obtained by running the electrolyte-anode interface, and also notably included mp-707412 corresponding to \ch{H22C10O3}. For the entire battery system configurations such as mp-1194779 and mp-698267 were found to be similar which correspond to \ch{CuH_3C_3O_4} and \ch{CoHCO_3} respectively. We provide:
\begin{itemize}
\item \verb|interface_chemiscope_input.json|
\item \verb|battery_chemiscope_input.json|
\item \verb|electrolyte_chemiscope_input.json|
\end{itemize}
to help visualize the atomic environments against the MP dataset at \url{chemiscope.org}.
\subsubsection*{Performance summary}
Solvent and electrolyte properties (density, diffusivity) well captured compared with custom-trained model. Full battery simulation stable at fixed volume but unstable with variable volume, and shows electrolyte breakdown at unpassivated electrode.
\clearpage
\subsection{Metal--organic frameworks}\label{sec:si_mofs}
\begin{figure}[http!]
\centering
\begin{subfigure}{0.495\textwidth}
\includegraphics[width=\textwidth,keepaspectratio]{figures_mofs/Mg-O-C-lowres.pdf}
\caption{}
\label{fig:mof_si_a}
\end{subfigure}
\begin{subfigure}{0.495\textwidth}
\includegraphics[width=\textwidth,keepaspectratio]{figures_mofs/mace-energy-error-atomic-density.pdf}
\caption{}
\label{fig:mof_si_b}
\end{subfigure}
\caption{(a) Top panel: instantaneous distance between the Mg center and the closest oxygen from \ch{CO2} in Mg-MOF-74, the red line shows the predicted average value from previous work. \cite{zeng2023deepmd} Bottom panel: instantaneous angle between Mg center, closest oxygen and carbon in the \ch{CO2}, red line indicates an average value from previous work from Ref.~\cite{zeng2023deepmd} (b) \Cref{fig:mof}a inset. Error between \MLFF{} predicted and PBE energies on 20,375 relaxed structures taken from the QMOF database, Refs. \cite{rosen2021machine,rosen2022high}. High-error, sparsely-populated region structures are mainly high density entries with elemental dependence shown in \Cref{fig:mof}b. }
\end{figure}
\subsubsection{QMOF}
Given that \MLFF{} is pretrained against PBE, whereas QMOF was constructed from PBE-D3(BJ) calculations to account for dispersion corrections \cite{grimme2010consistent,grimme2011effect}, in the comparison with QMOF, we compare \MLFF{} predicted energies with QMOF PBE energies by subtracting dispersion correction from total QMOF energies.
We noted that most of the high-energy error MOF structures are high in atomic density (\Cref{fig:mof_si_b}), and that the errors cannot be canceled by adding dispersion correction. Thus, we further analyze the element-wise energy error per atom by distributing the absolute energy error per atom to the constituent elements by the corresponding composition in each MOF structure. As presented in \Cref{fig:mof}b, there is a strong elemental dependence of energy error per atom. Most of high error elements can be attributed to the difference of chosen pseudopotentials used by MP and QMOF databases (in particular the choices of which electrons are treated as valence states), see \Cref{tab:mp-qmof-potcars}.
\begin{table}[htp!]
\centering
\begin{adjustbox}{width=\textwidth}
\renewcommand{\tabcolsep}{3pt}
\begin{tabular}{c|cccccccccccccccccc}
\textbf{MPtrj} & \verb|Be| & \verb|Bi| & \verb|Cr_pv| & \verb|Eu| & \verb|Fe_pv| & \verb|Gd| & \verb|Li_sv| & \verb|Mg_pv| & \verb|Mo_pv| & \verb|Nb_pv| & \verb|Ni_pv| & \verb|Os_pv| & \verb|Re_pv| & \verb|Ti_pv| & \verb|V_pv| & \verb|W_pv| & \verb|Yb_2| \\
\hline
\textbf{QMOF} & \verb|Be_v| & \verb|Bi_d| & \verb|Cu| & \verb|Eu_3| & \verb|Fe| & \verb|Gd_3| & \verb|Li| & \verb|Mg| & \verb|Mo_sv| & \verb|Nb_sv| & \verb|Ni| & \verb|Os| & \verb|Re| & \verb|Ti_sv| & \verb|V_sv| & \verb|W_sv| & \verb|Yb_3| \\
\end{tabular}%
\end{adjustbox}
\caption{Difference in VASP POTCARs used by MPtrj and QMOF}
\label{tab:mp-qmof-potcars}
\end{table}
\subsubsection{\ch{CO2} adsorption}
All the calculations for the \ch{CO2} dynamics with MOFs were performed with \MLFF{} by adding the D3 dispersion correction\cite{grimme2010consistent} to the \MLFF{} potential. The simulations were carried out with ASE \cite{larsen2017atomic} on a cell containing 165 atoms with one \ch{CO2} molecule, initialised at the centre of the pore, using NVT Langevin dynamics~\cite{VANDENEIJNDEN2006} with a friction factor of \SI{5e-3}{\per\femto\second}. The temperature was set to \SI{600}{K} with a time step of \SI{1}{\femto\second}. Twenty-four \SI{1}{\nano\second} trajectories were generated using different initial velocities, and all quantities presented were averaged over all of them, discarding the first \SI{2}{\pico\second} from each to account for equilibration. All structures had their cells and positions optimised at start using {\tt FrechetCellFilter} from ASE. The code used to generate the trajectories is available in the repo~\cite{elena2023}.
\subsubsection*{Similarity statement}
The MP dataset does not contain MOFs. There are 8 structures containing all \ch{MgOCH} elements, and 62 structures that have \ch{MgOHC} elements on their own or along with other elements. Based on UMAP analysis, we see that most atomic environments, both \ch{MgO} and linkers, in the example system, are similar to environments in the training data but none is Mg-MOF-74 specific.
The closest (most relevant) structures in the training set are \ch{CO2} (mp-556034 mp-20066 mp-995224 mp-11725 mp-644607 mp-1102227 mp-1190685 mp-995198 mp-1190699 mp-1077906 mp-1077316 mp-729728). \ch{CO2} alone matches 4896 structures with \ch{C}, \ch{O} and alongside other elements.
We provide
\begin{itemize}
\item \verb|co2_FilterType.exclusive_OC_chemiscope_input.json|
\item \verb|mg-mof-74-co2_FilterType.exclusive_MgOCH_chemiscope_input.json|
\item\verb|mg-mof-74-co2_FilterType.exclusive_MgOCH_chemiscope_input.json|
\end{itemize}
that contain exact matches of \ch{Mg}, \ch{O}, \ch{C} and \ch{H} and the inclusive versions
\begin{itemize}
\item \verb|mg-mof-74-co2_FilterType.exclusive_MgOCH_chemiscope_input.json|
\item \verb|mg-mof-74-co2_FilterType.inclusive_MgOCH_chemiscope_input.json|
\item \verb|mg-mof-74_FilterType.inclusive_MgOCH_chemiscope_input.json|
\end{itemize}
to help visualize the interactive UMAP on \url{chemiscope.org}.
\subsubsection*{Performance summary}
Excellent energy prediction for large database of MOFs. Correct prediction of binding structure and free energy of \ch{CO2} in Mg-MOF-74.
\clearpage
\subsection{Combinatorial Materials Discovery}\label{sec:materials discovery}
\subsubsection{Formation energy of hypothetical materials}\label{sec:formation}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.6\textwidth,keepaspectratio]{figures_mbd/parity-e-form-mace-IS2RE.pdf}
\caption{Formation energy parity plot showing the difference between the DFT-relaxed energy and the \MLFF{}-relaxed energy starting from WBM initial structures.}
\label{fig:parity-e-form-mace-IS2RE}
\end{figure}
\MLFF{}\ trained on the MPtrj dataset generalizes to out-of-distribution (OOD) chemistries as shown by its performance on the WBM dataset \cite{wang2021predicting} which was generated using elemental substitutions drawn according to a data-mined chemical similarity measure \cite{glawe_optimal_2016}.
The initial set of 9,524 structures for substitution were taken from the Materials Project convex hull.
The substituted structures were relaxed using the \texttt{MPRelaxSet} PBE DFT workflow.
After this, the convex hull was recalculated using the new structures and then subsequent rounds of substitution were carried out on the new structures that ended up on the combined MP plus growing WBM convex hull.
In total, 5 rounds of substitutions were carried out yielding a dataset of 257,487 inorganic crystals that are OOD with respect to MP and therefore well suited to benchmarking.
For this investigation, we use the cleaned version of the WBM dataset released in Matbench Discovery (MBD)\cite{riebesell_matbench_2023}, which first discards 524 crystals with unphysical or missing labels. Subsequently, all structures in WBM with composition+prototype matching a structure in MP are removed.
Within WBM composition+prototype, duplicates are dropped leaving only the lowest energy structure.
The final test set consists of \num{215 488} materials, and of these we obtained \MLFF{} relaxed structures for \num{208 750}.
Following the MBD protocol, we use \MLFF{}\ to relax the initial substituted structures and compare these predictions against the ground truth formation energy calculated with DFT.
Before calculating metrics, we drop 29 structures predicted to have formation formation enthalpies \textless \ \SI{-5}{eV/atom}.
The most prevalent elements in these dropped materials are H (36.4\%), Mn (11.7\%), Fe (10.5\%), Pu (10.1\%), and Cr (5.7\%), suggesting the existence of holes in the \MLFF{} PES for transition metal and f-block hydrides. The predictions on the \MLFF{}\ self-relaxed structures result in an MAE of \SI{57}{meV/atom} as shown in \cref{fig:parity-e-form-mace-IS2RE}.
The OOD nature of the WBM dataset can be seen in the increase in MAE between WBM batches from \SI{48}{meV/atom} for the first batch to \SI{78}{meV/atom} for the final batch as the structures become increasingly dissimilar to MP due to accumulated substitutions.
When these predictions are used to attempt to classify whether the structure lies above or below the MP convex hull training set, \MLFF{} achieves an F1 score of 0.67 and a discovery acceleration factor (DAF) of 3.76.
The DAF is the ratio of the precision (TP/PP) to the prevalence (P/N) of the test set. Here, TP = True Positives, PP = Predicted Positives, P = Total Positives and N is the test set size.
These results show that \MLFF{} can extrapolate to novel chemistries and is well-suited to high-throughput materials discovery.
\subsubsection*{Similarity statement}
By construction there is no overlap in terms of both composition and prototype together between MP and the WBM test set studied here. However there is overlap for both compositions and prototypes separately. MP contains \num{105 583} unique reduced formulae, whilst the WBM test set contains \num{160 055}. Of these \num{15 782} overlap MP albeit with all instances being examples of different prototype structures. MP contains \num{32 933} different isopointal prototypes, whilst the WBM test set contains just \num{2 816}. Of the prototypes found in WBM \num{1813} are also found in MP. Of the isopointal prototypes not seen in MP the 5 most common are (occurrences in parentheses): {ABC2\_oI8\_71\_a\_b\_f} (323), {ABC2\_hP12\_181\_c\_d\_i} (215), {AB2\_hP9\_189\_f\_adg} (156), {ABC2\_oI8\_44\_a\_b\_c} (117), {AB4C6\_mC22\_8\_a\_2ab\_2a2b} (106). These arise due to changes in symmetry during the relaxation of the substituted structures.
\clearpage
\subsubsection{Stoichiometric substitutions}\label{sec:element-substitution}
\begin{figure*}[!htbp]
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figures_materials_discovery/comb_mat_discover-lowres.pdf}
\caption{Left: Relative energies vs. volumes of ca. 150k relaxed structures generated via exhaustive element substitutions for 100 MP compositions. Center: Correlation between relative energies calculated with DFT and \MLFF{} for 6909 randomly drawn structures from the left plot. Right: DFT validated examples of newly discovered stable phases, along with their energies relative to the MP convex hull.}
\label{fig:element_substitution}
\end{figure*}
To test the interpolative and extrapolative capabilities of the model within MP chemistry, exhaustive element substitutions for 100 randomly selected compositions were performed. Specifically, MP was first filtered to remove all compositions with more than 16 atoms in the reduced formula, to ensure that a sufficient number of possible substitutions. This set was randomly split into host and target compositions with a 60:40 ratio. 100 compositions were randomly drawn from the target split and substituted into each stoichiometrically matching host, yielding 154,685 substituted structures. These were optimized with \MLFF{} using full unit cell relaxations (without the D3 correction). In \cref{fig:element_substitution}, the results are shown in terms of energies and volumes relative to the most stable structure of that composition within the MP. The distribution is sharply peaked at $\frac{V}{V_\mathrm{ref}}=1$ and ${E}-{E_\mathrm{ref}}=0$, indicating that the substituted cells often relax back to the known ground state structure from MP. This makes \MLFF{} potentially suitable for predicting the crystal structures of unknown materials. For a random sample of 6909 structures, the \MLFF{} relative energies were validated with MP-compatible PBE DFT calculations, yielding an RMSE of 0.097 eV/atom.
Beyond recovering the MP ground state, the wide range of substitutions tested also yields a large number of alternative structures for each composition. 2518 of these have relative energies of zero or lower and are thus predicted to be more stable than the MP reference. To validate these predictions, DFT relaxations were performed for all structures with negative relative energies. The results confirms that 2120 of them are indeed more stable than the corresponding MP reference. Importantly, not each of these is a new stable phase, as some relaxations converged to the same minimum and multiple lower energy structures are found for some compositions. Nonetheless, for nine of the 100 compositions considered here, new structures with energies below the current MP hull were discovered (for \ch{Bi3}Pb, \ch{Cs3BiF6}, \ch{Cs3TlF6}, \ch{LuCrO5}, \ch{NaNiIO6}, \ch{NaSrSnBiO6}, \ch{NdGaAu2}, \ch{Sr2CoO4}, and \ch{TiS2O8}). These constitute genuine predictions of new, thermodynamically stable phases relative to the MP convex hull. These phases were also not reported in the recent GNoME effort (which focused on compositions not included in MP). However, the WBM dataset does report slightly more stable structures for \ch{NaNiIO6} and \ch{NdGaAu2}. It also includes structures for \ch{Cs3BiF6} and \ch{Cs3TlF6}, which are less stable than the ones reported herein.
\subsubsection*{Similarity statement}
The MP contains at least one (\textit{e.g.} \ch{NaNiIO6}, \ch{NaSrSnBiO6}) and at most 423 (\ch{Sr2CoO4}) structures that contain all elements in the discovered stable phases. These matches are similar to the discovered structures, as host and reference structures are present in the MP. Yet, the discovered phases are unique and by definition not part of the training set. For each discovery, the closest structures in the training set are: mp-1106139 (\ch{Bi3Pb}), mp-559695 (\ch{Cs3BiF6}), mp-561827 (\ch{Cs3TlF6}), mp-1211553 (\ch{LuCrO5}), mp-545399 (\ch{NaNiIO6}), mp-1522253 (\ch{NaSrSnBiO6}), mp-1220399 (\ch{NdGaAu2}), mp-1100068 (\ch{Sr2CoO4}), mp-775149 (\ch{TiS2O8}). We provide json files to visualize interactive UMAPs on \url{chemiscope.org}.
\clearpage
\subsubsection{Analysis of highly-coordinated theoretical structures}\label{sec:coordination}
\begin{figure}[!htbp]
\centering
\includegraphics[width=4in,keepaspectratio]{figures_GNoME_materials/MACE_high_dens_relax.pdf}
\caption{
Comparison of the volume of the highly-coordinated structures in the GNoME database \cite{merchant_scaling_2023} (horizontal axis) against the same structures relaxed with the \MLFF{} UIP (vertical axis).
The line of perfect agreement is plotted as the solid black line.
The points are colored by the difference in the total \MLFF{} potential energy (in meV/atom) after relaxation and prior to relaxation.
The line of best fit is $V_\mathrm{\MLFF} \approx 1.009 \times V_\mathrm{GNoME} - 0.102$, with $R^2 \approx 0.998$.
+ }
\label{fig:GNoME_relax}
\end{figure}
The recent GNoME database \cite{merchant_scaling_2023} published approximately 381,000 stable materials which could not be matched to any experimental structure in the Inorganic Crystal Structure Database (ICSD) \cite{zagorac_JAC_2019}, nor any public repository of theoretical structures such as the Open Quantum Materials Database (OQMD) or Materials Project.
An analysis of the local coordination environment of these structures using the CrystalNN algorithm \cite{pan_IC_2021} reveals 21,300 structures with maximum coordination number greater than 16.
To evaluate the performance of \MLFF{} against the GNoME UIP which generated these structures, we have re-relaxed 1,199 structures with a predicted maximum coordination number greater than 20.
The {\tt FrechetCellFilter} class in the Atomic Simulation Environment (ASE) \cite{larsen2017atomic} was used to relax structures until the maximum (absolute) Cartesian component of any \MLFF{} inter-atomic force was less than \SI{e-5}{eV\per\Angstrom}.
No dispersion correction was used to augment MACE, consistent with the lack of a dispersion correction in GNoME.
All relaxations were successful, however two boron-dominated structures (GNoME identifiers 00edc694b5 and fc5cb3e024) took significantly longer than the other structures to reach this tolerance.
The MACE-relaxed structures are highly similar to the GNoME ones.
\Cref{fig:GNoME_relax} plots the volumes of these structures from the GNoME database before and after relaxation with MACE.
In virtually all cases, MACE relaxes structures to an indistinguishable configuration.
Only 19 structures (1.6\%) undergo an energy lowering of greater than \SI{10}{meV\per atom} during relaxation.
22 structures underwent a change of space group during relaxation, in all cases an increase in symmetry.
After relaxation, the maximum predicted coordination number changed for 135 structures (11\%), and decreased in all cases.
These observations cumulatively suggest that \MLFF{} performs comparably to the GNoME UIP for these highly-coordinated structures, and yields equilibrated structures of equal or higher symmetry than the GNoME UIP.
\subsubsection*{Similarity statement}
By construction, the materials in the GNoME set do not exist in the MP dataset.
The GNoME set was constructed in part by substituting elements on structures that originated in the Materials Project.
However, some of the GNoME materials exist in the same chemical space as those in MP.
For the 1,199 GNoME materials considered here, 131 (10.9\%) contain exactly the same elements as structures in MP.
The chemical space of maximum overlap, Cu-La-Zn, contains four GNoME (7ee54b7a37, 3422b9acb0, f36439bb45 and 68a5e7535f) and two MPtrj (mp-1223296 and mp-1093834) materials.
We provide two JSON-format dictionaries, \verb|MPtraj_chem_env.json.gz|, and \verb|GNoME_chem_env_w_ovlp.json.gz|, which tabulate the chemical environments spanned by MP and the GNoME subset, respectively.
The GNoME subset is further categorized by overlap with MP.
\subsubsection*{Performance summary}
Excellent energy prediction of WBM and GNoME hypothetical materials, and able to newly discover stable materials as validated by DFT.
\clearpage
\subsection{Alanine Tripeptide free energy surface}\label{sec:ala}
\begin{figure}[htbp!]
\centering
\begin{subfigure}[c]{0.4\linewidth}
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figures_ala_tripeptide/mace_mp_ala_render.pdf}
\end{subfigure}
\qquad
\begin{subfigure}[c]{0.4\linewidth}
\centering\includegraphics[width=\linewidth,keepaspectratio]{figures_ala_tripeptide/umace_ala_tripeptide_fes_vertical.pdf}
\label{fig:mace_vap}
\end{subfigure}
\label{fig:ala_tripeptide_fes}
\caption{Left: Alanine tripeptide solvated in explicit water. Right: Free energy surface calculated from \SI{300}{\pico\second} of biased sampling with \MLFF{} (Top) and the classical empirical forcefield AMBER14/TIP3P (Bottom)}
\end{figure}
In this section, we investigate the ability of \MLFF{} to construct the free energy surface of a simple peptide. We simulate a periodic box containing an alanine tripeptide solvated with explicit water for \SI{300}{\pico\second} without D3 correction. Sampling of the central backbone torsions was accelerated by metadynamics, as implemented in OpenMM, using an initial Gaussian height of \SI{1}{kJ\per mol} and a temperature scaling factor of 10. Whilst we find that the dynamics appear physically reasonable, \MLFF{} fails to accurately identify the correct minima. Gaussian fits to experimental NMR J-coupling indicate the secondary structure is predominantly a polyproline II conformation ($\phi=75^{\circ}$, $\psi=150^{\circ}$), whereas this region shows little population with \MLFF{} \cite{zhangMolecularDynamicsForce2020a}.
\subsubsection*{Similarity statement}
The MP dataset contains 99 structures containing exclusively the elements H, C, O and N. Based on UMAP analysis, we observe that all atomic environments in the example system are similar to environments in the training set. On closer inspection, we find that the most similar environments to the majority of the example configurations are clusters of water, ammonia and NO\textsubscript{2} molecules. Several examples contain oxygen-bearing molecules near water molecules, allowing for sampling of hydrogen bonding. We also find configurations containing clusters of large aromatic compounds containing carbon, nitrogen, oxygen and hydrogen. Interestingly, we also find a large cluster containing diverse configurations of small C,H,N,O-containing fragments, including carboxyl and amide fragments, \textit{e.g.} mp-997182.
\subsubsection*{Performance summary}
Stable MD at ambient conditions, but incorrect free energy of conformers.
\clearpage
\subsection{Molecule-Surface Interactions}\label{sec:molecule_surf_interactions}
\subsubsection{Adsorption energies - S24 dataset}
\begin{figure}[hbtp!]
\centering\includegraphics[width=0.9\linewidth]{figures_mol_surf_int/S24.pdf}
\caption{\label{fig:s24_dataset}Comparison between DFT (PBE-D3(BJ), red crosses) and MACE-MP-0 (blue circles) adsorption energies calculated for a diverse set of surfaces consisting of covalent, metallic, and ionic bonds as well as the porous material classes: MOFs and zeolites. Filled circles indicate surfaces where MACE-MP-0 reaches chemical accuracy ($43\,$meV) agreement to DFT. The inset shows that there is a strong correlation between DFT and MACE-MP-0 adsorption energies (Pearson correlation coefficient of 0.93). }
\end{figure}
Describing the interaction between a molecule and a surface with first-principles accuracy is central towards designing new and improved materials for heterogeneous catalysis, gas storage and separation, and many more~\cite{norskovComputationalDesignSolid2009a}.
Here, we test the accuracy of MACE-MP-0 (with Becke-Johnson D3 dispersion correction) for a set of prototypical systems found in surface chemistry, encompassing metallic, covalent and ionic-bonded surfaces, together with porous metal-organic frameworks (MOFs) and zeolites.
The structures used within this work - dubbed the S24 dataset - were taken from an amalgamation of published~\cite{al-hamdaniPropertiesWaterBoron2017a,brandenburgPhysisorptionWaterGraphene2019,ehlertCO2GrapheneBenchmarking2023a,tsatsoulisReactionEnergeticsHydrogen2018a,tsatsoulisComparisonQuantumChemistry2017,yeInitioSurfaceChemistry2023,lustembergVibrationalFrequenciesCeriumOxideBound2020c,shiManyBodyMethodsSurface2023a,doi:10.1126/science.abj0890,doi:10.1021/acscatal.2c05493,https://doi.org/10.1002/anie.202013671} and unpublished works.
Fig.~\ref{fig:s24_dataset} summarizes the computed PBE-D3(BJ) energies using \textsc{pymatgen} to generate VASP~\cite{VASP1,VASP2,VASP3} inputs with the \texttt{MPRelaxSet} settings.
We observe excellent performance of MACE-MP-0 across the range of surfaces, with a mean absolute deviation (MAD) of $138\,$meV.
Performance is particularly excellent for the ionic surfaces (with an MAD of $78\,$meV) and zeolites (with an MAD of $63\,$meV), with many systems reaching within chemical accuracy (${\sim}43\,$meV) against the DFT reference.
Performance is worse for the surfaces with metallic and covalent character, where the systems with the highest errors appear to be for H$_2$ adsorption onto Si(100) and Cu(111) surfaces with (absolute) differences of 710 and $519\,$meV to the DFT reference, respectively.
Rather than simple (molecular) physisorption, H$_2$ dissociates onto both of these surfaces and this appears to represent a challenging case for MACE-MP-0 to handle.
\subsubsection*{Similarity statement}
The S24 dataset contains 72 structures comprising 24 adsorbates, surfaces, and adsorbate-surface combinations. Based on the UMAP analysis, the elemental compositions of the adsorbates occur 6 to 27 times, the surfaces 0 to 9043 times and the adsorbate-surface combinations 0 to 4896 times on their own or along with other elements. Importantly, the training dataset does not contain any gas-phase molecules, surface-truncated models, or MOFs. For bare surfaces, on average, similar element compositions occur in the training data as follows: 3997 covalent, 5805 metallic, 2859 ionic, 46 MOFs, and 88 zeolites. For adsorbate-surface combinations, on average, the number of similar element compositions occurring in the training data are lowered to: 1492 covalent, 340 metallic, 499 ionic, 46 MOFs, and 59 zeolites. There is no clear correlation between the number of similar training data for a surface type and the accuracy of MACE-MP-0, with zeolites performing far better than metallic or covalent systems.
\subsubsection*{Performance summary}
Good agreement is generally achieved for adsorption energies, with excellent performance (close to chemical accuracy) for adsorption onto ionic surfaces and porous materials.
Performance appears to become poorer with dissociative chemisorption of H$_2$, albeit still reproducing the adsorption energy to the correct order of magnitude.
\clearpage
\subsubsection{Relative energies - OC157 dataset}
\begin{figure}[!ht]
\includegraphics[width=6.69in]{figures_mol_surf_int/OC.pdf}
\caption{\label{fig:oc_dataset}Comparison between the MACE-MP-0 and DFT (PBE-D3(BJ)) in predicting the relative energies between three structures across 157 molecule-surface combinations. This contains a highly diverse set of molecule-surface combinations which span 54 elements of the periodic table involving up to three elements per surface.}
\end{figure}
Identifying the most stable structures for a molecule-surface system is pivotal towards predicting the activity and selectivity of a catalyst, facilitating rational design of new catalyst materials~\cite{norskovDensityFunctionalTheory2011b}.
We compare MACE-MP-0 against \texttt{MPRelaxSet} DFT, both with Becke-Johnson damped D3 dispersion added, at predicting the relative energies between 3 structures for 157 molecule-surface combinations.
These surfaces were taken from the Open Catalyst Challenge 2023~\cite{chanussotOpenCatalyst20202021,tranOpenCatalyst20222023}, using structures generated by the baseline EquiformerV2 model trained on the \textsc{OC20-S2EF-2M} dataset.
While 200 molecule-surface combinations were originally provided, we have excluded those containing oxygen (O) in combination with several transition metals (Co, Cr, Fe, Mn, Mo, Ni, V and W) as this leads to complications with the Hubbard U correction (see main text) with \texttt{MPRelaxSet} settings.
Fig.~\ref{fig:oc_dataset} shows the 471 relative energies ($\Delta_{12}$, $\Delta_{13}$ and $\Delta_{23}$) for each of the remaining 157 molecule-surface systems, where $\Delta_{XY}$ is the relative energy between structure $X$ and $Y$, predicted with MACE-MP-0 and calculated with DFT.
Overall, we observe a moderately strong correlation between MACE-MP-0 and DFT on the relative energies, providing a Pearson correlation coefficient of 0.83 and an MAD of $0.42$ eV.
In particular, out of the 157 molecule-surface combinations, the lowest DFT energy configuration was correctly identified by MACE-MP-0 for 121 of the surfaces.
\subsubsection*{Similarity statement}
The molecule-surface combinations in the OC157 dataset cover all elements up to and including Bi (atomic number 83), except He, Li, Be, B, F, Ne, Mg, Cl, Ar, Br, Kr, I, Xe, Ba and all lanthanoids. The UMAP analysis shows that 126 of the 157 molecule-surface combinations have no similarity to the training dataset, with a further 20 having less than 10 and only 3 having more than 20 similar compositions.
\subsubsection*{Performance summary}
Moderate performance at predicting the absolute value of the relative energies between various molecule-surface configurations.
Lowest energy (DFT) structure correctly predicted in 77\% of the cases.
\clearpage
\subsection{Computational efficiency (twenty-element alloy)}\label{sec:timings}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\textwidth,keepaspectratio]{figures_HEA/hea-lowres.pdf}
\caption{Computational performance for NPT dynamics in LAMMPS, measured for the 20-element Cantor alloy at ambient pressure and \SI{300}{\kelvin}. The black curve indicates the performance on a single GPU, without domain decomposition, where the full structure is passed from LAMMPS to MACE as a periodic graph. The colored curves demonstrate weak scaling for 120 or 500 atoms per GPU; the dashed parts of those curves signify a transition to domain decomposition. The latter requires a different MACE evaluation strategy which (at present) incurs significant additional cost; the isolated colored dots show the performance when this strategy is used with a single GPU.}
\label{fig:HEA}
\end{figure}
This section demonstrates the current computational performance of the medium \MLFF{} (without dispersion correction), while also illustrating stable dynamics for a large, diverse system. High entropy alloys are multicomponent mixtures with at least four or five distinct elements, where each component appears in non-trace proportions. Here, we consider the 20-component alloy investigated by Cantor\cite{cantor2004microstructural}, which contains equal amounts of Mn, Cr, Fe, Co, Ni, Cu, Ag, W, Mo, Nb, Al, Cd, Sn, Pb, Bi, Zn, Ge, Si, Sb, and Mg. [See also recent computational work by Ceriotti and co-workers \cite{lopanitsyna2023modeling, mazitov2023surface}.] Our simulations were performed with LAMMPS \cite{thompson2022lammps}, utilizing its Kokkos extensions\cite{trott2022kokkos}, on NVIDIA A100 GPUs. For these tests we used the ``medium'' sized model (as for other applications) and 64-bit floating point operations. The ``small'' model is about a factor of two faster.
\Cref{fig:HEA} shows the computational results. For the single-GPU simulations (black curve), we began with supercells of a 20-atom FCC primitive cell, then performed a short melt-quench procedure over \SI{20}{\pico\second} at ambient pressure (which involved heating to \SI{4000}{\kelvin}, holding at \SI{4000}{\kelvin}, cooling to \SI{300}{\kelvin}, and holding at \SI{300}{\kelvin}). We then measured the computational throughput over \SI{5}{\pico\second} of NPT dynamics at ambient pressure and \SI{300}{\kelvin}. The colored curves demonstrate multi-GPU performance with domain decomposition (weak scaling). At present, this mode incurs significant additional cost, only becoming favorable for very large systems. These \SI{3}{\pico\second} simulations were initialized by replicating 120- or 500-atom cells. In all cases, the primary aim was to measure performance for a reasonably well-mixed system, and we did not attempt to reach full equilibration. Moreover, the performance quoted here is provisional: substantial improvements are expected when using, for example, custom CUDA kernels for expensive operations, which are under development.
\subsubsection*{Similarity statement}
Of the 150k MP structures, roughly 100k have at least one of the 20 elements used in this example, but only 3461 have compositions drawn exclusively from that set.
Of this latter group, 193 are single-component structures, 2079 are binaries, 1155 are ternaries, and 34 are quaternaries. Moreover, no structure in the entire database has five or more of the 20 elements considered here.
While it is encouraging that the dynamics appear stable for such a diverse, out-of-sample composition, we expect that incorporating dedicated datasets like that reported in \cite{lopanitsyna2023modeling} could enhance quantitative predictions.
\subsubsection*{Performance summary}
Stable molecular dynamics for solid and liquid. In parallelisation, perfect weak scaling up to 32,000 atoms and 64 GPUs.
\clearpage
\section{Benchmarks}
\subsection{Phonons}\label{sec:phonons}
\begin{figure}[http!]
\centering
\begin{subfigure}[b]{0.48\linewidth}
\includegraphics[width=\linewidth,keepaspectratio]{figures_phonons/mp-2758-bs-dos-pbe-vs-mace-y7uhwpje.pdf}
\caption{mp-2758 phonon bands and DOS}
\label{fig:bs-dos-pbe-vs-mace-mp-2758}
\end{subfigure}
\hfil
\begin{subfigure}[b]{0.48\linewidth}
\includegraphics[width=\linewidth,keepaspectratio]{figures_phonons/mp-2998-bs-dos-pbe-vs-mace-y7uhwpje.pdf}
\caption{mp-2998 phonon bands and DOS}
\label{fig:bs-dos-pbe-vs-mace-mp-2998}
\end{subfigure}
\begin{subfigure}[b]{0.48\linewidth}
\includegraphics[width=\linewidth,keepaspectratio]{figures_phonons/mp-27214-bs-dos-pbe-vs-mace-y7uhwpje.pdf}
\caption{mp-27214 phonon bands and DOS}
\label{fig:bs-dos-pbe-vs-mace-mp-27214}
\end{subfigure}
\hfil
\begin{subfigure}[b]{0.48\linewidth}
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figures_phonons/parity-pbe-vs-ml-band-width.pdf}
\put(-80,20){\includegraphics[width=0.36\linewidth,keepaspectratio]{figures_phonons/imaginary-freq-confusion-matrix.pdf}}
\caption{MACE vs PBE highest DOS frequency parity plot}
\label{fig:parity-highest-phonon-bands-freq}
\end{subfigure}
\caption{
Comparison of PBE and \MLFF{} phonon band structures (BS) and density of states (DOS).
PBE reference data was taken from PhononDB \cite{togo_firstprinciples_2023,togo_implementation_2023}.
\subref*{fig:bs-dos-pbe-vs-mace-mp-2758}) and \subref*{fig:bs-dos-pbe-vs-mace-mp-2998}) show examples of particularly good phonon bands while \subref*{fig:bs-dos-pbe-vs-mace-mp-27214}) is a particularly bad example.
\subref*{fig:parity-highest-phonon-bands-freq}) Parity plot of MACE vs PBE highest band frequency showing excellent agreement across diverse materials.
The inset shows the confusion matrix for \MLFF{} vs PBE presence of imaginary modes (at any $k$-point). \MLFF{} achieves dynamic stability classification accuracy of 80\% on our dataset of 97 materials of which 56.7\% are PBE-unstable.
}
\end{figure}
Accurate modeling of phonons is essential for determining dynamic stability of crystals, as well as entropic contributions to the free energy \cite{stoffel_ab_2010, bartel_review_2022}.
Both are important factors in the discovery of new materials.
Harmonic phonons are typically calculated from the restoring force on atomic displacements and require highly accurate force predictions to be physical.
An ML potential trained on PBE forces should be able to reproduce PBE lattice vibrations \cite{george_combining_2020, morrow_how_2023}.
To assess the accuracy of \MLFF{} restoring forces, we compare phonon modes predicted via the finite-displacement method as implemented in Phonopy\cite{togo_firstprinciples_2023,togo_implementation_2023} and atomate2~\cite{atomate2} with the PhononDB phonon database \cite{togo_firstprinciples_2023, togo_implementation_2023}.
We restrict our analysis to materials in which magnetism and U-correction do not play a role, since the phonon reference database was calculated at PBE level of theory without consideration of magnetism nor U-corrections.
Additional inaccuracies in the comparison may stem from not-fully-converged kinetic energy cutoffs of the plane-wave expansion or insufficient $k$-point density. Based on our analysis, the phonon reference calculations used the VASP-recommended and at the same time sufficiently high kinetic energy cutoff (1.3*ENMAX of the highest ENMAX indicated in the pseudopotential files for the PAW method for each compound). In most cases, the reciprocal $k$-point density used matches that in \texttt{MPRelaxSet} for structure optimization and force calculations. However, there are some cases where the maximum distance between adjacent points along a reciprocal axis used is slightly lower compared to \texttt{MPRelaxSet} by an average $14\pm13$ and $19\pm16$ \% for structure optimization and force calculations, respectively. As these are small deviations, the PhononDB results can be used as reasonable high-throughput reference data.
To minimize the discrepancy between \MLFF{} and Togo results, we simulate identical supercells as found in the phonon database.
In addition, we remove the non-analytic corrections derived from Born charges which are unavailable from electronic structure-less ML potentials.
While MP also computed phonons for several thousand compounds, we do not use it as reference data due to MP using PBEsol, a functional incompatible with the MPtrj training set.
As examples, we show two compounds out of our set of 97 materials where DFT and \MLFF{} agree well (\cref{fig:bs-dos-pbe-vs-mace-mp-2758,fig:bs-dos-pbe-vs-mace-mp-2998}), and
one of the compounds of largest qualitative disagreement (\cref{fig:bs-dos-pbe-vs-mace-mp-27214}).
The parity plot of \MLFF{} vs PBE highest-frequency phonon modes in \cref{fig:parity-highest-phonon-bands-freq} reveals that \MLFF{} typically slightly underestimates the highest overall frequency in the phonon band structure (henceforth referred to as band width, which correlates with the stiffness of the material's strongest bond) by \num{1}--\SI{2}{\tera\hertz} in comparison to DFT.
\MLFF{} band widths have excellent predictive power of $R^2 = 0.99$.
The largest absolute discrepancy between PBE vs \MLFF{} band width is around \SI{11}{\tera\hertz} (25\%) for \ch{Cs3HSeO8} (mp-23980).
Highly specialised models for one element can achieve much lower errors of \num{0.1}--\SI{0.2}{\tera\hertz} over the whole band structure\cite{george_combining_2020}. \MLFF{} is far from achieving such accuracy.
However, the deviations are close to results from models that specifically train only on the highest phonon peak on MatBench \cite{dunn_benchmarking_2020}.
Here, the best models arrive at mean absolute errors of \num{0.8}--\SI{1}{\tera\hertz} for the last phonon DOS peak, which is a similar but \textit{not} identical quantity to the band width.
We emphasize that neither \MLFF{} nor its training data was tailored in any way for phonon predictions.
Yet still, \MLFF{} band structures achieve at least qualitative agreement with PBE for most materials, which can be sufficient for many practical applications (\textit{e.g.} models that screen for low or high thermal conductivity \cite{agne_minimum_2018}).
As shown in the confusion matrix inset to \cref{fig:parity-highest-phonon-bands-freq}, the model also provides useful signal on the presence of imaginary modes.
The presence of such lattice vibrations indicates that the structure is dynamically unstable, meaning it can transform into a lower-energy structure by displacing atoms along the corresponding vibrational eigenvector.
\MLFF{} achieves 80\% accuracy on binary dynamic stability classification with a PBE unstable rate of 56.7\% (a nearly balanced data set). While this clearly outperforms dummy accuracy of 50\%, it also leaves room for significant improvement.
The parameter \texttt{Tol=0.1} in the inset caption determines how far below zero a band frequency (in THz) must reach to count as imaginary. For example, a mode at \SI{-0.09}{\tera\hertz} would not be considered dynamically unstable while \SI{-0.11}{\tera\hertz} would.
A caveat to the above analysis is that the PBE unstable labels we use as ground truth may themselves contain false positives. That is, any unconverged values of \texttt{ENCUT} and $k$-point density may result in incorrect dynamically unstable PBE labels that would stabilize by reaching numerical convergence.
As is, \MLFF{} can already serve as a useful pre-screening filter for dynamically unstable materials, especially given the lower false positive (7.2\%) than false negative rate (12.4\%).
This means \MLFF{} is less likely to predict materials as unstable that are stable than vice versa and hence is biased towards keeping materials in the candidate pool.
We also emphasize the low computational cost of these predictions.
Generating a complete phonon band structure on an Apple M2 Max CPU takes approximately \SI{30}{\second}.
Running \MLFF{} on a supercomputer could easily generate approximate phonon bands for every material in MP or other large databases.
However, the accuracy of \MLFF{} harmonic phonons is still insufficient to be a reliable substitute for DFT.
We share all code and data for this phonon analysis at \url{https://github.com/janosh/ffonons}.
The code was written in a model-agnostic way.
Subjecting other ML models that conform to the ASE calculator API to the same analysis should be straightforward.
\subsubsection*{Similarity statement}
The structures whose phonon predictions we analyzed are all part of the training set.
However, the supercells, including single-atom perturbations performed by the finite-displacement method are not.
\clearpage
\subsection{Bulk and Shear Moduli}\label{sec:bulk_moduli}
\begin{figure}[htbp!]
\centering \includegraphics[width=\textwidth,keepaspectratio]{figs/mace-yuan-199-epoch-bulk-and-shear-moduli-latest-lowres.pdf}
\caption{Comparison between \MLFF{}- and MP-calculated (a) bulk moduli and (b) shear moduli for approximately 10,000 materials stored in the MP Database. The dashed line is a parity line (\textit{i.e.} the target distribution of data). For bulk moduli, note that a single point for mp-721759 (face-centered cubic nitrogen) was excluded, as \MLFF{} predicted an unphysical high bulk modulus (\SI{\geq 600}{\giga\pascal}). For shear moduli, note that 26 points were excluded from this plot, as \MLFF{} predicted an unphysically high (\SI{\geq 600}{\giga\pascal}) or low (\SI{\leq -50}{\giga\pascal}) shear modulus; this includes mp-\{1002105, 1002117, 1002165, 1002206, 1002207, 10030, 1008669, 1009019, 1009485, 1009731, 1077353, 13174, 1428312, 14549, 2458, 30083, 570354, 631377, 631475, 631524, 972442, 976419, 980204, 989573, 999086, 999263\}.}
\label{fig:bulk-and-shear-moduli}
\end{figure}
\MLFF{} was benchmarked against the elastic properties of over 10,000 materials stored in the MP database. Being able to capture elastic properties such as bulk and shear modulus - which depend on the second derivatives of the energy with respect to strain - demonstrates a more precise ability to capture the potential energy surface.
Specifically, \MLFF{} was used to calculate the Voigt-Reuss-Hill average \cite{voigt_lehrbuch_1910, reuss_berechnung_1929, hill_elastic_1952} bulk modulus
and shear modulus
as derived from stress-strain relations. The initial structures used for these calculations were the relaxed PBE \cite{perdew1996generalized} structures from MP; these structures were then re-relaxed using the \MLFF{} model, and then deformed. Specifically, a total of 4 strain magnitudes were used along 6 independent strain modes (in Voigt notation): $\boldsymbol{\epsilon} \in [\epsilon_{11}, \epsilon_{22}, \epsilon_{33}, \epsilon_{44}, \epsilon_{55}, \epsilon_{66}]$. For $\epsilon_{11}$, $\epsilon_{22}$, and $\epsilon_{33}$, the strain magnitudes were $\pm 0.01$ and $\pm 0.005$. For $\epsilon_{44}$, $\epsilon_{55}$, and $\epsilon_{66}$, the strain magnitudes were $\pm 0.06$ and $\pm 0.03$. These calculations were performed using the \texttt{elasticity} module from the \texttt{MatCalc} package \cite{Riebesell_MatCalc_2023}. Hence, all of these predictions are based on equilibrium, bulk crystals alone. Moreover, to filter out likely unphysical DFT predictions, elastic properties from MP were excluded from this analysis if the DFT VRH average bulk or shear modulus are less than \SI{-50}{\giga\pascal} or greater than \SI{600}{\giga\pascal}. Note that the data excluded due to unphysical DFT-based properties are distinct from the data not plotted in \cref{fig:bulk-and-shear-moduli} due to poor \MLFF{} predictions.
Results comparing MP and \MLFF{} bulk moduli with MAE of \SI{15.73}{\giga\pascal} and $R^2$ of 0.85 are shown in \cref{fig:bulk-and-shear-moduli}(a). This compares favorably to the $R^2$ value of 0.757 reported for M3GNet \cite{chen_universal_2022}. Similarly, results for shear moduli are shown in \cref{fig:bulk-and-shear-moduli}(b). \MLFF{} struggles to predict shear properties, likely due to a lack of sheared structures in MP.
\subsubsection*{Similarity statement}
None of the DFT deformation calculations contained in MP are present in the MPtrj training set used for \MLFF{}.
\clearpage
\subsection{Cohesive energies} \label{sec:lattice-energies}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_cohesive_energies/S66_1.pdf}
\caption{Comparison between DFT (red circles) and \MLFF{} (blue squares) calculated binding energies of the S66 dimers. The binding energies are divided by the number of atoms in each dimer. The lines are guides for the eye.
}
\label{fig:s66}
\end{figure}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_cohesive_energies/X23_1.pdf}
\caption{Comparison between DFT (red circles) and \MLFF{}-calculated (blue squares) lattice energies of the X23 dataset. The lines are guides for the eye.
}
\label{fig:x23}
\end{figure}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figures_cohesive_energies/DMCICE13.pdf}
\caption{Comparison between DFT (red circles) and \MLFF{} -calculated (blue squares) lattice energies of the DMC-ICE13 dataset. We report both the absolute lattice energies (left), \textit{i.e.} the energy per molecule of each crystalline phase with respect to the gas phase, and the relative lattice energies (right), \textit{i.e.} the lattice energy relative to ice Ih.
}
\label{fig:dmcice13}
\end{figure}
In this section, we benchmark \MLFF{} against the cohesive energies of widely used data sets of molecules and molecular crystals, S66\cite{S66} and X23\cite{x23_rt}. S66 is a dataset comprising 66 molecular complexes at their reference equilibrium geometries, designed to cover the most common types of noncovalent interactions in biomolecules while keeping a balanced representation of dispersion and electrostatic contributions. X23 is a dataset of 23 organic molecular crystals. Furthermore, we analyze the relative stabilities of the ice polymorphs in DMC-ICE13\cite{dmcice13}. The DFT calculations were performed using VASP\cite{VASP1,VASP2,VASP3} with the PBE functional and D3 dispersion correction with Becke-Johnson damping. The energy cutoff is \SI{520}{eV}. Gas phase calculations were performed at the $\Gamma$ point in a \SI{25}{\Angstrom} cubic box. Solid phase calculations were performed with a $4\times 4 \times 4$ k-point grid. The results comparing \MLFF{} and DFT binding energies of the S66 dimers, and the lattice energies of X23 and DMC-ICE13 are shown in \cref{fig:s66}, \cref{fig:x23}, and \cref{fig:dmcice13}. The MAE is approximately \SI{4}{meV/atom} for S66, \SI{160}{meV} for X23, and \SI{46}{meV} for the DMC-ICE13 absolute lattice energies. The relative stabilities of the ice polymorphs are correctly captured with an MAE of \SI{4}{meV} on the relative lattice energies.
\subsubsection*{Similarity statement}
S66 and X23 comprise dimers and molecular crystals containing C, H, N, or O atoms. Ice polymorphs contain H and O atoms. The MP database contains 73799 structures with O atoms, 10312 structures with H atoms, 11356 structures with N atoms, and 9043 structures with C atoms. The database contains 6 structures matching an exact chemical formula in S66; these are \ch{H4O2}, \ch{C4HO8O4}, \ch{C8H8} and \ch{C4H4}. 16 structures match an exact chemical formula in X23; these are \ch{C8H16O8}, \ch{C20H32}, \ch{H12N4}, \ch{C4O8}, \ch{C8H16N16} and \ch{C2H8N4O2}. 9 structures match an exact chemical formula in DMC-ICE13; these are \ch{H24O12} and \ch{H16O8}. Overall, the database contains 630 structures with organic molecules and 1342 structures with water molecules. We provide \verb|s66_chemiscope_input.json|, \verb|x23_chemiscope_input.json|, and \verb|dmcice13_chemiscope_input.json| to help visualize the interactive UMAP on \url{chemiscope.org}.
\subsection{Atomization energies and lattice constants of solids}
\label{sec:atomization-energies}
In the following section, we benchmark \MLFF{} against the atomization energies and lattice constants of a set of solids. The details of the DFT calculations are the same as in \cref{sec:lattice-energies}. Solid phase DFT total energies were computed with a $16\times 16 \times 16$ k-point grid. \MLFF{} and DFT atomization energies (in eV/atom) are reported in \cref{tab:atomization-energies}. The MAE is \SI{0.732}{eV/atom}. Significant errors in the atomization energies of solids arise from the lack of isolated atom energies in the training set. For this reason, we also report the values of the atomization energies obtained by using the \MLFF{} energy of the solid and the DFT energies of the isolated atoms. We refer to these values as `corrected \MLFF{}'. The MAE for the corrected values is approximately \SI{0.055}{ eV/atom}. Lattice constants are computed on equilibrium structures obtained by geometry relaxation with a force convergence threshold of \SI{0.03}{eV\per\Angstrom}. \MLFF{} and DFT lattice constants (in \unit{\Angstrom}) are reported in \cref{tab:lattice-constants}. The MAE is \SI{0.03}{\Angstrom}.
\subsubsection*{Similarity statement}
All the tested solids are contained in the database, with an exact matching chemical formula for 71 structures. We provide \verb|solids_chemiscope_input.json| to help visualize the interactive UMAP on \url{chemiscope.org}.
\begin{table}
\centering
\begin{tabular}{cccccccc}
& \multicolumn{6}{c}{Atomization energies of solids} & \\ \hline \hline
& DFT & \MLFF{} & $\Delta$ & $\Delta/$DFT \% & corrected \MLFF{} & $\Delta$ & $\Delta$/DFT \% \\ \hline
Ag & 3.098 & 2.823 & 0.275 & 9 & 3.205 & -0.107 & 3.5 \\
Pd & 4.398 & 4.120 & 0.278 & 6 & 4.356 & 0.042 & 1.0 \\
Rh & 6.404 & 5.853 & 0.551 & 9 & 6.405 & -0.001 & 0.02 \\
Li & 1.782 & 1.157 & 0.625 & 35 & 1.784 & -0.002 & 0.1 \\
Na & 1.244 & 0.947 & 0.297 & 24 & 1.240 & 0.004 & 0.3 \\
K & 0.987 & 0.437 & 0.550 & 56 & 1.002 & -0.015 & 1.5 \\
Rb & 0.881 & 0.231 & 0.650 & 74 & 0.855 & 0.026 & 3.0 \\
Cs & 0.808 & 0.098 & 0.710 & 88 & 0.793 & 0.015 & 1.9 \\
Ca & 2.138 & 2.041 & 0.097 & 5 & 2.219 & -0.081 & 3.8 \\
Sr & 1.810 & 1.751 & 0.059 & 3 & 1.841 & -0.031 & 1.7 \\
Ba & 2.079 & 1.871 & 0.208 & 10 & 2.034 & 0.045 & 2.2 \\
Al & 3.891 & 3.373 & 0.518 & 13 & 3.862 & 0.029 & 0.7 \\
Cu & 4.075 & 3.749 & 0.326 & 8 & 4.435 & -0.359 & 9 \\
Si & 4.922 & 4.366 & 0.556 & 11 & 4.840 & 0.082 & 1.7 \\
Ge & 4.036 & 3.696 & 0.340 & 8 & 4.041 & -0.005 & 0.1 \\
C & 8.033 & 7.574 & 0.459 & 6 & 8.015 & 0.018 & 0.2 \\
LiF & 4.267 & 1.800 & 2.467 & 58 & 4.283 & -0.016 & 0.4 \\
NaF & 3.798 & 1.676 & 2.122 & 56 & 3.825 & -0.027 & 0.7 \\
NaCl & 3.129 & 1.799 & 1.330 & 43 & 3.122 & 0.007 & 0.2 \\
MgO & 4.710 & 4.196 & 0.514 & 11 & 4.726 & -0.016 & 0.3 \\
SiC & 5.762 & 4.847 & 0.915 & 16 & 5.761 & 0.001 & 0.02 \\
GaAs & 2.548 & 2.247 & 0.301 & 12 & 2.572 & -0.024 & 0.9 \\
LiCl & 3.377 & 1.716 & 1.661 & 49 & 3.374 & 0.003 & 0.1 \\ \hline
& & & MAE & & & MAE & \\
& & & 0.687 & & & 0.041 &
\end{tabular}
\caption{Comparison between DFT and \MLFF{}-calculated atomization energies of solids. The column $\Delta$ reports the difference between DFT and \MLFF{} values. Energies are reported in \unit{eV/atom}.}\label{tab:atomization-energies}
\end{table}
\begin{table}
\centering
\begin{tabular}{ccccc}
& \multicolumn{4}{c}{Lattice constants of solids} \\ \hline \hline
& DFT & \MLFF{} & $\Delta$ & $\Delta/$\% \\ \hline
Ag & 4.082 & 4.093 & -0.011 & 0.27 \\
Pd & 3.891 & 3.907 & -0.016 & 0.41 \\
Rh & 3.76 & 3.812 & -0.052 & 1.38 \\
Li & 3.352 & 3.302 & 0.05 & 1.49 \\
Na & 4.107 & 4.068 & 0.039 & 0.95 \\
K & 5.191 & 5.145 & 0.046 & 0.89 \\
Rb & 5.572 & 5.463 & 0.109 & 1.96 \\
Cs & 6.106 & 6.07 & 0.036 & 0.59 \\
Ca & 5.463 & 5.421 & 0.042 & 0.77 \\
Sr & 5.908 & 5.948 & -0.04 & 0.68 \\
Ba & 4.976 & 4.938 & 0.038 & 0.76 \\
Al & 4.002 & 3.993 & 0.009 & 0.22 \\
LiF & 3.995 & 4.017 & -0.022 & 0.55 \\
NaF & 4.619 & 4.628 & -0.009 & 0.19 \\
NaCl & 5.585 & 5.575 & 0.01 & 0.18 \\
MgO & 4.203 & 4.208 & -0.005 & 0.12 \\
Si & 5.434 & 5.399 & 0.035 & 0.64 \\
Ge & 5.719 & 5.703 & 0.016 & 0.28 \\
GaAs & 5.69 & 5.66 & 0.03 & 0.53 \\
Cu & 3.568 & 3.561 & 0.007 & 0.20 \\
C & 3.562 & 3.556 & 0.006 & 0.17 \\
LiCl & 5.056 & 5.019 & 0.037 & 0.73 \\
SiC (a) & 3.072 & 3.075 & -0.003 & 0.10 \\
SiC (c) & 5.029 & 5.025 & 0.004 & 0.08 \\ \hline
& & & MAE & \\
& & & 0.03 &
\end{tabular}
\caption{Comparison between DFT and \MLFF{}-calculated lattice constants of solids. The column $\Delta$ reports the difference between DFT and \MLFF{} values. Lattice constants are in $\text{Å}$.}\label{tab:lattice-constants}
\end{table}
\clearpage
\subsection{Reaction barrier heights}
\label{sec:reaction-barrier}
In the following section, we benchmark \MLFF{} against the reaction barrier heights of the CRBH20 datase\cite{crbh20}, comprising 20 barrier heights for the cycloreversion of heterocyclic rings. The set-up of the DFT calculations is the same as in \cref{sec:lattice-energies}. \MLFF{} and DFT barrier heights (in \unit{eV}) are reported in \cref{tab:barrier-heights}. The MAE is approximately \SI{0.3}{eV}.
\subsubsection*{Similarity statement} The CRBH20 dataset comprises barrier heights of organic molecules, containing H, O, C, N, S, and F atoms. We report the list of atoms with the number of structures in which they are contained in parentheses: O (73799), S (11972), H (10312), N (11356), F (11277), C (9043).
Overall, the database contains no structures matching an exact chemical formula in CRBH20. We provide \verb|crbh20_chemiscope_input.json| to help visualize the interactive UMAP on \url{chemiscope.org}.
\begin{table}
\centering
\begin{tabular}{ccccc}
\multicolumn{5}{c}{Barrier heights of CRBH20} \\ \hline \hline
& DFT & \MLFF{} & $\Delta$ & $\Delta/$DFT \% \\ \hline
1 & 1.7194 & 2.0593 & -0.3399 & 20 \\
2 & 1.9241 & 2.5655 & -0.6414 & 33 \\
3 & 1.7499 & 2.2417 & -0.4918 & 28 \\
4 & 1.8238 & 2.0893 & -0.2655 & 15 \\
5 & 1.7237 & 2.2944 & -0.5707 & 33 \\
6 & 1.5653 & 1.1929 & 0.3724 & 24 \\
7 & 1.0911 & 1.5189 & -0.4278 & 39 \\
8 & 1.8983 & 1.5607 & 0.3376 & 18 \\
9 & 1.5477 & 2.0802 & -0.5325 & 34 \\
10 & 1.7115 & 1.7171 & -0.0056 & 0 \\
11 & 1.7379 & 1.8844 & -0.1465 & 8 \\
12 & 2.0361 & 2.0365 & -0.0004 & 0 \\
13 & 1.8739 & 1.7503 & 0.1236 & 7 \\
14 & 1.976 & 1.6933 & 0.2827 & 14 \\
15 & 1.8865 & 1.8492 & 0.0373 & 2 \\
16 & 1.5741 & 0.9392 & 0.6349 & 40 \\
17 & 1.2587 & 1.0375 & 0.2212 & 18 \\
18 & 1.7497 & 1.3102 & 0.4395 & 25 \\
19 & 1.6989 & 1.6871 & 0.0118 & 1 \\
20 & 1.7654 & 1.2114 & 0.5540 & 31 \\ \hline
\multicolumn{3}{c}{} & MAE & \\
\multicolumn{3}{c}{} & 0.3 &
\end{tabular}
\caption{Comparison between DFT and \MLFF{}-calculated barrier heights for CRBH20. The column $\Delta$ reports the difference between DFT and \MLFF{} values. Energies are in eV.}\label{tab:barrier-heights}
\end{table}
\clearpage
\subsection{Homonuclear diatomics}\label{sec:dia}
\begin{figure*}[htbp!]
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figures_tks/homonuclear-medium-periodic-table.pdf}
\caption{Energies of homonuclear diatomics in vacuum. For the majority of the periodic table there is clear repulsion at small distances, except for some elements.}
\label{fig:homonuclear-2b}
\end{figure*}
Core repulsion is essential for stable modeling of atomic interactions, so atoms are prevented from coming too close together, especially when modeling high temperatures and pressures. Energies of all pairs of atoms have been evaluated with the model in vacuum, to test the 2-body interaction. The resultant curves for homo-nuclear diatomics are plotted in \cref{fig:homonuclear-2b}. Some elements have non-repulsive potential at small distances, and the noble gases \ch{He}, \ch{Ne}, \ch{Ar} -- which had negligible presence in the training set (\cref{fig:element-counts-ratio-by-occurrence}) -- are flat.
\clearpage
\section{Training Methods and Data Exploration}
\subsection{Training protocol}\label{sec:trainingprotocol}
The pretrained \MLFF{} interatomic potentials consists of multi-body message passing (interaction) layers. In each layer, the message is encoded in the irreducible representation basis with $C$ channels up to an angular frequency of order $L$. This is specified as \verb|(128x0e+128x1o+128x2e)| for 128 channels and $L=2$.
In each batch updating step, the weighted sum of Huber losses~\cite{Huber1992} of energy, forces, and stress incurred by all structures in a batch are averaged and back-propagated into the neural networks: \begin{equation}
\begin{aligned}
\mathcal{L} & = \frac{\lambda_E}{N_b} \sum_{b=1}^{N_b}\mathcal{L}_\text{Huber}\biggl(\frac{\hat{E}_b}{N_a}, \frac{E_b}{N_a}, \delta_E\biggr) + \frac{\lambda_F}{3\sum_{b=1}^{N_b}N_a} \sum_{b=1}^{N_b} \sum_{a=1}^{N_a}\sum_{i=1}^{3}\mathcal{L}^\star_\text{Huber}\biggl(\frac{\partial\hat{E}_b}{\partial {r}_{b,a,i}}, F_{b,a,i}, \delta_F\biggr) \\
& \hphantom{=}+ \frac{\lambda_\sigma}{9N_b} \sum_{b=1}^{N_b}\sum_{i=1}^3\sum_{j=1}^3\mathcal{L}_\text{Huber}\biggl(\frac{1}{V_b}\frac{\partial\hat{E}_b}{\partial {\varepsilon}_{b,ij}}, \sigma_{b,ij}, \delta_\sigma\biggr),
\end{aligned}
\end{equation} where $\lambda_E, \lambda_F, \lambda_\sigma$ are predetermined weights of energy, forces, and stress losses. $(\lambda_E, \lambda_F, \lambda_\sigma) = (1, 10, 100)$ is adopted. The energy fine-tuned small model (``small-energy'') is trained for additional 50 epochs with $(\lambda_E, \lambda_F, \lambda_\sigma) = (10, 1, 100)$. $N_b$ and $N_a$ are batch size and the number of atoms in each structure. Huber deltas $\delta_E = \delta_F = \delta_\sigma = 0.01$ are used. In particular, we use conditional Huber loss $\mathcal{L}^\star_\text{Huber}$ for forces, where the Huber delta $\delta_F$ is adaptive to the force magnitude on each atom. To be specific, the Huber delta $\delta_F$ decreases step-wise by a factor from 1.0 to 0.1 as the atomic force increases from \num{0} to \SI{300}{eV\per\Angstrom\per atom}. The Huber loss for forces can therefore be equivalently represented as:
\begin{equation}
\mathcal{L}^\star_\text{Huber}\left(\frac{\partial\hat{E}_b}{\partial {r}_{b,a,i}}, F_{b,a,i}, \delta_F\right) = \left\lbrace\begin{aligned}
\mathcal{L}_\text{Huber}(\dots, \delta_F)\,, & \qquad F_{b,a,i} \textup{ \textless } \ 100 \\
\mathcal{L}_\text{Huber}(\dots, 0.7\delta_F)\,, & \qquad 100 \le F_{b,a,i} \textup{ \textless } \ 200 \\
\mathcal{L}_\text{Huber}(\dots, 0.4\delta_F)\,, & \qquad 200 \le F_{b,a,i} \textup{ \textless } \ 300 \\
\mathcal{L}_\text{Huber}(\dots, 0.1\delta_F)\,, & \qquad F_{b,a,i} \ge 300 \\
\end{aligned}\right.
\end{equation}
Standardization of target variables (here energies, forces and stresses) with different scales has been proven to be important for weight initialization and training stability \cite{bishop1995neural}, in that a large spread of input or output will result in large and uneven weight values and cause model instability. After each message passing layer $k$, the node energies $\epsilon_a$ are scaled and shifted before sum pooling. The energy prediction of each structure therefore reads: \begin{equation}
\hat{E} = \sum_{a=1}^{N} \left[\sigma \left(\sum_{k=1}^K
\epsilon_a^{(k)}\right) + \mu_{Z_a}\right],
\end{equation} where $K$ denotes the total number of message passing layers and $\epsilon_a^{(k)}$ is the node energy of atom $a$ at $k$-th layer. $\mu_Z$ and $\sigma$ are the mean of atomic energies and the root mean square of the atomic forces computed over the training dataset. The predicted forces and stress are computed through PyTorch's automatic differentiation \texttt{torch.autograd} of total energy with respect to atomic positions and lattice strain tensor.
The pretrained \MLFF{} models are trained for 200 epochs with \numrange{40}{80} NVIDIA A100 GPUs across \numrange{10}{20} nodes on HPE (Hewlett Packard Enterprise) Cray EX supercomputer Perlmutter, maintained by National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory (LBNL).
\begin{figure}[htbp!]
\centering
\includegraphics[width=\textwidth,keepaspectratio]{figs/mace-mp-metrics.pdf}
\caption{Training curves of \MLFF{} models fitted to MPtrj data. (a) Loss (\cref{eq:loss}). (b-d) Mean absolute error (MAE) of (b) energy per atom, (c) force, and (d) stress per atom. All curves are evaluated on the validation set.}
\label{fig:metrics}
\end{figure}
\clearpage
\subsection{Exploration of the training data}
\label{sec:eda}
\begin{figure*}[htbp!]
\centering
\begin{subfigure}[b]{0.99\textwidth}
\includegraphics[width=\textwidth,keepaspectratio]{figs/mp-trj-element-counts-by-occurrence-log.pdf}
\caption{MPtrj training set element occurrence}
\label{fig:mp-trj-element-counts-by-occurrence-log}
\end{subfigure}
\begin{subfigure}[b]{0.99\textwidth}
\includegraphics[width=\textwidth,keepaspectratio]{figs/mp-trj-mp-ratio-element-counts-by-occurrence-normalized.pdf}
\caption{Normalized ratio of elements in MPtrj to MP, $\frac{\text{MPtrj / len(MPtrj)}}{\text{MP / len(MP)}}$}
\label{fig:mp-trj-mp-ratio-element-counts-by-occurrence}
\end{subfigure}
\caption{
The number of structures containing a given element in the MPtrj training set \cite{deng_chgnet_2023}.
MPtrj consists of multiple configurations from every relaxation trajectory in MP.
Some elements can require more ionic steps to relax than others.
To visualize this, \subref*{fig:mp-trj-mp-ratio-element-counts-by-occurrence} shows the overabundance of elements relative to the number of structures containing a given element in MP ground states (after normalizing by dividing each dataset by its number of structures).
That is, the factor 1.33 for hydrogen in \subref*{fig:mp-trj-mp-ratio-element-counts-by-occurrence} indicates that structures containing hydrogen were selected 33\% more frequently than the base prevalence of hydrogen in MP ground states.
}
\label{fig:element-counts-ratio-by-occurrence}
\end{figure*}
\begin{figure*}[htbp!]
\centering
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth,keepaspectratio]{figs/mp-trj-e-form-hist.pdf}
\caption{MPtrj formation energies}
\label{fig:mp-trj-e-form-hist}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth,keepaspectratio]{figs/mp-trj-forces-hist.pdf}
\caption{MPtrj forces}
\label{fig:mp-trj-forces-hist}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth,keepaspectratio]{figs/mp-trj-stresses-hist.pdf}
\caption{MPtrj stresses}
\label{fig:mp-trj-stresses-hist}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\includegraphics[width=\linewidth,keepaspectratio]{figs/mp-trj-magmoms-hist.pdf}
\caption{MPtrj magnetic moments}
\label{fig:mp-trj-magmoms-hist}
\end{subfigure}
\caption{
Distribution of energies, forces, stresses and magnetic moments in the MPtrj dataset \cite{deng_chgnet_2023,riebesell_matbench_2023}.
The bimodality in the formation energy distribution is due to the MP anion correction scheme \cite{wang_framework_2021,kingsbury2022flexible} which significantly lowers oxide formation energies.
}
\label{fig:mp-trj-hists}
\end{figure*}
\begin{figure*}[htbp!]
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figs/mp-trj-magmoms-ptable-hists.pdf}
\caption{
Distribution of magnetic moments for each element in the MPtrj dataset \cite{deng_chgnet_2023,riebesell_matbench_2023,riebesell_pymatviz_2022}.
The $y$-axis is log-scaled to allow visualization of the tail of high magnetic moments in some elements with a sharp peak at 0.
The number in the top right corner of each element tile counts magnetic moments for that element in the MPtrj dataset.
This plot reveals rare erroneous data points in MPtrj.
For instance, \ch{Cr} has a single-point calculation with a highly unphysical magnetic moment of \num{17}{$\mu_\text{B}$}.
}
\label{fig:mp-trj-magmoms-ptable-hists}
\end{figure*}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.7\columnwidth,keepaspectratio]{figs/E_range_histogram.pdf}
\caption{Distribution (PDF) of approximate deviation of energy within
each material due to variation in magnetic order (non-magnetic calculation, calculation
converges to moment zero, ferromagnetic, and other magnetic orders), with cumulative distribution (CDF) on right axis. The vast majority
of materials have very small variation (y axis range does not show full extent of first PDF bar), but a few hundred include different magnetic orders with energies that vary by more than \SI{0.1}{eV/atoms}}
\label{fig:E_magmom_range}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.495\linewidth}
\includegraphics[width=\linewidth,keepaspectratio]{figs/mp-vs-mp-trj-arity-hist.pdf}
\caption{}
\label{fig:mp-vs-mp-trj-arity-hist}
\end{subfigure}
\hfil
\begin{subfigure}[b]{0.495\linewidth}
\includegraphics[width=\linewidth,keepaspectratio]{figs/mp-trj-n-sites-hist.pdf}
\caption{}
\label{fig:mp-trj-n-sites-hist}
\end{subfigure}
\caption{
\subref*{fig:mp-vs-mp-trj-arity-hist}) Distribution of the number of elements in the compositions of MP structures compared to MPtrj.
We observe a slight overabundance of small numbers of elements in MPtrj relative to MP.
\subref*{fig:mp-trj-n-sites-hist}) Distribution of a number of sites in MPtrj.
The inset shows the same distribution log-scaled to visualize the tail of high site counts.
The green cumulative line in the inset shows that 82\% have less than 50 sites and 97\% of structures in MPtrj have less than 100 atoms.
}
\label{fig:mp-trj-arity-and-n-sites-hist}
\end{figure}
\begin{figure*}[htbp!]
\centering
\includegraphics[width=\linewidth,keepaspectratio]{figs/mptrj-umap-manhattan-100-1.0.pdf}
\caption{
UMAP projection of MACE descriptors for atoms in MPtrj. Each point represents the averaged feature vector of a single element in one structure and is colored by atomic number (left) and group (right) in the periodic table. The features of \MLFF{} model are 256-dimensional vectors (concatenation from both first and second layer of 128 channels). Manhattan distance is used for the construction of a high-dimensional UMAP manifold.
}
\label{fig:mp-trj-umap}
\end{figure*}
\clearpage
\subsection{Similarity analysis}
In the examples above, we have shown that \MLFF{}\ is capable of surprising degrees of extrapolation. The use of semi-local features (as a result of message passing)~\cite{batatia2022design} and element mixing~\cite{Darby2023Trace} within the MACE architecture are key components underlying \MLFF{}'s capabilities. These components allow \MLFF{}\ to extrapolate to systems that globally seem completely different from the training data but have close matches locally.
To quantify the similarity, we compare atomic environments from test systems to filtered portions of the training data using the following procedure:
\begin{enumerate}
\item Filter training data to a subset with elemental compositions similar or exactly matching the test system.
\item Use \MLFF{}\ to extract invariant descriptors for all atoms in both the test system and the filtered training subset.
\item Calculate the cosine similarity between the atoms in the test system and each filtered training structure. For each atom, we use the maximum cosine similarity found this way. This is essentially a best-match structure kernel~\cite{deComparingMoleculesSolids2016a} that allows many-to-one mappings.
\item Average these maximum atomic similarities by element and then combine them by averaging again, yielding an element-stratified similarity.
\end{enumerate}
Through this procedure, we identify training set structures that contain the most similar local environments to those in any given test system. In addition, we create \texttt{chemiscope}\cite{frauxChemiscopeInteractiveStructureproperty2020} (\url{https://chemiscope.org/}) input files containing UMAP~\cite{mcinnesUMAPUniformManifold2020} projections of the atomic descriptors (fitted only on the training environments), allowing a more granular and interactive inspection of the environments in the test and training data.
The code for analysing the data and generating \texttt{chemiscope} inputs is available as a Python package~\cite{rokasel_2023_10426282}.
\end{document}
\documentclass[12pt]{article}
\usepackage{scicite}
\usepackage{times}
\topmargin 0.0cm
\oddsidemargin 0.2cm
\textwidth 16cm
\textheight 21cm
\footskip 1.0cm
\newenvironment{sciabstract}{%
\begin{quote} \bf}
{\end{quote}}
\title{A simple {\it Science\/} Template}
\author
{John Smith,$^{1\ast}$ Jane Doe,$^{1}$ Joe Scientist$^{2}$\\
\\
\normalsize{$^{1}$Department of Chemistry, University of Wherever,}\\
\normalsize{An Unknown Address, Wherever, ST 00000, USA}\\
\normalsize{$^{2}$Another Unknown Address, Palookaville, ST 99999, USA}\\
\\
\normalsize{$^\ast$To whom correspondence should be addressed; E-mail: jsmith@wherever.edu.}
}
\date{}
\begin{document}
\baselineskip24pt
\maketitle
\begin{sciabstract}
This document presents a number of hints about how to set up your
{\it Science\/} paper in \LaTeX\ . We provide a template file,
\texttt{scifile.tex}, that you can use to set up the \LaTeX\ source
for your article. An example of the style is the special
\texttt{\{sciabstract\}} environment used to set up the abstract you
see here.
\end{sciabstract}
\section*{Introduction}
In this file, we present some tips and sample mark-up to assure your
\LaTeX\ file of the smoothest possible journey from review manuscript
to published {\it Science\/} paper. We focus here particularly on
issues related to style files, citation, and math, tables, and
figures, as those tend to be the biggest sticking points. Please use
the source file for this document, \texttt{scifile.tex}, as a template
for your manuscript, cutting and pasting your content into the file at
the appropriate places.
{\it Science\/}'s publication workflow relies on Microsoft Word. To
translate \LaTeX\ files into Word, we use an intermediate MS-DOS
routine \cite{tth} that converts the \TeX\ source into HTML\@. The
routine is generally robust, but it works best if the source document
is clean \LaTeX\ without a significant freight of local macros or
\texttt{.sty} files. Use of the source file \texttt{scifile.tex} as a
template, and calling {\it only\/} the \texttt{.sty} and \texttt{.bst}
files specifically mentioned here, will generate a manuscript that
should be eminently reviewable, and yet will allow your paper to
proceed quickly into our production flow upon acceptance \cite{use2e}.
\section*{Formatting Citations}
Citations can be handled in one of three ways. The most
straightforward (albeit labor-intensive) would be to hardwire your
citations into your \LaTeX\ source, as you would if you were using an
ordinary word processor. Thus, your code might look something like
this:
\begin{quote}
\begin{verbatim}
However, this record of the solar nebula may have been
partly erased by the complex history of the meteorite
parent bodies, which includes collision-induced shock,
thermal metamorphism, and aqueous alteration
({\it 1, 2, 5--7\/}).
\end{verbatim}
\end{quote}
\noindent Compiled, the last two lines of the code above, of course, would give notecalls in {\it Science\/} style:
\begin{quote}
\ldots thermal metamorphism, and aqueous alteration ({\it 1, 2, 5--7\/}).
\end{quote}
Under the same logic, the author could set up his or her reference list as a simple enumeration,
\begin{quote}
\begin{verbatim}
{\bf References and Notes}
\begin{enumerate}
\item G. Gamow, {\it The Constitution of Atomic Nuclei
and Radioactivity\/} (Oxford Univ. Press, New York, 1931).
\item W. Heisenberg and W. Pauli, {\it Zeitschr.\ f.\
Physik\/} {\bf 56}, 1 (1929).
\end{enumerate}
\end{verbatim}
\end{quote}
\noindent yielding
\begin{quote}
{\bf References and Notes}
\begin{enumerate}
\item G. Gamow, {\it The Constitution of Atomic Nuclei and
Radioactivity\/} (Oxford Univ. Press, New York, 1931).
\item W. Heisenberg and W. Pauli, {\it Zeitschr.\ f.\ Physik} {\bf 56},
1 (1929).
\end{enumerate}
\end{quote}
That's not a solution that's likely to appeal to everyone, however ---
especially not to users of B{\small{IB}}\TeX\ \cite{inclme}. If you
are a B{\small{IB}}\TeX\ user, we suggest that you use the
\texttt{Science.bst} bibliography style file and the
\texttt{scicite.sty} package, both of which are downloadable from our author help site.
{\bf While you can use B{\small{IB}}\TeX\ to generate the reference list, please don't submit
your .bib and .bbl files; instead, paste the generated .bbl file into the .tex file, creating
\texttt{\{thebibliography\}} environment.}
You can also
generate your reference lists directly by using
\texttt{\{thebibliography\}} at the end of your source document; here
again, you may find the \texttt{scicite.sty} file useful.
Whatever you use, be
very careful about how you set up your in-text reference calls and
notecalls. In particular, observe the following requirements:
\begin{enumerate}
\item Please follow the style for references outlined at our author
help site and embodied in recent issues of {\it Science}. Each
citation number should refer to a single reference; please do not
concatenate several references under a single number.
\item The reference numbering continues from the
main text to the Supplementary Materials (e.g. this main
text has references 1-3; the numbering of references in the
Supplementary Materials should start with 4).
\item Please cite your references and notes in text {\it only\/} using
the standard \LaTeX\ \verb+\cite+ command, not another command
driven by outside macros.
\item Please separate multiple citations within a single \verb+\cite+
command using commas only; there should be {\it no space\/}
between reference keynames. That is, if you are citing two
papers whose bibliography keys are \texttt{keyname1} and
\texttt{keyname2}, the in-text cite should read
\verb+\cite{keyname1,keyname2}+, {\it not\/}
\verb+\cite{keyname1, keyname2}+.
\end{enumerate}
\noindent Failure to follow these guidelines could lead
to the omission of the references in an accepted paper when the source
file is translated to Word via HTML.
\section*{Handling Math, Tables, and Figures}
Following are a few things to keep in mind in coding equations,
tables, and figures for submission to {\it Science}.
\paragraph*{In-line math.} The utility that we use for converting
from \LaTeX\ to HTML handles in-line math relatively well. It is best
to avoid using built-up fractions in in-line equations, and going for
the more boring ``slash'' presentation whenever possible --- that is,
for \verb+$a/b$+ (which comes out as $a/b$) rather than
\verb+$\frac{a}{b}$+ (which compiles as $\frac{a}{b}$).
Please do not code arrays or matrices as
in-line math; display them instead. And please keep your coding as
\TeX-y as possible --- avoid using specialized math macro packages
like \texttt{amstex.sty}.
\paragraph*{Tables.} The HTML converter that we use seems to handle
reasonably well simple tables generated using the \LaTeX\
\texttt{\{tabular\}} environment. For very complicated tables, you
may want to consider generating them in a word processing program and
including them as a separate file.
\paragraph*{Figures.} Figure callouts within the text should not be
in the form of \LaTeX\ references, but should simply be typed in ---
that is, \verb+(Fig. 1)+ rather than \verb+\ref{fig1}+. For the
figures themselves, treatment can differ depending on whether the
manuscript is an initial submission or a final revision for acceptance
and publication. For an initial submission and review copy, you can
use the \LaTeX\ \verb+{figure}+ environment and the
\verb+\includegraphics+ command to include your PostScript figures at
the end of the compiled file. For the final revision,
however, the \verb+{figure}+ environment should {\it not\/} be used;
instead, the figure captions themselves should be typed in as regular
text at the end of the source file (an example is included here), and
the figures should be uploaded separately according to the Art
Department's instructions.
\section*{What to Send In}
What you should send to {\it Science\/} will depend on the stage your manuscript is in:
\begin{itemize}
\item {\bf Important:} If you're sending in the initial submission of
your manuscript (that is, the copy for evaluation and peer review),
please send in {\it only\/} a PDF version of the
compiled file (including figures). Please do not send in the \TeX\
source, \texttt{.sty}, \texttt{.bbl}, or other associated files with
your initial submission. (For more information, please see the
instructions at our Web submission site.)
\item When the time comes for you to send in your revised final
manuscript (i.e., after peer review), we require that you include
source files and generated files in your upload. {\bf The .tex file should include
the reference list as an itemized list (see "Formatting citations" for the various options). The bibliography should not be in a separate file.}
Thus, if the
name of your main source document is \texttt{ltxfile.tex}, you
need to include:
\begin{itemize}
\item \texttt{ltxfile.tex}.
\item \texttt{ltxfile.aux}, the auxilliary file generated by the
compilation.
\item A PDF file generated from
\texttt{ltxfile.tex}.
\end{itemize}
\end{itemize}
\bibliography{scibib}
\bibliographystyle{Science}
\section*{Acknowledgments}
Include acknowledgments of funding, any patents pending, where raw data for the paper are deposited, etc.
\section*{Supplementary materials}
Materials and Methods\\
Supplementary Text\\
Figs. S1 to S3\\
Tables S1 to S4\\
References \textit{(4-10)}
\clearpage
\noindent {\bf Fig. 1.} Please do not use figure environments to set
up your figures in the final (post-peer-review) draft, do not include graphics in your
source code, and do not cite figures in the text using \LaTeX\
\verb+\ref+ commands. Instead, simply refer to the figure numbers in
the text per {\it Science\/} style, and include the list of captions at
the end of the document, coded as ordinary paragraphs as shown in the
\texttt{scifile.tex} template file. Your actual figure files should
be submitted separately.
\end{document}
```
4. **Bibliographic Information:**
```bbl
\begin{thebibliography}{100}
\bibitem{KohnDFT}
W.~Kohn and L.~J. Sham, ``Self-consistent equations including exchange and
correlation effects,'' {\em Phys. Rev.}, vol.~140, pp.~A1133--A1138, Nov
1965.
\bibitem{qe2020}
P.~Giannozzi, O.~Baseggio, P.~Bonfà, D.~Brunato, R.~Car, I.~Carnimeo,
C.~Cavazzoni, S.~de~Gironcoli, P.~Delugas, F.~Ferrari~Ruffino, A.~Ferretti,
N.~Marzari, I.~Timrov, A.~Urru, and S.~Baroni, ``{Quantum ESPRESSO toward the
exascale},'' {\em The Journal of Chemical Physics}, vol.~152, p.~154105, 04
2020.
\bibitem{kuhne2020cp2k}
T.~D. K{\"u}hne, M.~Iannuzzi, M.~Del~Ben, V.~V. Rybkin, P.~Seewald, F.~Stein,
T.~Laino, R.~Z. Khaliullin, O.~Sch{\"u}tt, F.~Schiffmann, {\em et~al.},
``Cp2k: An electronic structure and molecular dynamics software
package-quickstep: Efficient and accurate electronic structure
calculations,'' {\em The Journal of Chemical Physics}, vol.~152, no.~19,
2020.
\bibitem{VASP1}
G.~Kresse and J.~Hafner, ``Ab initio molecular dynamics for liquid metals,''
{\em Phys. Rev. B}, vol.~47, p.~558, 1993.
\bibitem{Hasnip2014Density}
P.~J. Hasnip, K.~Refson, M.~I.~J. Probert, J.~R. Yates, S.~J. Clark, and C.~J.
Pickard, ``Density functional theory in the solid state,'' {\em Phil. Trans.
R. Soc. A.}, vol.~372, p.~20130270, 2014.
\bibitem{Jain2016Computational}
A.~Jain, Y.~Shin, and K.~A. Persson, ``Computational predictions of energy
materials using density functional theory,'' {\em Nat Rev Mater}, vol.~1,
p.~714, 2016.
\bibitem{Neugebauer2013Density}
J.~Neugebauer and T.~Hickel, ``Density functional theory in materials
science,'' {\em WIREs Comput Mol Sci}, vol.~3, pp.~438--448, 2013.
\bibitem{finnis2003interatomic}
M.~Finnis, {\em Interatomic forces in condensed matter}, vol.~1.
\newblock Oxford Series on Materials Mod, 2003.
\bibitem{behler2007}
J.~Behler and M.~Parrinello, ``Generalized neural-network representation of
high-dimensional potential-energy surfaces,'' {\em Phys. Rev. Lett.},
vol.~98, p.~146401, Apr 2007.
\bibitem{gap}
A.~P. Bart\'ok, M.~C. Payne, R.~Kondor, and G.~Cs\'anyi, ``Gaussian
approximation potentials: The accuracy of quantum mechanics, without the
electrons,'' {\em Phys. Rev. Lett.}, vol.~104, p.~136403, Apr 2010.
\bibitem{THOMPSON2015316}
A.~Thompson, L.~Swiler, C.~Trott, S.~Foiles, and G.~Tucker, ``Spectral neighbor
analysis method for automated generation of quantum-accurate interatomic
potentials,'' {\em Journal of Computational Physics}, vol.~285, pp.~316--330,
2015.
\bibitem{schnet}
K.~T. Sch{\"u}tt, H.~E. Sauceda, P.-J. Kindermans, A.~Tkatchenko, and K.-R.
M{\"u}ller, ``{SchNet -- A deep learning architecture for molecules and
materials},'' {\em The Journal of Chemical Physics}, vol.~148, p.~241722, 03
2018.
\bibitem{deringer_machine_2019}
V.~L. Deringer, M.~A. Caro, and G.~Cs\'anyi, ``Machine {Learning} {Interatomic}
{Potentials} as {Emerging} {Tools} for {Materials} {Science},'' {\em Advanced
Materials}, vol.~31, no.~46, p.~1902765, 2019.
\bibitem{drautz2019}
R.~Drautz, ``Atomic cluster expansion for accurate and transferable interatomic
potentials,'' {\em Phys. Rev. B}, vol.~99, p.~014104, Jan 2019.
\bibitem{lilienfeld2020}
O.~A. von Lilienfeld and K.~Burke, ``Retrospective on a decade of machine
learning for chemical discovery,'' {\em Nature Communications}, vol.~11,
no.~1, p.~4895, 2020.
\bibitem{batzner2022}
S.~Batzner, A.~Musaelian, L.~Sun, M.~Geiger, J.~P. Mailoa, M.~Kornbluth,
N.~Molinari, T.~E. Smidt, and B.~Kozinsky, ``E (3)-equivariant graph neural
networks for data-efficient and accurate interatomic potentials,'' {\em
Nature communications}, vol.~13, no.~1, p.~2453, 2022.
\bibitem{ko2023recent}
T.~W. Ko and S.~P. Ong, ``Recent advances and outstanding challenges for
machine learning interatomic potentials,'' {\em Nature Computational
Science}, pp.~1--3, 2023.
\bibitem{deringer_chemrev2021}
V.~L. Deringer, A.~P. Bart{\'o}k, N.~Bernstein, D.~M. Wilkins, M.~Ceriotti, and
G.~Cs{\'a}nyi, ``Gaussian process regression for materials and molecules,''
{\em Chemical Reviews}, vol.~121, no.~16, pp.~10073--10141, 2021.
\newblock PMID: 34398616.
\bibitem{jain2013commentary}
A.~Jain, S.~P. Ong, G.~Hautier, W.~Chen, W.~D. Richards, S.~Dacek, S.~Cholia,
D.~Gunter, D.~Skinner, G.~Ceder, {\em et~al.}, ``Commentary: The materials
project: A materials genome approach to accelerating materials innovation,''
{\em APL materials}, vol.~1, no.~1, 2013.
\bibitem{Stocker2022robust}
S.~Stocker, J.~Gasteiger, F.~Becker, S.~G\"unnemann, and J.~T. Margraf, ``How
robust are modern graph neural network potentials in long and hot molecular
dynamics simulations?,'' {\em Mach. Learn.: Sci. Technol.}, vol.~3,
p.~045010, 2022.
\bibitem{megnet2019}
C.~Chen, W.~Ye, Y.~Zuo, C.~Zheng, and S.~P. Ong, ``Graph networks as a
universal machine learning framework for molecules and crystals,'' {\em
Chemistry of Materials}, vol.~31, no.~9, pp.~3564--3572, 2019.
\bibitem{perdew1996generalized}
J.~P. Perdew, K.~Burke, and M.~Ernzerhof, ``Generalized gradient approximation
made simple,'' {\em Physical review letters}, vol.~77, no.~18, p.~3865, 1996.
\bibitem{chen_universal_2022}
C.~Chen and S.~P. Ong, ``A universal graph deep learning interatomic potential
for the periodic table,'' {\em Nat Comput Sci}, vol.~2, no.~11, pp.~718--728,
2022.
\newblock Number: 11 Publisher: Nature Publishing Group.
\bibitem{deng_chgnet_2023}
B.~Deng, P.~Zhong, K.~Jun, J.~Riebesell, K.~Han, C.~J. Bartel, and G.~Ceder,
``{{CHGNet}} as a pretrained universal neural network potential for
charge-informed atomistic modeling,'' {\em Nature Machine Intelligence},
vol.~5, no.~9, pp.~1031--1041, 2023.
\bibitem{choudhary2023unified}
K.~Choudhary, B.~DeCost, L.~Major, K.~Butler, J.~Thiyagalingam, and F.~Tavazza,
``Unified graph neural network force-field for the periodic table: solid
state applications,'' {\em Digital Discovery}, vol.~2, no.~2, pp.~346--355,
2023.
\bibitem{jarvis}
K.~Choudhary, K.~F. Garrity, A.~C.~E. Reid, B.~DeCost, A.~J. Biacchi, A.~R.
Hight~Walker, Z.~Trautt, J.~Hattrick-Simpers, A.~G. Kusne, A.~Centrone,
A.~Davydov, J.~Jiang, R.~Pachter, G.~Cheon, E.~Reed, A.~Agrawal, X.~Qian,
V.~Sharma, H.~Zhuang, S.~V. Kalinin, B.~G. Sumpter, G.~Pilania, P.~Acar,
S.~Mandal, K.~Haule, D.~Vanderbilt, K.~Rabe, and F.~Tavazza, ``The joint
automated repository for various integrated simulations (jarvis) for
data-driven materials design,'' {\em npj Computational Materials}, vol.~6,
no.~1, p.~173, 2020.
\bibitem{optB88vdW}
J.~Klime{\v s}, D.~R. Bowler, and A.~Michaelides, ``Chemical accuracy for the
van der waals density functional,'' {\em Journal of Physics: Condensed
Matter}, vol.~22, no.~2, p.~022201, 2010.
\bibitem{merchant_scaling_2023}
A.~Merchant, S.~Batzner, S.~S. Schoenholz, M.~Aykol, G.~Cheon, and E.~D. Cubuk,
``Scaling deep learning for materials discovery,'' {\em Nature}, pp.~1--6,
2023.
\newblock Publisher: Nature Publishing Group.
\bibitem{dpa1}
D.~Zhang, H.~Bi, F.-Z. Dai, W.~Jiang, L.~Zhang, and H.~Wang, ``Dpa-1:
Pretraining of attention-based deep potential model for molecular
simulation,'' {\em tbd}, 2023.
\bibitem{dpa2}
D.~Zhang, X.~Liu, X.~Zhang, C.~Zhang, C.~Cai, H.~Bi, Y.~Du, X.~Qin, J.~Huang,
B.~Li, Y.~Shan, J.~Zeng, Y.~Zhang, S.~Liu, Y.~Li, J.~Chang, X.~Wang, S.~Zhou,
J.~Liu, X.~Luo, Z.~Wang, W.~Jiang, J.~Wu, Y.~Yang, J.~Yang, M.~Yang, F.-Q.
Gong, L.~Zhang, M.~Shi, F.-Z. Dai, D.~M. York, S.~Liu, T.~Zhu, Z.~Zhong,
J.~Lv, J.~Cheng, W.~Jia, M.~Chen, G.~Ke, W.~E, L.~Zhang, and H.~Wang,
``Dpa-2: Towards a universal large atomic model for molecular and material
simulation,'' {\em tbd}, 2023.
\bibitem{takamoto2022towards}
S.~Takamoto, C.~Shinagawa, D.~Motoki, K.~Nakago, W.~Li, I.~Kurata, T.~Watanabe,
Y.~Yayama, H.~Iriguchi, Y.~Asano, {\em et~al.}, ``Towards universal neural
network potential for material discovery applicable to arbitrary combination
of 45 elements,'' {\em Nature Communications}, vol.~13, no.~1, p.~2991, 2022.
\bibitem{takamoto2022teanet}
S.~Takamoto, S.~Izumi, and J.~Li, ``Teanet: Universal neural network
interatomic potential inspired by iterative electronic relaxations,'' {\em
Computational Materials Science}, vol.~207, p.~111280, 2022.
\bibitem{Takamoto2023}
S.~Takamoto, D.~Okanohara, Q.-J. Li, and J.~Li, ``Towards universal neural
network interatomic potential,'' {\em Journal of Materiomics}, vol.~9, no.~3,
pp.~447--454, 2023.
\bibitem{smith_ani-1_2017}
J.~S. Smith, O.~Isayev, and A.~E. Roitberg, ``{ANI}-1: an extensible neural
network potential with {DFT} accuracy at force field computational cost,''
{\em Chem. Sci.}, vol.~8, no.~4, pp.~3192--3203, 2017.
\newblock Publisher: The Royal Society of Chemistry.
\bibitem{Smith2019}
J.~S. Smith, B.~T. Nebgen, R.~Zubatyuk, N.~Lubbers, C.~Devereux, K.~Barros,
S.~Tretiak, O.~Isayev, and A.~E. Roitberg, ``Approaching coupled cluster
accuracy with a general-purpose neural network potential through transfer
learning,'' {\em Nat. Commun.}, vol.~10, no.~1, p.~2903, 2019.
\bibitem{aimnet2019}
R.~Zubatyuk, J.~S. Smith, J.~Leszczynski, and O.~Isayev, ``Accurate and
transferable multitask prediction of chemical properties with an
atoms-in-molecules neural network,'' {\em Science Advances}, vol.~5, no.~8,
p.~eaav6490, 2019.
\bibitem{kovacs2023maceoff}
D.~P. Kov{\'a}cs, J.~H. Moore, N.~J. Browning, I.~Batatia, J.~T. Horton,
V.~Kapil, I.-B. Magd{\u{a}}u, D.~J. Cole, and G.~Cs{\'a}nyi, ``Mace-off23:
Transferable machine learning force fields for organic molecules,'' {\em
arXiv preprint arXiv:2312.15211}, 2023.
\bibitem{lopanitsyna2023modeling}
N.~Lopanitsyna, G.~Fraux, M.~A. Springer, S.~De, and M.~Ceriotti, ``Modeling
high-entropy transition metal alloys with alchemical compression,'' {\em
Physical Review Materials}, vol.~7, no.~4, p.~045802, 2023.
\bibitem{batatia_mace_2023}
I.~Batatia, D.~P. Kovacs, G.~Simm, C.~Ortner, and G.~Cs{\'a}nyi, ``Mace: Higher
order equivariant message passing neural networks for fast and accurate force
fields,'' {\em Advances in Neural Information Processing Systems}, vol.~35,
pp.~11423--11436, 2022.
\bibitem{dusson2022atomic}
G.~Dusson, M.~Bachmayr, G.~Cs{\'a}nyi, R.~Drautz, S.~Etter, C.~van~der Oord,
and C.~Ortner, ``Atomic cluster expansion: Completeness, efficiency and
stability,'' {\em Journal of Computational Physics}, vol.~454, p.~110946,
2022.
\bibitem{lysogorskiy2021performant}
Y.~Lysogorskiy, C.~v.~d. Oord, A.~Bochkarev, S.~Menon, M.~Rinaldi,
T.~Hammerschmidt, M.~Mrovec, A.~Thompson, G.~Cs{\'a}nyi, C.~Ortner, {\em
et~al.}, ``Performant implementation of the atomic cluster expansion (pace)
and application to copper and silicon,'' {\em npj computational materials},
vol.~7, no.~1, p.~97, 2021.
\bibitem{witt2023acepotentials}
W.~C. Witt, C.~van~der Oord, E.~Gel{\v{z}}inyt{\.e}, T.~J{\"a}rvinen, A.~Ross,
J.~P. Darby, C.~H. Ho, W.~J. Baldwin, M.~Sachs, J.~Kermode, {\em et~al.},
``Acepotentials. jl: A julia implementation of the atomic cluster
expansion,'' {\em The Journal of Chemical Physics}, vol.~159, no.~16, 2023.
\bibitem{batatia2022design}
I.~Batatia, S.~Batzner, D.~P. Kovács, A.~Musaelian, G.~N.~C. Simm, R.~Drautz,
C.~Ortner, B.~Kozinsky, and G.~Csányi, ``The design space of
e(3)-equivariant atom-centered interatomic potentials,'' 2022.
\bibitem{Darby2023Trace}
J.~P. Darby, D.~P. Kov\'acs, I.~Batatia, M.~A. Caro, G.~L.~W. Hart, C.~Ortner,
and G.~Cs\'anyi, ``Tensor-reduced atomic density representations,'' {\em
Phys. Rev. Lett.}, vol.~131, p.~028001, Jul 2023.
\bibitem{Skinner2013/10.1063/1.4790861}
L.~B. Skinner, C.~Huang, D.~Schlesinger, L.~G.~M. Pettersson, A.~Nilsson, and
C.~J. Benmore, ``{Benchmark oxygen-oxygen pair-distribution function of
ambient water from x-ray diffraction measurements with a wide Q-range},''
{\em J. Chem. Phys.}, vol.~138, p.~074506, 02 2013.
\bibitem{Bertie1996}
J.~E. Bertie and Z.~Lan, ``Infrared intensities of liquids xx: The intensity of
the \ch{OH} stretching band of liquid water revisited, and the best current
values of the optical constants of \ch{H2O}(l) at 25$^{\circ}$c between
15,000 and 1\,cm$^{-1}$,'' {\em Appl. Spectrosc.}, vol.~50, no.~8,
pp.~1047--1057, 1996.
\bibitem{Moberg2017}
D.~R. Moberg, S.~C. Straight, C.~Knight, and F.~Paesani, ``Molecular {O}rigin
of the {V}ibrational {S}tructure of {I}ce {I}h,'' {\em J. Phys. Chem. Lett.},
vol.~8, no.~12, pp.~2579--2583, 2017.
\newblock PMID: 28541703.
\bibitem{grimme2010consistent}
S.~Grimme, J.~Antony, S.~Ehrlich, and H.~Krieg, ``A consistent and accurate ab
initio parametrization of density functional dispersion correction (dft-d)
for the 94 elements h-pu,'' {\em The Journal of chemical physics}, vol.~132,
no.~15, 2010.
\bibitem{oneill2022crumbling}
N.~O'Neill, C.~Schran, S.~J. Cox, and A.~Michaelides, ``Crumbling crystals: On
the dissolution mechanism of nacl in water,'' 2022.
\bibitem{gillan2016perspective}
M.~J. Gillan, D.~Alfe, and A.~Michaelides, ``Perspective: How good is dft for
water?,'' {\em The Journal of chemical physics}, vol.~144, no.~13, 2016.
\bibitem{Marx1999/10.1038/17579}
D.~Marx, M.~E. Tuckerman, J.~Hutter, and M.~Parrinello, ``{The nature of the
hydrated excess proton in water},'' {\em Nature}, vol.~397, pp.~601--604, feb
1999.
\bibitem{Tuckerman2002/10.1038/nature00797}
M.~E. Tuckerman, D.~Marx, and M.~Parrinello, ``{The nature and transport
mechanism of hydrated hydroxide ions in aqueous solution},'' {\em Nature},
vol.~417, pp.~925--929, jun 2002.
\bibitem{Agmon2016/10.1021/acs.chemrev.5b00736}
N.~Agmon, H.~J. Bakker, R.~K. Campen, R.~H. Henchman, P.~Pohl, S.~Roke,
M.~Th{\"{a}}mer, and A.~Hassanali, ``{Protons and Hydroxide Ions in Aqueous
Systems},'' {\em Chem. Rev.}, vol.~116, pp.~7642--7672, jul 2016.
\bibitem{dmcice13}
F.~Della~Pia, A.~Zen, D.~Alfè, and A.~Michaelides, ``{DMC-ICE13: Ambient and
high pressure polymorphs of ice from diffusion Monte Carlo and density
functional theory},'' {\em The Journal of Chemical Physics}, vol.~157,
p.~134701, 10 2022.
\bibitem{Algara-Siller2015/10.1038/nature14295}
G.~Algara-Siller, O.~Lehtinen, F.~C. Wang, R.~R. Nair, U.~Kaiser, H.~A. Wu,
A.~K. Geim, and I.~V. Grigorieva, ``{Square ice in graphene
nanocapillaries},'' {\em Nature}, vol.~519, pp.~443--445, mar 2015.
\bibitem{Fumagalli2018/10.1126/science.aat4191}
L.~Fumagalli, A.~Esfandiar, R.~Fabregas, S.~Hu, P.~Ares, A.~Janardanan,
Q.~Yang, B.~Radha, T.~Taniguchi, K.~Watanabe, G.~Gomila, K.~S. Novoselov, and
A.~K. Geim, ``{Anomalously low dielectric constant of confined water},'' {\em
Science}, vol.~360, pp.~1339--1342, jun 2018.
\bibitem{kapil_first-principles_2023}
V.~Kapil, C.~Schran, A.~Zen, J.~Chen, C.~J. Pickard, and A.~Michaelides, ``The
first-principles phase diagram of monolayer nanoconfined water,'' {\em
Nature}, vol.~609, no.~7927, pp.~512--516, 2022.
\bibitem{Hansen2008-zz}
H.~A. Hansen, J.~Rossmeisl, and J.~K. N{\o}rskov, ``Surface pourbaix diagrams
and oxygen reduction activity of {Pt}, {Ag} and {Ni}(111) surfaces studied by
{DFT},'' {\em Phys. Chem. Chem. Phys.}, vol.~10, pp.~3722--3730, July 2008.
\bibitem{norskov2022nonlinear}
S.~Vijay, G.~Kastlunger, K.~Chan, and J.~K. Nørskov, ``Limits to scaling
relations between adsorption energies?,'' {\em The Journal of Chemical
Physics}, vol.~156, June 2022.
\bibitem{Tiwari2020}
A.~Tiwari, H.~H. Heenen, A.~S. Bj{\o}rnlund, D.~Hochfilzer, K.~Chan, and
S.~Horch, ``Electrochemical oxidation of co on cu single crystals under
alkaline conditions,'' {\em ACS Energy Letters}, vol.~5, no.~11,
pp.~3437--3442, 2020.
\bibitem{schaaf2023accurate}
L.~L. Schaaf, E.~Fako, S.~De, A.~Sch{\"a}fer, and G.~Cs{\'a}nyi, ``Accurate
energy barriers for catalytic reaction pathways: an automatic training
protocol for machine learning force fields,'' {\em npj Computational
Materials}, vol.~9, no.~1, p.~180, 2023.
\bibitem{mcinnes2018umap}
L.~McInnes, J.~Healy, and J.~Melville, ``Umap: Uniform manifold approximation
and projection for dimension reduction,'' {\em arXiv preprint
arXiv:1802.03426}, 2018.
\bibitem{rokasel_2023_10426282}
R.~Elijošius, ``{MACE-MP UMAP} analysis,'' {\em github.com}, 2023.
\newblock 10.5281/zenodo.10426282 - https://github.com/RokasEl/mace-mp-umap.
\bibitem{Nrskov2009Towards}
J.~K. N{\o}rskov, T.~Bligaard, J.~Rossmeisl, and C.~H. Christensen, ``Towards
the computational design of solid catalysts,'' {\em Nature Chem}, vol.~1,
pp.~37--46, 2009.
\bibitem{Medford2015From}
A.~J. Medford, A.~Vojvodic, J.~S. Hummelsh{\o}j, J.~Voss, F.~Abild-pedersen,
F.~Studt, T.~Bligaard, A.~Nilsson, and J.~K. N{\o}rskov, ``From the sabatier
principle to a predictive theory of transition-metal heterogeneous
catalysis,'' {\em Journal of Catalysis}, vol.~328, pp.~36--42, 2015.
\bibitem{Bruix2019First-principles-based}
A.~Bruix, J.~T. Margraf, M.~Andersen, and K.~Reuter, ``First-principles-based
multiscale modelling of heterogeneous catalysis,'' {\em Nat Catal}, vol.~2,
pp.~659--670, 2019.
\bibitem{Qin2023Cation-Coordinated}
X.~Qin, T.~Vegge, and H.~A. Hansen, ``Cation-coordinated inner-sphere
co2electroreduction at au--water interfaces,'' {\em J. Am. Chem.
Soc.}, vol.~145, pp.~1897--1905, 2023.
\bibitem{Man2011Universality}
I.~C. Man, H.~Su, F.~Calle‐vallejo, H.~A. Hansen, J.~I. Mart\'{\i}nez, N.~G.
Inoglu, J.~Kitchin, T.~F. Jaramillo, J.~K. N{\o}rskov, and J.~Rossmeisl,
``Universality in oxygen evolution electrocatalysis on oxide surfaces,'' {\em
ChemCatChem}, vol.~3, pp.~1159--1165, 2011.
\bibitem{Auer2020Self-activation}
A.~Auer, M.~Andersen, E.-M. Wernig, N.~G. H\"ormann, N.~Buller, K.~Reuter, and
J.~Kunze-liebh\"auser, ``Self-activation of copper electrodes during co
electro-oxidation in alkaline electrolyte,'' {\em Nat Catal}, vol.~3,
pp.~797--803, 2020.
\bibitem{Wang2021Ternary}
Q.~Wang, J.~Pan, J.~Guo, H.~A. Hansen, H.~Xie, L.~Jiang, L.~Hua, H.~Li,
Y.~Guan, P.~Wang, W.~Gao, L.~Liu, H.~Cao, Z.~Xiong, T.~Vegge, and P.~Chen,
``Ternary ruthenium complex hydrides for ammonia synthesis via the
associative mechanism,'' {\em Nat Catal}, vol.~4, pp.~959--967, 2021.
\bibitem{Norskov2014}
J.~K. N{\o}rskov, F.~Studt, F.~Abild‐Pedersen, and T.~Bligaard, {\em
{Fundamental Concepts in Heterogeneous Catalysis}}.
\newblock John Wiley \& Sons, Inc., 2014.
\bibitem{Margraf2023Exploring}
J.~T. Margraf, H.~Jung, C.~Scheurer, and K.~Reuter, ``Exploring catalytic
reaction networks with machine learning,'' {\em Nat Catal}, vol.~6,
pp.~112--121, 2023.
\bibitem{Yang2023Neural}
X.~Yang, A.~Bhowmik, T.~Vegge, and H.~A. Hansen, ``Neural network potentials
for accelerated metadynamics of oxygen reduction kinetics at au--water
interfaces,'' {\em Chem. Sci.}, vol.~14, pp.~3913--3922, 2023.
\bibitem{Tran2018Active}
K.~Tran and Z.~W. Ulissi, ``Active learning across intermetallics to guide
discovery of electrocatalysts for co2 reduction and h2 evolution,'' {\em Nat
Catal}, vol.~1, pp.~696--703, 2018.
\bibitem{Foppa_Sandip_2022}
L.~Foppa, C.~Sutton, L.~M. Ghiringhelli, S.~De, P.~Löser, S.~A. Schunk,
A.~Schäfer, and M.~Scheffler, ``Learning design rules for selective
oxidation catalysts from high-throughput experimentation and artificial
intelligence,'' {\em ACS Catalysis}, vol.~12, no.~4, pp.~2223--2232, 2022.
\bibitem{Khatamirad2023}
M.~Khatamirad, E.~Fako, C.~Boscagli, M.~M\"{u}ller, F.~Ebert,
R.~Naumann~d’Alnoncourt, A.~Schaefer, S.~A. Schunk, I.~Jevtovikj,
F.~Rosowski, and S.~De, ``A data-driven high-throughput workflow applied to
promoted in-oxide catalysts for co2 hydrogenation to methanol,'' {\em
Catalysis Science \& Technology}, vol.~13, no.~9, p.~2656–2661, 2023.
\bibitem{Stocker2023Estimating}
S.~Stocker, H.~Jung, G.~Cs\'anyi, C.~F. Goldsmith, K.~Reuter, and J.~T.
Margraf, ``Estimating free energy barriers for heterogeneous catalytic
reactions with machine learning potentials and umbrella integration,'' {\em
J. Chem. Theory Comput.}, vol.~19, pp.~6796--6804, 2023.
\bibitem{Tran2023Open}
R.~Tran, J.~Lan, M.~Shuaibi, B.~M. Wood, S.~Goyal, A.~Das, J.~Heras-domingo,
A.~Kolluru, A.~Rizvi, N.~Shoghi, A.~Sriram, F.~Therrien, J.~Abed, O.~Voznyy,
E.~H. Sargent, Z.~Ulissi, and C.~L. Zitnick, ``The open catalyst 2022 (oc22)
dataset and challenges for oxide electrocatalysts,'' {\em ACS Catal.},
vol.~13, pp.~3066--3084, 2023.
\bibitem{pourbaix1966}
M.~Pourbaix, {\em ATLAS of Electrochemical Equilibria in Aqueous Solutions}.
\newblock Pergamon Press, Oxford, 1966.
\bibitem{pourbaix1973}
M.~Pourbaix, {\em Lectures on Electrochemical Corrosion}.
\newblock Springer Science and Business Media, 1973.
\bibitem{Norskov2004-ow}
J.~K. N{\o}rskov, J.~Rossmeisl, A.~Logadottir, L.~Lindqvist, J.~R. Kitchin,
T.~Bligaard, and H.~J{\'o}nsson, ``Origin of the overpotential for oxygen
reduction at a fuel-cell cathode,'' {\em J. Phys. Chem. B}, vol.~108,
pp.~17886--17892, Nov. 2004.
\bibitem{Persson2012-kg}
K.~A. Persson, B.~Waldwick, P.~Lazic, and G.~Ceder, ``Prediction of
solid-aqueous equilibria: Scheme to combine first-principles calculations of
solids with experimental aqueous states,'' {\em Phys. Rev. B Condens. Matter
Mater. Phys.}, vol.~85, June 2012.
\bibitem{Singh2017-uu}
A.~K. Singh, L.~Zhou, A.~Shinde, S.~K. Suram, J.~H. Montoya, D.~Winston, J.~M.
Gregoire, and K.~A. Persson, ``Electrochemical stability of metastable
materials,'' {\em Chem. Mater.}, vol.~29, pp.~10159--10167, Dec. 2017.
\bibitem{norskov2007linear}
F.~Abild-Pedersen, J.~Greeley, F.~Studt, J.~Rossmeisl, T.~R. Munter, P.~G.
Moses, E.~Skúlason, T.~Bligaard, and J.~K. Nørskov, ``Scaling properties of
adsorption energies for hydrogen-containing molecules on transition-metal
surfaces,'' {\em Physical Review Letters}, vol.~99, July 2007.
\bibitem{Lopez2019lsr}
J.~Pérez-Ramírez and N.~López, ``Strategies to break linear scaling
relationships,'' {\em Nature Catalysis}, vol.~2, p.~971–976, Oct. 2019.
\bibitem{GarciaMuelas2019redox}
R.~Garc{\'\i}a-Muelas and N.~L{\'o}pez, ``Statistical learning goes beyond the
d-band model providing the thermochemistry of adsorbates on transition
metals,'' {\em Nat. Commun.}, vol.~10, p.~4687, Oct. 2019.
\bibitem{dang2020rationally}
S.~Dang, B.~Qin, Y.~Yang, H.~Wang, J.~Cai, Y.~Han, S.~Li, P.~Gao, and Y.~Sun,
``Rationally designed indium oxide catalysts for \ch{CO2} hydrogenation to
methanol with high activity and selectivity,'' {\em Science advances},
vol.~6, no.~25, p.~eaaz2060, 2020.
\bibitem{rosen2021machine}
A.~S. Rosen, S.~M. Iyer, D.~Ray, Z.~Yao, A.~Aspuru-Guzik, L.~Gagliardi, J.~M.
Notestein, and R.~Q. Snurr, ``Machine learning the quantum-chemical
properties of metal--organic frameworks for accelerated materials
discovery,'' {\em Matter}, vol.~4, no.~5, pp.~1578--1597, 2021.
\bibitem{rosen2022high}
A.~S. Rosen, V.~Fung, P.~Huck, C.~T. O’Donnell, M.~K. Horton, D.~G. Truhlar,
K.~A. Persson, J.~M. Notestein, and R.~Q. Snurr, ``High-throughput
predictions of metal--organic framework electronic properties: theoretical
challenges, graph neural networks, and data exploration,'' {\em npj
Computational Materials}, vol.~8, no.~1, p.~112, 2022.
\bibitem{zeng2023deepmd}
J.~Zeng, D.~Zhang, D.~Lu, P.~Mo, Z.~Li, Y.~Chen, M.~Rynik, L.~Huang, Z.~Li,
S.~Shi, Y.~Wang, H.~Ye, P.~Tuo, J.~Yang, Y.~Ding, Y.~Li, D.~Tisi, Q.~Zeng,
H.~Bao, Y.~Xia, J.~Huang, K.~Muraoka, Y.~Wang, J.~Chang, F.~Yuan, S.~L. Bore,
C.~Cai, Y.~Lin, B.~Wang, J.~Xu, J.-X. Zhu, C.~Luo, Y.~Zhang, R.~E.~A.
Goodall, W.~Liang, A.~K. Singh, S.~Yao, J.~Zhang, R.~Wentzcovitch, J.~Han,
J.~Liu, W.~Jia, D.~M. York, W.~E, R.~Car, L.~Zhang, and H.~Wang,
``{DeePMD-kit v2: A software package for deep potential models},'' {\em The
Journal of Chemical Physics}, vol.~159, p.~054801, 08 2023.
\bibitem{rappe1992uff}
A.~K. Rapp{\'e}, C.~J. Casewit, K.~Colwell, W.~A. Goddard~III, and W.~M. Skiff,
``Uff, a full periodic table force field for molecular mechanics and
molecular dynamics simulations,'' {\em Journal of the American chemical
society}, vol.~114, no.~25, pp.~10024--10035, 1992.
\bibitem{manz2016introducing}
T.~A. Manz and N.~G. Limas, ``Introducing ddec6 atomic population analysis:
part 1. charge partitioning theory and methodology,'' {\em RSC advances},
vol.~6, no.~53, pp.~47771--47801, 2016.
\bibitem{potoff2001vapor}
J.~J. Potoff and J.~I. Siepmann, ``Vapor--liquid equilibria of mixtures
containing alkanes, carbon dioxide, and nitrogen,'' {\em AIChE journal},
vol.~47, no.~7, pp.~1676--1682, 2001.
\bibitem{yaghi2019introduction}
O.~M. Yaghi, M.~J. Kalmutzki, and C.~S. Diercks, {\em Introduction to reticular
chemistry: metal-organic frameworks and covalent organic frameworks}.
\newblock John Wiley \& Sons, 2019.
\bibitem{britt2009highly}
D.~Britt, H.~Furukawa, B.~Wang, T.~G. Glover, and O.~M. Yaghi, ``Highly
efficient separation of carbon dioxide by a metal-organic framework replete
with open metal sites,'' {\em Proceedings of the National Academy of
Sciences}, vol.~106, no.~49, pp.~20637--20640, 2009.
\bibitem{queen2014comprehensive}
W.~L. Queen, M.~R. Hudson, E.~D. Bloch, J.~A. Mason, M.~I. Gonzalez, J.~S. Lee,
D.~Gygi, J.~D. Howe, K.~Lee, T.~A. Darwish, {\em et~al.}, ``Comprehensive
study of carbon dioxide adsorption in the metal--organic frameworks {M} 2
(dobdc)({M= Mg, Mn, Fe, Co, Ni, Cu, Zn}),'' {\em Chemical Science}, vol.~5,
no.~12, pp.~4569--4581, 2014.
\bibitem{choe2021mof}
J.~H. Choe, H.~Kim, and C.~S. Hong, ``Mof-74 type variants for co 2 capture,''
{\em Materials Chemistry Frontiers}, vol.~5, no.~14, pp.~5172--5185, 2021.
\bibitem{kokccam2020coordinatively}
{\"U}.~K{\"o}k{\c{c}}am-Demir, A.~Goldman, L.~Esrafili, M.~Gharib, A.~Morsali,
O.~Weingart, and C.~Janiak, ``Coordinatively unsaturated metal sites (open
metal sites) in metal--organic frameworks: design and applications,'' {\em
Chemical Society Reviews}, vol.~49, no.~9, pp.~2751--2798, 2020.
\bibitem{zheng2023quantum}
B.~Zheng, F.~L. Oliveira, R.~Neumann Barros~Ferreira, M.~Steiner, H.~Hamann,
G.~X. Gu, and B.~Luan, ``Quantum informed machine-learning potentials for
molecular dynamics simulations of \ch{CO2}’s chemisorption and diffusion in
{Mg-MOF-74},'' {\em ACS nano}, vol.~17, no.~6, pp.~5579--5587, 2023.
\bibitem{valenzano2010computational}
L.~Valenzano, B.~Civalleri, S.~Chavan, G.~T. Palomino, C.~O. Are{\'a}n, and
S.~Bordiga, ``Computational and experimental studies on the adsorption of
\ch{CO}, \ch{N2}, and \ch{CO2} on {Mg-MOF-74},'' {\em The Journal of Physical
Chemistry C}, vol.~114, no.~25, pp.~11185--11191, 2010.
\bibitem{witman2016silico}
M.~Witman, S.~Ling, S.~Anderson, L.~Tong, K.~C. Stylianou, B.~Slater, B.~Smit,
and M.~Haranczyk, ``In silico design and screening of hypothetical mof-74
analogs and their experimental synthesis,'' {\em Chemical science}, vol.~7,
no.~9, pp.~6263--6272, 2016.
\bibitem{bennett2018liquid}
T.~D. Bennett and S.~Horike, ``Liquid, glass and amorphous solid states of
coordination polymers and metal--organic frameworks,'' {\em Nature Reviews
Materials}, vol.~3, no.~11, pp.~431--440, 2018.
\bibitem{castel2022atomistic}
N.~Castel and F.-X. Coudert, ``Atomistic {M}odels of {A}morphous
{M}etal--{O}rganic {F}rameworks,'' {\em The Journal of Physical Chemistry C},
vol.~126, no.~16, pp.~6905--6914, 2022.
\bibitem{evans2020four}
J.~D. Evans, V.~Bon, I.~Senkovska, H.-C. Lee, and S.~Kaskel, ``Four-dimensional
metal-organic frameworks,'' {\em Nature Communications}, vol.~11, no.~1,
p.~2690, 2020.
\bibitem{taylor2018near}
M.~K. Taylor, T.~Runcevski, J.~Oktawiec, J.~E. Bachman, R.~L. Siegelman,
H.~Jiang, J.~A. Mason, J.~D. Tarver, and J.~R. Long, ``Near-perfect
\ch{CO2}/\ch{CH4} selectivity achieved through reversible guest templating in
the flexible metal--organic framework co (bdp),'' {\em Journal of the
American Chemical Society}, vol.~140, no.~32, pp.~10324--10331, 2018.
\bibitem{van_speybroeck_towards_2021}
V.~Van~Speybroeck, S.~Vandenhaute, A.~E. Hoffman, and S.~M. Rogge, ``Towards
modeling spatiotemporal processes in metal–organic frameworks,'' {\em
Trends Chem.}, vol.~3, pp.~605--619, 2021.
\bibitem{S66}
J.~Řezáč, K.~E. Riley, and P.~Hobza, ``S66: A well-balanced database of
benchmark interaction energies relevant to biomolecular structures,'' {\em
Journal of Chemical Theory and Computation}, vol.~7, no.~8, pp.~2427--2438,
2011.
\newblock PMID: 21836824.
\bibitem{x23_rt}
A.~M. Reilly and A.~Tkatchenko, ``{Understanding the role of vibrations, exact
exchange, and many-body van der Waals interactions in the cohesive properties
of molecular crystals},'' {\em The Journal of Chemical Physics}, vol.~139,
p.~024705, 07 2013.
\bibitem{crbh20}
T.~W. Keal and D.~J. Tozer, ``{Semiempirical hybrid functional with improved
performance in an extensive chemical assessment},'' {\em The Journal of
Chemical Physics}, vol.~123, p.~121103, 09 2005.
\bibitem{Furness2020Accurate}
J.~W. Furness, A.~D. Kaplan, J.~Ning, J.~P. Perdew, and J.~Sun, ``Accurate and
numerically efficient r$^2${SCAN} meta-generalized gradient approximation,''
{\em J. Phys. Chem. Lett.}, vol.~11, pp.~8208--8215, 2020.
\bibitem{Kingsbury2022Performance}
R.~Kingsbury, A.~S. Gupta, C.~J. Bartel, J.~M. Munro, S.~Dwaraknath, M.~Horton,
and K.~A. Persson, ``Performance comparison of r$^2${SCAN} and {SCAN}
{metaGGA} density functionals for solid materials via an automated,
high-throughput computational workflow,'' {\em Phys. Rev. Materials}, vol.~6,
2022.
\bibitem{henderson_accurate_2011}
T.~M. Henderson, J.~Paier, and G.~E. Scuseria, ``Accurate treatment of solids
with the {HSE} screened hybrid,'' {\em physica status solidi (b)}, vol.~248,
no.~4, pp.~767--774, 2011.
\bibitem{harl_assessing_2010}
J.~Harl, L.~Schimka, and G.~Kresse, ``Assessing the quality of the random phase
approximation for lattice constants and atomization energies of solids,''
{\em Phys. Rev. B}, vol.~81, p.~115126, Mar. 2010.
\bibitem{Ghasemi2015Interatomic}
S.~A. Ghasemi, A.~Hofstetter, S.~Saha, and S.~Goedecker, ``Interatomic
potentials for ionic systems with density functional accuracy based on charge
densities obtained by a neural network,'' {\em Phys. Rev. B}, vol.~92, 2015.
\bibitem{Grisafi2019Incorporating}
A.~Grisafi and M.~Ceriotti, ``Incorporating long-range physics in atomic-scale
machine learning,'' {\em J. Chem. Phys.}, vol.~151, p.~12828, 2019.
\bibitem{Ko2021fourth-generation}
T.~W. Ko, J.~A. Finkler, S.~Goedecker, and J.~Behler, ``A fourth-generation
high-dimensional neural network potential with accurate electrostatics
including non-local charge transfer,'' {\em Nat Commun}, vol.~12, p.~585,
2021.
\bibitem{Vondrak2023q-pac}
M.~Vondr\'ak, K.~Reuter, and J.~T. Margraf, ``q-pac: A python package for
machine learned charge equilibration models,'' {\em J. Chem. Phys.},
vol.~159, p.~10037, 2023.
\bibitem{mtpmagnetic2022}
I.~Novikov, B.~Grabowski, F.~K{\"o}rmann, and A.~Shapeev, ``Magnetic moment
tensor potentials for collinear spin-polarized materials reproduce different
magnetic states of bcc fe,'' {\em npj Computational Materials}, vol.~8,
no.~1, p.~13, 2022.
\bibitem{rinaldi2023noncollinear}
M.~Rinaldi, M.~Mrovec, A.~Bochkarev, Y.~Lysogorskiy, and R.~Drautz,
``Non-collinear magnetic atomic cluster expansion for iron,'' 2023.
\bibitem{magduau2023machine}
I.-B. Magd{\u{a}}u, D.~J. Arismendi-Arrieta, H.~E. Smith, C.~P. Grey,
K.~Hermansson, and G.~Cs{\'a}nyi, ``Machine learning force fields for
molecular liquids: Ethylene carbonate/ethyl methyl carbonate binary
solvent,'' {\em npj Computational Materials}, vol.~9, no.~1, p.~146, 2023.
\bibitem{byggmastar2019}
J.~Byggm\"astar, A.~Hamedani, K.~Nordlund, and F.~Djurabekova,
``Machine-learning interatomic potential for radiation damage and defects in
tungsten,'' {\em Phys. Rev. B}, vol.~100, p.~144105, Oct 2019.
\bibitem{morrow_how_2023}
J.~D. Morrow, J.~L.~A. Gardner, and V.~L. Deringer, ``How to validate
machine-learned interatomic potentials,'' {\em J. Chem. Phys.}, vol.~158,
p.~121501, Mar. 2023.
\bibitem{Gardner2023}
J.~L.~A. Gardner, K.~T. Baker, and V.~L. Deringer, ``Synthetic pre-training for
neural-network interatomic potentials,'' {\em Mach. Learn.: Sci. Technol.},
pp.~in press, DOI: 10.1088/2632--2153/ad1626, 2023.
\bibitem{bartok_water_2013}
A.~P. Bart\'ok, M.~J. Gillan, F.~R. Manby, and G.~Cs\'anyi, ``Machine-learning
approach for one- and two-body corrections to density functional theory:
Applications to molecular and condensed water,'' {\em Phys. Rev. B}, vol.~88,
p.~054104, Aug 2013.
\bibitem{dral2020}
P.~O. Dral, A.~Owens, A.~Dral, and G.~Cs{\'a}nyi, ``{Hierarchical machine
learning of potential energy surfaces},'' {\em The Journal of Chemical
Physics}, vol.~152, p.~204110, 05 2020.
\bibitem{Paszke2019PyTorchAI}
A.~Paszke, S.~Gross, F.~Massa, A.~Lerer, J.~Bradbury, G.~Chanan, T.~Killeen,
Z.~Lin, N.~Gimelshein, L.~Antiga, A.~Desmaison, A.~K{\"o}pf, E.~Yang,
Z.~DeVito, M.~Raison, A.~Tejani, S.~Chilamkurthy, B.~Steiner, L.~Fang,
J.~Bai, and S.~Chintala, ``Pytorch: An imperative style, high-performance
deep learning library,'' in {\em Neural Information Processing Systems},
2019.
\bibitem{geiger2022e3nn}
M.~Geiger and T.~Smidt, ``e3nn: Euclidean neural networks,'' 2022.
\bibitem{KovacsBenchmark2023}
D.~P. Kovács, I.~Batatia, E.~S. Arany, and G.~Csányi, ``{Evaluation of the
MACE force field architecture: From medicinal chemistry to materials
science},'' {\em The Journal of Chemical Physics}, vol.~159, p.~044118, 07
2023.
\bibitem{Reddi2019}
S.~J. Reddi, S.~Kale, and S.~Kumar, ``On the convergence of adam and beyond,''
{\em arXiv preprint arXiv:1904.09237}, 2019.
\bibitem{Kingma2014}
D.~P. Kingma and J.~Ba, ``Adam: A method for stochastic optimization,'' {\em
arXiv preprint arXiv:1412.6980}, 2014.
\bibitem{MP_calc_details}
``Materials project calculation details.''
\url{https://docs.materialsproject.org/methodology/materials-methodology/calculation-details}.
\newblock Accessed: 2023-12-18.
\bibitem{MP_Hubbard_U}
``Materials project.''
\url{https://docs.materialsproject.org/methodology/materials-methodology/calculation-details/gga+u-calculations/hubbard-u-values}.
\newblock Accessed: 2023-12-12.
\bibitem{Jain_PRB_2011}
A.~Jain, G.~Hautier, S.~P. Ong, C.~J. Moore, C.~C. Fischer, K.~A. Persson, and
G.~Ceder, ``Formation enthalpies by mixing {GGA} and {GGA} $+$ ${U}$
calculations,'' {\em Phys. Rev. B}, vol.~84, p.~045115, July 2011.
\bibitem{Horton_npjCM_2019}
M.~K. Horton, J.~H. Montoya, M.~Liu, and K.~A. Persson, ``High-throughput
prediction of the ground-state collinear magnetic order of inorganic
materials using {Density} {Functional} {Theory},'' {\em npj Comput Mater},
vol.~5, pp.~1--11, June 2019.
\bibitem{sabatini2013rVV10}
R.~Sabatini, T.~Gorni, and S.~de~Gironcoli, ``Nonlocal van der waals density
functional made simple and efficient,'' {\em Phys. Rev. B}, vol.~87,
p.~041108, Jan 2013.
\bibitem{lin2012water}
I.-C. Lin, A.~P. Seitsonen, I.~Tavernelli, and U.~Rothlisberger, ``Structure
and dynamics of liquid water from ab initio molecular dynamics—comparison
of blyp, pbe, and revpbe density functionals with and without van der waals
corrections,'' {\em J. Chem. Theory Comput.}, vol.~8, no.~10, pp.~3902--3910,
2012.
\bibitem{terentjev2018layered}
A.~V. Terentjev, L.~A. Constantin, and J.~M. Pitarke, ``Dispersion-corrected
pbesol exchange-correlation functional,'' {\em Phys. Rev. B}, vol.~98,
p.~214108, Dec 2018.
\bibitem{formalik2018MOFvdW}
F.~Formalik, M.~Fischer, J.~Rogacka, L.~Firlej, and B.~Kuchta, ``{Benchmarking
of GGA density functionals for modeling structures of nanoporous, rigid and
flexible MOFs},'' {\em J. Chem. Phys.}, vol.~149, p.~064110, 08 2018.
\bibitem{axilrod1943threebody}
B.~M. Axilrod and E.~Teller, ``{Interaction of the van der Waals Type Between
Three Atoms},'' {\em J. Chem. Phys.}, vol.~11, pp.~299--300, 12 2004.
\bibitem{grimme2011effect}
S.~Grimme, S.~Ehrlich, and L.~Goerigk, ``Effect of the damping function in
dispersion corrected density functional theory,'' {\em Journal of
computational chemistry}, vol.~32, no.~7, pp.~1456--1465, 2011.
\bibitem{bartok2018machine}
A.~P. Bart{\'o}k, J.~Kermode, N.~Bernstein, and G.~Cs{\'a}nyi, ``Machine
learning a general-purpose interatomic potential for silicon,'' {\em Physical
Review X}, vol.~8, no.~4, p.~041048, 2018.
\bibitem{yoshida2015defects}
Y.~Yoshida and G.~Langouche, {\em Defects and Impurities in Silicon Materials}.
\newblock Springer, 2015.
\bibitem{richie2004complexity}
D.~Richie, J.~Kim, S.~A. Barr, K.~R. Hazzard, R.~Hennig, and J.~W. Wilkins,
``Complexity of small silicon self-interstitial defects,'' {\em Physical
review letters}, vol.~92, no.~4, p.~045501, 2004.
\bibitem{sahli2005ab}
B.~Sahli and W.~Fichtner, ``Ab initio molecular dynamics simulation of
self-interstitial diffusion in silicon,'' {\em Physical Review B}, vol.~72,
no.~24, p.~245210, 2005.
\bibitem{posselt2005migration}
M.~Posselt, F.~Gao, and D.~Zwicker, ``Migration of di-and tri-interstitials in
silicon,'' {\em Nuclear Instruments and Methods in Physics Research Section
B: Beam Interactions with Materials and Atoms}, vol.~228, no.~1-4,
pp.~212--217, 2005.
\bibitem{du2005fast}
Y.~A. Du, S.~A. Barr, K.~R. Hazzard, T.~J. Lenosky, R.~G. Hennig, and J.~W.
Wilkins, ``Fast diffusion mechanism of silicon tri-interstitial defects,''
{\em Physical Review B}, vol.~72, no.~24, p.~241306, 2005.
\bibitem{pichler2012intrinsic}
P.~Pichler, {\em Intrinsic point defects, impurities, and their diffusion in
silicon}.
\newblock Springer Science \& Business Media, 2012.
\bibitem{dorner2018melting}
F.~Dorner, Z.~Sukurma, C.~Dellago, and G.~Kresse, ``Melting si: beyond density
functional theory,'' {\em Physical Review Letters}, vol.~121, no.~19,
p.~195701, 2018.
\bibitem{deringer2021origins}
V.~L. Deringer, N.~Bernstein, G.~Cs{\'a}nyi, C.~Ben~Mahmoud, M.~Ceriotti,
M.~Wilson, D.~A. Drabold, and S.~R. Elliott, ``Origins of structural and
electronic transitions in disordered silicon,'' {\em Nature}, vol.~589,
no.~7840, pp.~59--64, 2021.
\bibitem{bernstein_quantifying_2019}
N.~Bernstein, B.~Bhattarai, G.~Cs\'a{}nyi, D.~A. Drabold, S.~R. Elliott, and
V.~L. Deringer, ``Quantifying {Chemical} {Structure} and {Machine}-{Learned}
{Atomic} {Energies} in {Amorphous} and {Liquid} {Silicon},'' {\em Angew.
Chem. Int. Ed.}, vol.~58, pp.~7057--7061, 2019.
\bibitem{morrow_indirect_2022}
J.~D. Morrow and V.~L. Deringer, ``Indirect learning and physically guided
validation of interatomic potential models,'' {\em J. Chem. Phys.}, vol.~157,
no.~10, p.~104105, 2022.
\bibitem{deringer2018realistic}
V.~L. Deringer, N.~Bernstein, A.~P. Bart{\'o}k, M.~J. Cliffe, R.~N. Kerber,
L.~E. Marbella, C.~P. Grey, S.~R. Elliott, and G.~Cs{\'a}nyi, ``Realistic
atomistic structure of amorphous silicon from machine-learning-driven
molecular dynamics,'' {\em The journal of physical chemistry letters},
vol.~9, no.~11, pp.~2879--2885, 2018.
\bibitem{plimpton1995fast}
S.~Plimpton, ``Fast parallel algorithms for short-range molecular dynamics,''
{\em Journal of computational physics}, vol.~117, no.~1, pp.~1--19, 1995.
\bibitem{roorda1991structural}
S.~Roorda, W.~Sinke, J.~Poate, D.~Jacobson, S.~Dierker, B.~Dennis,
D.~Eaglesham, F.~Spaepen, and P.~Fuoss, ``Structural relaxation and defect
annihilation in pure amorphous silicon,'' {\em Physical review B}, vol.~44,
no.~8, p.~3702, 1991.
\bibitem{grigorev2023matscipy}
P.~Grigorev, L.~Fr{\'e}rot, F.~Birks, A.~Gola, J.~Golebiowski, J.~Grie{\ss}er,
J.~L. H{\"o}rmann, A.~Klemenz, G.~Moras, W.~G. N{\"o}hring, {\em et~al.},
``matscipy: materials science at the atomic scale with python,'' {\em Journal
of Open Source Software}, vol.~9, no.~93, p.~5668, 2024.
\bibitem{DeringerC2017}
V.~L. Deringer and G.~Cs\'anyi, ``Machine learning based interatomic potential
for amorphous carbon,'' {\em Phys. Rev. B}, vol.~95, p.~094203, Mar 2017.
\bibitem{Jana2019}
R.~Jana, D.~Savio, V.~L. Deringer, and L.~Pastewka, ``Structural and elastic
properties of amorphous carbon from simulated quenching at low rates,'' {\em
Modeling and Simulation in Materials Science and Engineering}, vol.~27,
p.~085009, Oct. 2019.
\bibitem{Qamar2023}
M.~Qamar, M.~Mrovec, Y.~Lysogorskiy, A.~Bochkarev, and R.~Drautz, ``Atomic
cluster expansion for quantum-accurate large-scale simulations of carbon,''
{\em Journal of Chemical Theory and Computation}, vol.~19, p.~5151–5167,
June 2023.
\bibitem{Stenczel2023}
T.~K. Stenczel, Z.~El-Machachi, G.~Liepuoniute, J.~D. Morrow, A.~P. Bart\'o{}k,
M.~I.~J. Probert, G.~Cs\'a{}nyi, and V.~L. Deringer, ``{Machine-learned
acceleration for molecular dynamics in CASTEP},'' {\em J. Chem. Phys.},
vol.~159, p.~044803, 07 2023.
\bibitem{deTomas2019}
C.~{de Tomas}, A.~Aghajamali, J.~L. Jones, D.~J. Lim, M.~J. López,
I.~Suarez-Martinez, and N.~A. Marks, ``Transferability in interatomic
potentials for carbon,'' {\em Carbon}, vol.~155, pp.~624--634, 2019.
\bibitem{marchant_exploring_2023}
G.~A. Marchant, M.~A. Caro, B.~Karasulu, and L.~B. P\'a{}rtay, ``Exploring the
configuration space of elemental carbon with empirical and machine learned
interatomic potentials,'' {\em npj Comput. Mater.}, vol.~9, p.~131, 2023.
\bibitem{Caro2018}
M.~A. Caro, V.~L. Deringer, J.~Koskinen, T.~Laurila, and G.~Cs\'anyi, ``Growth
mechanism and origin of high $s{p}^{3}$ content in tetrahedral amorphous
carbon,'' {\em Phys. Rev. Lett.}, vol.~120, p.~166101, Apr 2018.
\bibitem{broqvist2015reaxff}
P.~Broqvist, J.~Kullgren, M.~J. Wolf, A.~C. van Duin, and K.~Hermansson,
``Reaxff force-field for ceria bulk, surfaces, and nanoparticles,'' {\em The
Journal of Physical Chemistry C}, vol.~119, no.~24, pp.~13598--13609, 2015.
\bibitem{Senftle2016Reaxff}
T.~P. Senftle, S.~Hong, M.~M. Islam, S.~B. Kylasa, Y.~Zheng, Y.~K. Shin,
C.~Junkermeier, R.~Engel-Herbert, M.~J. Janik, H.~M. Aktulga, T.~Verstraelen,
A.~Grama, and A.~C.~T. van Duin, ``The reaxff reactive force-field:
development, applications and future directions,'' {\em npj Computational
Materials}, vol.~2, p.~15011, 2016.
\bibitem{kullgren2013supercharged}
J.~Kullgren, K.~Hermansson, and P.~Broqvist, ``Supercharged low-temperature
oxygen storage capacity of ceria at the nanoscale,'' {\em The journal of
physical chemistry letters}, vol.~4, no.~4, pp.~604--608, 2013.
\bibitem{Even2018}
A.~Marronnier, G.~Roma, S.~Boyer-Richard, L.~Pedesseau, J.~M. Jancu,
Y.~Bonnassieux, C.~Katan, C.~C. Stoumpos, M.~G. Kanatzidis, and J.~Even,
``Anharmonicity and disorder in the black phases of cesium lead iodide used
for stable inorganic perovskite solar cells,'' {\em ACS Nano}, vol.~12,
pp.~3477--3486, 4 2018.
\bibitem{ACESmall2023}
W.~J. Baldwin, X.~Liang, J.~Klarbring, M.~Dubajic, D.~Dell'Angelo, C.~Sutton,
C.~Caddeo, S.~D. Stranks, A.~Mattoni, A.~Walsh, and G.~Csányi, ``Dynamic
local structure in caesium lead iodide: Spatial correlation and transient
domains,'' {\em Small}, vol.~n/a, no.~n/a, p.~2303565, 2023.
\bibitem{fransson2023}
E.~Fransson, J.~Wiktor, and P.~Erhart, ``Phase transitions in inorganic halide
perovskites from machine-learned potentials,'' {\em The Journal of Physical
Chemistry C}, vol.~127, no.~28, pp.~13773--13781, 2023.
\bibitem{Jinnouchi2019}
R.~Jinnouchi, J.~Lahnsteiner, F.~Karsai, G.~Kresse, and M.~Bokdam, ``Phase
transitions of hybrid perovskites simulated by machine-learning force fields
trained on the fly with bayesian inference,'' {\em Phys. Rev. Lett.},
vol.~122, p.~225701, Jun 2019.
\bibitem{hoip}
D.~H. Cao, C.~C. Stoumpos, O.~K. Farha, J.~T. Hupp, and M.~G. Kanatzidis, ``2d
homologous perovskites as light-absorbing materials for solar cell
applications,'' {\em Journal of the American Chemical Society}, vol.~137,
no.~24, pp.~7843--7850, 2015.
\newblock PMID: 26020457.
\bibitem{Wang2023}
T.~Wang, X.~He, M.~Li, B.~Shao, and T.-Y. Liu, ``Aimd-chig: Exploring the
conformational space of a 166-atom protein chignolin with ab initio molecular
dynamics,'' {\em Scientific Data}, vol.~10, Aug. 2023.
\bibitem{agrawalla_reaxff}
S.~Agrawalla and A.~C.~T. van Duin, ``Development and application of a reaxff
reactive force field for hydrogen combustion,'' {\em The Journal of Physical
Chemistry A}, vol.~115, no.~6, pp.~960--972, 2011.
\newblock PMID: 21261320.
\bibitem{baulch_2005}
D.~L. Baulch, C.~T. Bowman, C.~J. Cobos, R.~A. Cox, T.~Just, J.~A. Kerr, M.~J.
Pilling, D.~Stocker, J.~Troe, W.~Tsang, R.~W. Walker, and J.~Warnatz,
``{Evaluated Kinetic Data for Combustion Modeling: Supplement II},'' {\em
Journal of Physical and Chemical Reference Data}, vol.~34, pp.~757--1397, 07
2005.
\bibitem{martinez2009packmol}
L.~Mart{\'\i}nez, R.~Andrade, E.~G. Birgin, and J.~M. Mart{\'\i}nez, ``Packmol:
A package for building initial configurations for molecular dynamics
simulations,'' {\em Journal of computational chemistry}, vol.~30, no.~13,
pp.~2157--2164, 2009.
\bibitem{stukowski2009visualization}
A.~Stukowski, ``Visualization and analysis of atomistic simulation data with
ovito--the open visualization tool,'' {\em Modeling and simulation in
materials science and engineering}, vol.~18, no.~1, p.~015012, 2009.
\bibitem{steudel2003elemental}
R.~Steudel, {\em Elemental sulfur and sulfur-rich compounds {II}}, vol.~2.
\newblock Springer Science \& Business Media, 2003.
\bibitem{Lindgren2019}
P.~Lindgren, G.~Kastlunger, and A.~A. Peterson, ``Scaled and dynamic
optimizations of nudged elastic bands,'' {\em Journal of Chemical Theory and
Computation}, vol.~15, no.~11, p.~5787–5793, 2019.
\newblock PMID: 31600078.
\bibitem{Makri2019}
S.~Makri, C.~Ortner, and J.~R. Kermode, ``{A preconditioning scheme for minimum
energy path finding methods},'' {\em The Journal of Chemical Physics},
vol.~150, p.~094109, 03 2019.
\bibitem{Trachta2023}
M.~Trachta, O.~Bludsk{\'y}, J.~Vacul{\'i}k, R.~Bul{\'a}nek, and M.~Rube{\v{s}},
``Investigation of br{\o}nsted acidity in zeolites through adsorbates with
diverse proton affinities,'' {\em Scientific Reports}, vol.~13, p.~12380, Jul
2023.
\bibitem{melchionna1993}
S.~Melchionna, G.~Ciccotti, and B.~L. Holian, ``Hoover npt dynamics for systems
varying in shape and size,'' {\em Molecular Physics}, vol.~78, no.~3,
pp.~533--544, 1993.
\bibitem{melchionna2000}
S.~Melchionna, ``Constrained systems and statistical distribution,'' {\em Phys.
Rev. E}, vol.~61, pp.~6165--6170, Jun 2000.
\bibitem{Cucinotta2006}
C.~S. Cucinotta, A.~Ruini, A.~Catellani, and A.~Stirling, ``Ab initio molecular
dynamics study of the keto–enol tautomerism of acetone in solution,'' {\em
ChemPhysChem}, vol.~7, no.~6, p.~1229–1234, 2006.
\bibitem{elena2023}
A.~M. Elena, ``md driver,'' {\em gitlab.com}, 2023.
\newblock 10.5281/zenodo.10432005 - https://gitlab.com/drFaustroll/lavello.
\bibitem{stevensdahn2001}
D.~A. Stevens and J.~R. Dahn, ``The mechanisms of lithium and sodium insertion
in carbon materials,'' {\em Journal of The Electrochemical Society},
vol.~148, p.~A803, jun 2001.
\bibitem{huang2019anode}
J.-X. Huang, G.~Csányi, J.-B. Zhao, J.~Cheng, and V.~L. Deringer,
``First-principles study of alkali-metal intercalation in disordered carbon
anode materials,'' {\em J. Mater. Chem. A}, vol.~7, pp.~19070--19080, 2019.
\bibitem{babar2021graphite}
M.~Babar, H.~L. Parks, G.~Houchins, and V.~Viswanathan, ``An accurate machine
learning calculator for the lithium-graphite system,'' {\em Journal of
Physics: Energy}, vol.~3, p.~014005, dec 2020.
\bibitem{GenreithSchrieverChemrxiv}
A.~Genreith-Schriever, A.~Alexiu, G.~Phillips, C.~Coates, L.~Nagle-Cocco,
J.~Bocarsly, F.~Sayed, S.~Dutton, and C.~Grey, ``Jahn-{T}eller distortions
and phase transitions in \ch{LiNiO2}: Insights from {\it ab initio} molecular
dynamics and variable-temperature x-ray diffraction.,'' {\em ChemRrxiv
preprint}, 2023.
\bibitem{Hiremath2022}
P.~Hiremath, S.~Melin, E.~Bitzek, and P.~A. Olsson, ``Effects of interatomic
potential on fracture behaviour in single- and bicrystalline tungsten,'' {\em
Computational Materials Science}, vol.~207, 2022.
\bibitem{Smirnova2020}
D.~Smirnova, S.~Starikov, G.~D. Leines, Y.~Liang, N.~Wang, M.~N. Popov, I.~A.
Abrikosov, D.~G. Sangiovanni, R.~Drautz, and M.~Mrovec, ``{Atomistic
description of self-diffusion in molybdenum: A comparative theoretical study
of non-Arrhenius behavior},'' {\em Physical Review Materials}, vol.~4, no.~1,
p.~13605, 2020.
\bibitem{Yang2019_Nb}
C.~Yang and L.~Qi, ``{Modified embedded-atom method potential of niobium for
studies on mechanical properties},'' {\em Computational Materials Science},
vol.~161, pp.~351--363, 2019.
\bibitem{Cak2014}
M.~{\v{C}}{\'{a}}k, T.~Hammerschmidt, J.~Rogal, V.~Vitek, and R.~Drautz,
``{Analytic bond-order potentials for the bcc refractory metals Nb, Ta, Mo
and W},'' {\em Journal of Physics Condensed Matter}, vol.~26, no.~19,
p.~195501, 2014.
\bibitem{Starikov2024}
S.~Starikov, P.~Grigorev, and P.~A. Olsson, ``Angular-dependent interatomic
potential for large-scale atomistic simulation of w-mo-nb ternary alloys,''
{\em Computational Materials Science}, vol.~233, p.~112734, 2024.
\bibitem{Ma2019}
P.~W. Ma and S.~L. Dudarev, ``{Universality of point defect structure in
body-centered cubic metals},'' {\em Physical Review Materials}, vol.~3,
no.~1, p.~13605, 2019.
\bibitem{rodney2017ab}
L.~Ventelon, E.~Clouet, and F.~Willaime, ``Ab initio modeling of dislocation
core properties in metals and semiconductors,'' {\em Acta Materialia},
vol.~124, pp.~633--659, 2017.
\bibitem{Goryaeva2021}
A.~M. Goryaeva, J.~D{\'e}r{\`e}s, C.~Lapointe, P.~Grigorev, T.~D. Swinburne,
J.~R. Kermode, L.~Ventelon, J.~Baima, and M.-C. Marinica, ``Efficient and
transferable machine learning potentials for the simulation of crystal
defects in bcc fe and {W},'' {\em Phys. Rev. Materials}, vol.~5, p.~103803,
Oct. 2021.
\bibitem{Ventelon2010}
L.~Ventelon and F.~Willaime, ``Generalized stacking-faults and
screw-dislocation core-structure in bcc iron: A comparison between ab initio
calculations and empirical potentials,'' {\em Philosophical Magazine},
vol.~90, 2010.
\bibitem{Dezerald2014}
L.~Dezerald, L.~Ventelon, E.~Clouet, C.~Denoual, D.~Rodney, and F.~Willaime,
``Ab initio modeling of the two-dimensional energy landscape of screw
dislocations in bcc transition metals,'' {\em Physical Review B}, vol.~89,
p.~024104, 1 2014.
\bibitem{grigorev2023calculation}
P.~Grigorev, A.~M. Goryaeva, M.-C. Marinica, J.~R. Kermode, and T.~D.
Swinburne, ``Calculation of dislocation binding to helium-vacancy defects in
tungsten using hybrid ab initio-machine learning methods,'' {\em Acta
Materialia}, vol.~247, p.~118734, 2023.
\bibitem{Swinburne2017}
T.~D. Swinburne and J.~R. Kermode, ``Computing energy barriers for rare events
from hybrid quantum/classical simulations through the virtual work
principle,'' {\em Phys. Rev. B}, vol.~96, p.~144102, Oct 2017.
\bibitem{bitzek2006structural}
E.~Bitzek, P.~Koskinen, F.~G{\"a}hler, M.~Moseler, and P.~Gumbsch, ``Structural
relaxation made simple,'' {\em Physical review letters}, vol.~97, no.~17,
p.~170201, 2006.
\bibitem{grigorev2020hybrid}
P.~Grigorev, T.~D. Swinburne, and J.~R. Kermode, ``Hybrid quantum/classical
study of hydrogen-decorated screw dislocations in tungsten: Ultrafast pipe
diffusion, core reconstruction, and effects on glide mechanism,'' {\em
Physical Review Materials}, vol.~4, no.~2, p.~023601, 2020.
\bibitem{diffusion_in_alumina_2006}
R.~H. Doremus, ``{Diffusion in alumina},'' {\em Journal of Applied Physics},
vol.~100, p.~101301, 11 2006.
\bibitem{neb_aseneb}
E.~L. Kolsbjerg, M.~N. Groves, and B.~Hammer, ``{An automated nudged elastic
band method},'' {\em The Journal of Chemical Physics}, vol.~145, p.~094107,
09 2016.
\bibitem{neb_jk_optimiser}
S.~Makri, C.~Ortner, and J.~R. Kermode, ``{A preconditioning scheme for minimum
energy path finding methods},'' {\em The Journal of Chemical Physics},
vol.~150, p.~094109, 03 2019.
\bibitem{clark2005first}
S.~J. Clark, M.~D. Segall, C.~J. Pickard, P.~J. Hasnip, M.~I. Probert,
K.~Refson, and M.~C. Payne, ``First principles methods using castep,'' {\em
Zeitschrift f{\"u}r kristallographie-crystalline materials}, vol.~220,
no.~5-6, pp.~567--570, 2005.
\bibitem{pickard2011ab}
C.~J. Pickard and R.~Needs, ``Ab initio random structure searching,'' {\em
Journal of Physics: Condensed Matter}, vol.~23, no.~5, p.~053201, 2011.
\bibitem{bernstein2019novo}
N.~Bernstein, G.~Cs{\'a}nyi, and V.~L. Deringer, ``De novo exploration and
self-guided learning of potential-energy surfaces,'' {\em npj Computational
Materials}, vol.~5, no.~1, p.~99, 2019.
\bibitem{podryabinkin2019accelerating}
E.~V. Podryabinkin, E.~V. Tikhonov, A.~V. Shapeev, and A.~R. Oganov,
``Accelerating crystal structure prediction by machine-learning interatomic
potentials with active learning,'' {\em Physical Review B}, vol.~99, no.~6,
p.~064114, 2019.
\bibitem{pickard2022ephemeral}
C.~J. Pickard, ``Ephemeral data derived potentials for random structure
search,'' {\em Physical Review B}, vol.~106, no.~1, p.~014102, 2022.
\bibitem{vanderbilt1990soft}
D.~Vanderbilt, ``Soft self-consistent pseudopotentials in a generalized
eigenvalue formalism,'' {\em Physical review B}, vol.~41, no.~11, p.~7892,
1990.
\bibitem{hart2018one}
M.~Hart, J.~Chen, A.~Michaelides, A.~Sella, M.~S. Shaffer, and C.~G. Salzmann,
``One-dimensional arsenic allotropes: Polymerization of yellow arsenic inside
single-wall carbon nanotubes,'' {\em Angewandte Chemie}, vol.~130, no.~36,
pp.~11823--11827, 2018.
\bibitem{schiferl1969crystal}
D.~Schiferl and C.~Barrett, ``The crystal structure of arsenic at 4.2, 78 and
299 k,'' {\em Journal of Applied Crystallography}, vol.~2, no.~1, pp.~30--36,
1969.
\bibitem{smith1975structures}
P.~Smith, A.~Leadbetter, and A.~Apling, ``The structures of orthorhombic and
vitreous arsenic,'' {\em Philosophical Magazine}, vol.~31, no.~1, pp.~57--64,
1975.
\bibitem{silas2008density}
P.~Silas, J.~R. Yates, and P.~D. Haynes, ``Density-functional investigation of
the rhombohedral to simple-cubic phase transition of arsenic,'' {\em Physical
Review B}, vol.~78, no.~17, p.~174101, 2008.
\bibitem{southard2018perry}
M.~Southard and D.~Green, {\em Perry's Chemical Engineers' Handbook, 9th
Edition}.
\newblock {McGraw-Hill Education}, 2018.
\bibitem{liu2008density}
L.-M. Liu, M.~Krack, and A.~Michaelides, ``Density oscillations in a nanoscale
water film on salt: Insight from ab initio molecular dynamics,'' {\em Journal
of the American Chemical Society}, vol.~130, pp.~8572--8573, 11 2008.
\bibitem{Chiang_muse_2023}
Y.~Chiang, ``Muse: A python package for fast building amorphous solids and
liquid mixtures,'' {\em github.com}, Dec. 2023.
\newblock 10.5281/zenodo.10369245.
\bibitem{van2021coupled}
G.~van Oudenaren, J.~Ocadiz-Flores, and A.~Smith, ``Coupled
structural-thermodynamic modeling of the molten salt system nacl-ucl3,'' {\em
Journal of Molecular Liquids}, vol.~342, p.~117470, 2021.
\bibitem{andersson2022ab}
D.~Andersson and B.~W. Beeler, ``Ab initio molecular dynamics (aimd)
simulations of nacl, ucl3 and nacl-ucl3 molten salts,'' {\em Journal of
Nuclear Materials}, vol.~568, p.~153836, 2022.
\bibitem{rebeloDetailedThermodynamicAnalysis2004}
L.~P.~N. Rebelo, V.~Najdanovic-Visak, Z.~P. Visak, d.~P. M.~Nunes,
J.~Szydlowski, C.~A. Cerdeiriña, J.~Troncoso, L.~Romaní, J.~M. S.~S.
Esperança, H.~J.~R. Guedes, and d.~S. H.~C., ``A detailed thermodynamic
analysis of [{{C4mim}}][{{BF4}}] + water as a case study to model ionic
liquid aqueous solutions,'' {\em Green Chemistry}, vol.~6, no.~8,
pp.~369--381, 2004.
\bibitem{perltFindingBestDensity2018}
E.~Perlt, P.~Ray, A.~Hansen, F.~Malberg, S.~Grimme, and B.~Kirchner, ``Finding
the best density functional approximation to describe interaction energies
and structures of ionic liquids in molecular dynamics studies,'' {\em The
Journal of Chemical Physics}, vol.~148, no.~19, p.~193835, 2018.
\bibitem{gregoryanz2020everything}
E.~Gregoryanz, C.~Ji, P.~Dalladay-Simpson, B.~Li, R.~T. Howie, and H.-K. Mao,
``Everything you always wanted to know about metallic hydrogen but were
afraid to ask,'' {\em Matter and Radiation at Extremes}, vol.~5, no.~3, 2020.
\bibitem{magduau2017simple}
I.~B. Magd{\u{a}}u, M.~Marqu{\'e}s, B.~Borgulya, and G.~J. Ackland, ``Simple
thermodynamic model for the hydrogen phase diagram,'' {\em Physical Review
B}, vol.~95, no.~9, p.~094107, 2017.
\bibitem{magduau2017theory}
I.~B. Magd{\u{a}}u, F.~Balm, and G.~J. Ackland, ``Theory of high pressure
hydrogen, made simple,'' in {\em Journal of Physics: Conference Series},
vol.~950, 4, p.~042059, IOP Publishing, 2017.
\bibitem{pickard2012density}
C.~J. Pickard, M.~Martinez-Canales, and R.~J. Needs, ``Density functional
theory study of phase iv of solid hydrogen,'' {\em Physical Review B},
vol.~85, no.~21, p.~214114, 2012.
\bibitem{magduau2013identification}
I.~B. Magd{\u{a}}u and G.~J. Ackland, ``Identification of high-pressure phases
iii and iv in hydrogen: Simulating raman spectra using molecular dynamics,''
{\em Physical Review B}, vol.~87, no.~17, p.~174110, 2013.
\bibitem{howie2014phonon}
R.~T. Howie, I.~B. Magd{\u{a}}u, A.~F. Goncharov, G.~J. Ackland, and
E.~Gregoryanz, ``Phonon localization by mass disorder in dense
hydrogen-deuterium binary alloy,'' {\em Physical review letters}, vol.~113,
no.~17, p.~175501, 2014.
\bibitem{magduau2017infrared}
I.~B. Magd{\u{a}}u and G.~J. Ackland, ``Infrared peak splitting from phonon
localization in solid hydrogen,'' {\em Physical review letters}, vol.~118,
no.~14, p.~145701, 2017.
\bibitem{cooke2020raman}
P.~I. Cooke, I.~B. Magd{\u{a}}u, M.~Pe{\~n}a-Alvarez, V.~Afonina,
P.~Dalladay-Simpson, X.-D. Liu, R.~T. Howie, E.~Gregoryanz, and G.~J.
Ackland, ``Raman signal from a hindered hydrogen rotor,'' {\em Physical
Review B}, vol.~102, no.~6, p.~064102, 2020.
\bibitem{cheng2020evidence}
B.~Cheng, G.~Mazzola, C.~J. Pickard, and M.~Ceriotti, ``Evidence for
supercritical behaviour of high-pressure liquid hydrogen,'' {\em Nature},
vol.~585, no.~7824, pp.~217--220, 2020.
\bibitem{zong2020understanding}
H.~Zong, H.~Wiebe, and G.~J. Ackland, ``Understanding high pressure molecular
hydrogen with a hierarchical machine-learned potential,'' {\em Nature
communications}, vol.~11, no.~1, p.~5014, 2020.
\bibitem{Frueh2011}
S.~Frueh, R.~Kellett, C.~Mallery, T.~Molter, W.~S. Willis, C.~King’ondu, and
S.~L. Suib, ``Pyrolytic decomposition of ammonia borane to boron nitride,''
{\em Inorganic Chemistry}, vol.~50, pp.~783--792, 2 2011.
\newblock doi: 10.1021/ic101020k.
\bibitem{Marfavi2022}
A.~Marfavi, P.~Kavianpour, and L.~M. Rendina, ``Carboranes in drug discovery,
chemical biology and molecular imaging,'' {\em Nature Reviews Chemistry},
vol.~6, pp.~486--504, 2022.
\bibitem{Sivaev2000}
I.~B. Sivaev and V.~I. Bregadze, ``Chemistry of nickel and iron
bis(dicarbollides). a review,'' {\em Journal of Organometallic Chemistry},
vol.~614-615, pp.~27--36, 2000.
\bibitem{Brown2006}
C.~A. Brown and M.~L. McKee, ``Rearrangements in icosahedral boranes and
carboranes revisited,'' {\em Journal of Molecular Modeling}, vol.~12,
pp.~653--664, 2006.
\bibitem{Neese2022}
F.~Neese, ``Software update: The orca program system—version 5.0,'' {\em
WIREs Computational Molecular Science}, vol.~12, p.~e1606, 9 2022.
\bibitem{rosen2018comprehensive}
A.~S. Rosen, J.~M. Notestein, and R.~Q. Snurr, ``Comprehensive phase diagrams
of mos2 edge sites using dispersion-corrected dft free energy calculations,''
{\em The Journal of Physical Chemistry C}, vol.~122, no.~27,
pp.~15318--15329, 2018.
\bibitem{kieczka2023defects}
D.~Kieczka, T.~Durrant, K.~Milton, K.~E.~J. Goh, M.~Bosman, and A.~Shluger,
``Defects in ws2 monolayer calculated with a nonlocal functional: any
difference from gga?,'' {\em Electronic Structure}, vol.~5, no.~2, p.~024001,
2023.
\bibitem{fang2021formation}
Z.~Fang, M.~P. Confer, Y.~Wang, Q.~Wang, M.~R. Kunz, E.~J. Dufek, B.~Liaw,
T.~M. Klein, D.~A. Dixon, and R.~Fushimi, ``Formation of surface impurities
on lithium--nickel--manganese--cobalt oxides in the presence of co2 and
h2o,'' {\em Journal of the American Chemical Society}, vol.~143, no.~27,
pp.~10261--10274, 2021.
\bibitem{larsen2017atomic}
A.~H. Larsen, J.~J. Mortensen, J.~Blomqvist, I.~E. Castelli, R.~Christensen,
M.~Du{\l}ak, J.~Friis, M.~N. Groves, B.~Hammer, C.~Hargus, {\em et~al.},
``The atomic simulation environment—a python library for working with
atoms,'' {\em Journal of Physics: Condensed Matter}, vol.~29, no.~27,
p.~273002, 2017.
\bibitem{VANDENEIJNDEN2006}
E.~Vanden-Eijnden and G.~Ciccotti, ``Second-order integrators for langevin
equations with holonomic constraints,'' {\em Chemical Physics Letters},
vol.~429, no.~1, pp.~310--316, 2006.
\bibitem{wang2021predicting}
H.-C. Wang, S.~Botti, and M.~A. Marques, ``Predicting stable crystalline
compounds using chemical similarity,'' {\em npj Computational Materials},
vol.~7, no.~1, p.~12, 2021.
\bibitem{glawe_optimal_2016}
H.~Glawe, A.~Sanna, E.~K.~U. Gross, and M.~A.~L. Marques, ``The optimal one
dimensional periodic table: A modified {{Pettifor}} chemical scale from data
mining,'' {\em New Journal of Physics}, vol.~18, no.~9, p.~093011, 2016.
\bibitem{riebesell_matbench_2023}
J.~Riebesell, R.~E.~A. Goodall, Y.~Chiang, A.~Jain, P.~Benner, K.~A. Persson,
and A.~A. Lee, ``Matbench {{Discovery}} -- {{An}} evaluation framework for
machine learning crystal stability prediction,'' {\em arXiv}, 2023.
\bibitem{zagorac_JAC_2019}
D.~Zagorac, H.~M{\"{u}}ller, S.~Ruehl, J.~Zagorac, and S.~Rehme, ``{Recent
developments in the Inorganic Crystal Structure Database: theoretical crystal
structure data and related features},'' {\em J. Appl. Crystallography},
vol.~52, pp.~918--925, Oct 2019.
\bibitem{pan_IC_2021}
H.~Pan, A.~M. Ganose, M.~Horton, M.~Aykol, K.~A. Persson, N.~E.~R. Zimmermann,
and A.~Jain, ``Benchmarking coordination number prediction algorithms on
inorganic crystal structures,'' {\em Inorganic Chem.}, vol.~60, no.~3,
pp.~1590--1603, 2021.
\bibitem{zhangMolecularDynamicsForce2020a}
S.~Zhang, R.~{Schweitzer-Stenner}, and B.~Urbanc, ``Do {{Molecular Dynamics
Force Fields Capture Conformational Dynamics}} of {{Alanine}} in
{{Water}}?,'' {\em Journal of Chemical Theory and Computation}, vol.~16,
pp.~510--527, Jan. 2020.
\bibitem{norskovComputationalDesignSolid2009a}
J.~K. N{\o}rskov, T.~Bligaard, J.~Rossmeisl, and C.~H. Christensen, ``Towards
the computational design of solid catalysts,'' {\em Nat. Chem.}, vol.~1,
pp.~37--46, Apr. 2009.
\bibitem{al-hamdaniPropertiesWaterBoron2017a}
Y.~S. {Al-Hamdani}, M.~Rossi, D.~Alf{\`e}, T.~Tsatsoulis, B.~Ramberger, J.~G.
Brandenburg, A.~Zen, G.~Kresse, A.~Gr{\"u}neis, A.~Tkatchenko, and
A.~Michaelides, ``Properties of the water to boron nitride interaction:
{{From}} zero to two dimensions with benchmark accuracy,'' {\em J. Chem.
Phys.}, vol.~147, p.~044710, July 2017.
\bibitem{brandenburgPhysisorptionWaterGraphene2019}
J.~G. Brandenburg, A.~Zen, M.~Fitzner, B.~Ramberger, G.~Kresse, T.~Tsatsoulis,
A.~Gr{\"u}neis, A.~Michaelides, and D.~Alf{\`e}, ``Physisorption of water on
graphene: {{Subchemical}} accuracy from many-body electronic structure
methods,'' {\em J. Phys. Chem. Lett.}, vol.~10, pp.~358--368, Feb. 2019.
\bibitem{ehlertCO2GrapheneBenchmarking2023a}
C.~Ehlert, A.~Piras, and G.~Gryn'ova, ``{{CO$_2$}} on graphene: {Benchmarking}
computational approaches to noncovalent interactions,'' {\em ACS Omega},
vol.~8, pp.~35768--35778, Oct. 2023.
\bibitem{tsatsoulisReactionEnergeticsHydrogen2018a}
T.~Tsatsoulis, S.~Sakong, A.~Gro{\ss}, and A.~Gr{\"u}neis, ``Reaction
energetics of hydrogen on {{Si}}(100) surface: {{A}} periodic many-electron
theory study,'' {\em J. Chem. Phys.}, vol.~149, p.~244105, Dec. 2018.
\bibitem{tsatsoulisComparisonQuantumChemistry2017}
T.~Tsatsoulis, F.~Hummel, D.~Usvyat, M.~Sch{\"u}tz, G.~H. Booth, S.~S. Binnie,
M.~J. Gillan, D.~Alf{\`e}, A.~Michaelides, and A.~Gr{\"u}neis, ``A comparison
between quantum chemistry and quantum {{Monte Carlo}} techniques for the
adsorption of water on the (001) {{LiH}} surface,'' {\em J. Chem. Phys.},
vol.~146, p.~204108, May 2017.
\bibitem{yeInitioSurfaceChemistry2023}
H.-Z. Ye and T.~C. Berkelbach, ``Ab initio surface chemistry with chemical
accuracy,'' {\em arXiv preprint arXiv:2309.14640}, 2023.
\bibitem{lustembergVibrationalFrequenciesCeriumOxideBound2020c}
P.~G. Lustemberg, P.~N. Plessow, Y.~Wang, C.~Yang, A.~Nefedov, F.~Studt,
C.~W{\"o}ll, and M.~V. {Ganduglia-Pirovano}, ``Vibrational frequencies of
cerium-oxide-bound {CO}: {A} challenge for conventional dft methods,'' {\em
Phys. Rev. Lett.}, vol.~125, p.~256101, Dec. 2020.
\bibitem{shiManyBodyMethodsSurface2023a}
B.~X. Shi, A.~Zen, V.~Kapil, P.~R. Nagy, A.~Gr{\"u}neis, and A.~Michaelides,
``Many-body methods for surface chemistry come of age: {Achieving} consensus
with experiments,'' {\em J. Am. Chem. Soc.}, vol.~145, pp.~25372--25381, Nov.
2023.
\bibitem{doi:10.1126/science.abj0890}
N.~Hanikel, X.~Pei, S.~Chheda, H.~Lyu, W.~Jeong, J.~Sauer, L.~Gagliardi, and
O.~M. Yaghi, ``Evolution of water structures in metal-organic frameworks for
improved atmospheric water harvesting,'' {\em Science}, vol.~374,
pp.~454--459, 2021.
\bibitem{doi:10.1021/acscatal.2c05493}
F.~Berger, M.~Rybicki, and J.~Sauer, ``Molecular dynamics with chemical
accuracy--{Alkane} adsorption in acidic zeolites,'' {\em ACS Catal.},
vol.~13, pp.~2011--2024, 2023.
\bibitem{https://doi.org/10.1002/anie.202013671}
F.~Berger and J.~Sauer, ``Dimerization of linear butenes and pentenes in an
acidic zeolite {(H-MFI)},'' {\em Angew. Chem., Int. Ed.}, vol.~60,
pp.~3529--3533, 2021.
\bibitem{VASP2}
G.~Kresse and J.~Hafner, ``Ab initio molecular-dynamics simulation of the
liquid-metal-amorphous-semiconductor transition in germanium,'' {\em Phys.
Rev. B}, vol.~49, p.~14251, 1994.
\bibitem{VASP3}
G.~Kresse and J.~Furthmüller, ``Efficiency of ab-initio total energy
calculations for metals and semiconductors using a plane-wave basis set,''
{\em Comput. Mat. Sci.}, vol.~6, p.~15, 1996.
\bibitem{norskovDensityFunctionalTheory2011b}
J.~K. N{\o}rskov, F.~{Abild-Pedersen}, F.~Studt, and T.~Bligaard, ``Density
functional theory in surface chemistry and catalysis,'' {\em Proceedings of
the National Academy of Sciences}, vol.~108, pp.~937--943, Jan. 2011.
\bibitem{chanussotOpenCatalyst20202021}
L.~Chanussot, A.~Das, S.~Goyal, T.~Lavril, M.~Shuaibi, M.~Riviere, K.~Tran,
J.~{Heras-Domingo}, C.~Ho, W.~Hu, A.~Palizhati, A.~Sriram, B.~Wood, J.~Yoon,
D.~Parikh, C.~L. Zitnick, and Z.~Ulissi, ``Open {{Catalyst}} 2020 ({{OC20}})
dataset and community challenges,'' {\em ACS Catal.}, vol.~11,
pp.~6059--6072, May 2021.
\bibitem{tranOpenCatalyst20222023}
R.~Tran, J.~Lan, M.~Shuaibi, B.~M. Wood, S.~Goyal, A.~Das, J.~{Heras-Domingo},
A.~Kolluru, A.~Rizvi, N.~Shoghi, A.~Sriram, F.~Therrien, J.~Abed, O.~Voznyy,
E.~H. Sargent, Z.~Ulissi, and C.~L. Zitnick, ``The {{Open Catalyst}} 2022
({{OC22}}) dataset and challenges for oxide electrocatalysts,'' {\em ACS
Catal.}, vol.~13, pp.~3066--3084, Mar. 2023.
\bibitem{cantor2004microstructural}
B.~Cantor, I.~Chang, P.~Knight, and A.~Vincent, ``Microstructural development
in equiatomic multicomponent alloys,'' {\em Materials Science and
Engineering: A}, vol.~375, pp.~213--218, 2004.
\bibitem{mazitov2023surface}
A.~Mazitov, M.~A. Springer, N.~Lopanitsyna, G.~Fraux, S.~De, and M.~Ceriotti,
``Surface segregation in high-entropy alloys from alchemical machine
learning,'' {\em arXiv preprint arXiv:2310.07604}, 2023.
\bibitem{thompson2022lammps}
A.~P. Thompson, H.~M. Aktulga, R.~Berger, D.~S. Bolintineanu, W.~M. Brown,
P.~S. Crozier, P.~J. in't Veld, A.~Kohlmeyer, S.~G. Moore, T.~D. Nguyen, {\em
et~al.}, ``Lammps-a flexible simulation tool for particle-based materials
modeling at the atomic, meso, and continuum scales,'' {\em Computer Physics
Communications}, vol.~271, p.~108171, 2022.
\bibitem{trott2022kokkos}
C.~R. Trott, D.~Lebrun-Grandié, D.~Arndt, J.~Ciesko, V.~Dang, N.~Ellingwood,
R.~Gayatri, E.~Harvey, D.~S. Hollman, D.~Ibanez, N.~Liber, J.~Madsen,
J.~Miles, D.~Poliakoff, A.~Powell, S.~Rajamanickam, M.~Simberg,
D.~Sunderland, B.~Turcksin, and J.~Wilke, ``Kokkos 3: Programming model
extensions for the exascale era,'' {\em IEEE Transactions on Parallel and
Distributed Systems}, vol.~33, no.~4, pp.~805--817, 2022.
\bibitem{togo_firstprinciples_2023}
A.~Togo, ``First-principles {{Phonon Calculations}} with {{Phonopy}} and
{{Phono3py}},'' {\em Journal of the Physical Society of Japan}, vol.~92,
no.~1, p.~012001, 2023.
\bibitem{togo_implementation_2023}
A.~Togo, L.~Chaput, T.~Tadano, and I.~Tanaka, ``Implementation strategies in
phonopy and phono3py,'' {\em Journal of Physics: Condensed Matter}, vol.~35,
no.~35, p.~353001, 2023.
\bibitem{stoffel_ab_2010}
R.~P. Stoffel, C.~Wessel, M.-W. Lumey, and R.~Dronskowski, ``Ab {Initio}
{Thermochemistry} of {Solid}-{State} {Materials},'' {\em Angew. Chem. Int.
Ed.}, vol.~49, no.~31, p.~5242, 2010.
\bibitem{bartel_review_2022}
C.~J. Bartel, ``Review of computational approaches to predict the thermodynamic
stability of inorganic solids,'' {\em J Mater Sci}, Feb. 2022.
\bibitem{george_combining_2020}
J.~George, G.~Hautier, A.~P. Bartók, G.~Csányi, and V.~L. Deringer,
``Combining phonon accuracy with high transferability in {Gaussian}
approximation potential models,'' {\em J. Chem. Phys.}, vol.~153, p.~044104,
July 2020.
\newblock Publisher: American Institute of Physics.
\bibitem{atomate2}
``atomate2.'' \url{https://github.com/materialsproject/atomate2}.
\bibitem{dunn_benchmarking_2020}
A.~Dunn, Q.~Wang, A.~Ganose, D.~Dopp, and A.~Jain, ``Benchmarking materials
property prediction methods: the {Matbench} test set and {Automatminer}
reference algorithm,'' {\em Npj Comput. Mater.}, vol.~6, pp.~1--10, Sept.
2020.
\newblock Number: 1 Publisher: Nature Publishing Group.
\bibitem{agne_minimum_2018}
M.~T. Agne, R.~Hanus, and G.~J. Snyder, ``Minimum thermal conductivity in the
context of \textit{diffuson} -mediated thermal transport,'' {\em Energy
Environ. Sci.}, vol.~11, no.~3, pp.~609--616, 2018.
\bibitem{voigt_lehrbuch_1910}
W.~Voigt, {\em Lehrbuch der kristallphysik: (mit ausschluss der
kristalloptik)}.
\newblock B.G. Teubner, 1910.
\newblock Google-Books-{ID}: 9GISAAAAIAAJ.
\bibitem{reuss_berechnung_1929}
A.~Reuss, ``Berechnung der fließgrenze von mischkristallen auf grund der
plastizitätsbedingung für einkristalle .,'' {\em {ZAMM} - Journal of
Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik
und Mechanik}, vol.~9, no.~1, pp.~49--58, 1929.
\newblock https://onlinelibrary.wiley.com/doi/pdf/10.1002/zamm.19290090104.
\bibitem{hill_elastic_1952}
R.~Hill, ``The elastic behaviour of a crystalline aggregate,'' {\em Proc. Phys.
Soc. A}, vol.~65, no.~5, p.~349, 1952.
\bibitem{Riebesell_MatCalc_2023}
J.~Riebesell, E.~Liu, J.~Qi, T.~W. Ko, and S.~P. Ong, ``{MatCalc: A Python
library for calculating materials properties},'' {\em github.com}, 2023.
\newblock released July 2023, https://github.com/materialsvirtuallab/matcalc.
\bibitem{Huber1992}
P.~J. Huber, ``Robust estimation of a location parameter,'' in {\em
Breakthroughs in statistics: Methodology and distribution}, pp.~492--518,
Springer, 1992.
\bibitem{bishop1995neural}
C.~M. Bishop, {\em Neural networks for pattern recognition}.
\newblock Oxford university press, 1995.
\bibitem{wang_framework_2021}
A.~Wang, R.~Kingsbury, M.~McDermott, M.~Horton, A.~Jain, S.~P. Ong,
S.~Dwaraknath, and K.~A. Persson, ``A framework for quantifying uncertainty
in {{DFT}} energy corrections,'' {\em Scientific Reports}, vol.~11, p.~15496,
July 2021.
\bibitem{kingsbury2022flexible}
R.~S. Kingsbury, A.~S. Rosen, A.~S. Gupta, J.~M. Munro, S.~P. Ong, A.~Jain,
S.~Dwaraknath, M.~K. Horton, and K.~A. Persson, ``A flexible and scalable
scheme for mixing computed formation energies from different levels of
theory,'' {\em npj Computational Materials}, vol.~8, no.~1, p.~195, 2022.
\bibitem{riebesell_pymatviz_2022}
J.~Riebesell, ``Pymatviz: visualization toolkit for materials informatics,''
{\em github.com}, 2022.
\newblock 10.5281/zenodo.7486816 - https://github.com/janosh/pymatviz.
\bibitem{deComparingMoleculesSolids2016a}
S.~De, A.~P. Bart{\'o}k, G.~Cs{\'a}nyi, and M.~Ceriotti, ``Comparing molecules
and solids across structural and alchemical space,'' {\em Physical Chemistry
Chemical Physics}, vol.~18, no.~20, pp.~13754--13769, 2016.
\bibitem{frauxChemiscopeInteractiveStructureproperty2020}
G.~Fraux, R.~K. Cersonsky, and M.~Ceriotti, ``Chemiscope: Interactive
structure-property explorer for materials and molecules,'' {\em Journal of
Open Source Software}, vol.~5, no.~51, p.~2117, 2020.
\bibitem{mcinnesUMAPUniformManifold2020}
L.~McInnes, J.~Healy, and J.~Melville, ``{{UMAP}}: {{Uniform Manifold
Approximation}} and {{Projection}} for {{Dimension Reduction}},'' 2020.
\end{thebibliography}
```
5. **Author Information:**
- Lead Author: {'name': 'Ilyes Batatia'}
- Full Authors List:
```yaml
Ilyes Batatia: {}
Philipp Benner: {}
Yuan Chiang: {}
Alin M. Elena: {}
"D\xE1vid P. Kov\xE1cs": {}
Janosh Riebesell: {}
Xavier R. Advincula: {}
Mark Asta: {}
Matthew Avaylon: {}
William J. Baldwin: {}
Fabian Berger: {}
Noam Bernstein: {}
Arghya Bhowmik: {}
Samuel M. Blau: {}
"Vlad C\u0103rare": {}
James P. Darby: {}
Sandip De: {}
Flaviano Della Pia: {}
Volker L. Deringer: {}
"Rokas Elijo\u0161ius": {}
Zakariya El-Machachi: {}
Fabio Falcioni: {}
Edvin Fako: {}
Andrea C. Ferrari: {}
Annalena Genreith-Schriever: {}
Janine George: {}
Rhys E. A. Goodall: {}
Clare P. Grey: {}
Petr Grigorev: {}
Shuang Han: {}
Will Handley:
pi:
start: 2020-10-01
thesis: null
postdoc:
start: 2016-10-01
end: 2020-10-01
thesis: null
phd:
start: 2012-10-01
end: 2016-09-30
supervisors:
- Anthony Lasenby
- Mike Hobson
thesis: 'Kinetic initial conditions for inflation: theory, observation and methods'
original_image: images/originals/will_handley.jpeg
image: /assets/group/images/will_handley.jpg
links:
Webpage: https://willhandley.co.uk
Hendrik H. Heenen: {}
Kersti Hermansson: {}
Christian Holm: {}
Jad Jaafar: {}
Stephan Hofmann: {}
Konstantin S. Jakob: {}
Hyunwook Jung: {}
Venkat Kapil: {}
Aaron D. Kaplan: {}
Nima Karimitari: {}
James R. Kermode: {}
Namu Kroupa:
phd:
start: 2023-10-01
supervisors:
- "G\xE1bor Cs\xE1nyi"
- Will Handley
thesis: null
summer:
start: 2023-06-12
end: 2023-09-30
thesis: null
original_image: images/originals/namu_kroupa.jpg
image: /assets/group/images/namu_kroupa.jpg
Jolla Kullgren: {}
Matthew C. Kuner: {}
Domantas Kuryla: {}
Guoda Liepuoniute: {}
Johannes T. Margraf: {}
"Ioan-Bogdan Magd\u0103u": {}
Angelos Michaelides: {}
J. Harry Moore: {}
Aakash A. Naik: {}
Samuel P. Niblett: {}
Sam Walton Norwood: {}
Niamh O'Neill: {}
Christoph Ortner: {}
Kristin A. Persson: {}
Karsten Reuter: {}
Andrew S. Rosen: {}
Lars L. Schaaf: {}
Christoph Schran: {}
Benjamin X. Shi: {}
Eric Sivonxay: {}
"Tam\xE1s K. Stenczel": {}
Viktor Svahn: {}
Christopher Sutton: {}
Thomas D. Swinburne: {}
Jules Tilly: {}
Cas van der Oord: {}
Eszter Varga-Umbrich: {}
Tejs Vegge: {}
"Martin Vondr\xE1k": {}
Yangshuai Wang: {}
William C. Witt: {}
Fabian Zills: {}
"G\xE1bor Cs\xE1nyi":
coi:
start: 2023-10-01
thesis: null
image: https://www.eng.cam.ac.uk/sites/www.eng.cam.ac.uk/files/styles/small_events_search_results_profile/public/uploads/profiles/images/Gabor-kalapos-square-400.png?itok=GszDCAc9
links:
Department webpage: http://www.eng.cam.ac.uk/profiles/gc121
```
This YAML file provides a concise snapshot of an academic research group. It lists members by name along with their academic roles—ranging from Part III and summer projects to MPhil, PhD, and postdoctoral positions—with corresponding dates, thesis topics, and supervisor details. Supplementary metadata includes image paths and links to personal or departmental webpages. A dedicated "coi" section profiles senior researchers, highlighting the group’s collaborative mentoring network and career trajectories in cosmology, astrophysics, and Bayesian data analysis.
====================================================================================
Final Output Instructions
====================================================================================
- Combine all data sources to create a seamless, engaging narrative.
- Follow the exact Markdown output format provided at the top.
- Do not include any extra explanation, commentary, or wrapping beyond the specified Markdown.
- Validate that every bibliographic reference with a DOI or arXiv identifier is converted into a Markdown link as per the examples.
- Validate that every Markdown author link corresponds to a link in the author information block.
- Before finalizing, confirm that no LaTeX citation commands or other undesired formatting remain.
- Before finalizing, confirm that the link to the paper itself [2401.00096](https://arxiv.org/abs/2401.00096) is featured in the first sentence.
Generate only the final Markdown output that meets all these requirements.
{% endraw %}