Foundations of computational materials science

Subjects

Eight books, from the foundations to current research methods. Each opens as a single page with its own contents.

Mathematics for scienceCalculus, linear algebra and probability

Computers, Linux and compiled codeProcessors, Linux and compiled code

Python for scientific computingFrom the interpreter to NumPy

Machine learningFrom one neuron to transformers

Physics and statistical mechanicsFrom Newton to free energy

Molecular simulationMolecular dynamics and Monte Carlo

Rare events, free energies and enhanced samplingFree energies and rare events

Machine learning for materialsInteratomic potentials and generative models Machine learning for materials A history of interatomic potentialsLearning a potential energy surfaceSymmetry and atomic descriptorsRepresenting materialsKernel and linear potentialsAtomic neural-network potentialsMessage-passing and equivariant potentialsTraining data and validationBuilding a potential from scratchUniversal potentialsActive learningGenerative models for crystals and moleculesDifferentiable simulation Open the book →

What these notes cover

The notes run from the foundations to current research methods, in eight subjects.

Mathematics, computing and Python provide the working tools: calculus, linear algebra and probability; how a processor, the Linux shell and compiled code work; and Python from the interpreter and its object model up to NumPy and performance.

Machine learning and physics and statistical mechanics provide the two theories that everything else depends on. The first covers learning from data, from a single artificial neuron to deep networks. The second covers microstates and entropy, the Boltzmann distribution, ensembles, free energy, fluctuations, phase transitions and the structure of liquids.

Molecular simulation and enhanced sampling turn the physics into practice: molecular dynamics and Monte Carlo, thermostats and barostats, transport coefficients and trajectory analysis, then energy landscapes and rate theory, free-energy perturbation, umbrella sampling, metadynamics, OPES, replica exchange, path sampling and Markov state models.

Machine learning for materials brings the threads together: symmetry and descriptors, the history and theory of machine-learned interatomic potentials from Behler–Parrinello networks to MACE and the universal models, generative models for crystals and differentiable simulation. It includes a complete tutorial that builds, trains and validates an interatomic potential.

How I prepared these notes

I came to molecular simulation from materials engineering, and to machine learning through my PhD research. Almost every method here I first learned the hard way: from textbooks written for physicists, papers that skipped the steps I needed, and software manuals that assumed I already knew why a setting mattered.

Preparing these chapters meant going back to the original papers, re-deriving the equations line by line, and rebuilding each method from scratch until it behaved. It was slow, and much of it failed first: descriptors too coarse to learn copper forces, free energies that refused to converge, a machine-learned potential that passed its test set and still got the lattice constant wrong. Those failures are in the chapters, because they teach as much as the successes. Every figure is a small simulation written from scratch, and every number in the text comes from code I ran.

The notes are for students and early-career scientists who want to understand the methods they use, not only run them. If you find a mistake, an unclear passage or a missing topic, please write to me at sibasankarpanigrahy08@gmail.com.

How to read a chapter

Every chapter is built from the same elements. The excerpt below is live: move the slider.

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4.2 Deriving the Boltzmann factor

A system in contact with a heat bath at temperature T is found in a state of energy E_n with a probability that falls exponentially with energy. Ludwig Boltzmann reached this result in 1877 by counting the number of ways the energy of a gas can be shared among its molecules.

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Populations of six evenly spaced levels. At low temperature almost everything is in the ground state; as k_BT grows past the spacing, the populations even out.

5 Key pointsSelf-check questionsReferences

One page per subject. Each book is a single scrolling page. The contents panel tracks your position and jumps to any chapter or section; on a phone it opens from the button at the top.
Text first. Chapters are continuous prose with numbered sections. They begin with the question that motivated an idea and the people and papers that answered it, then develop the theory step by step.
Equation cards. Key equations are set apart with every symbol explained and a one-line reading in words. Colours in the equation match the colours in the figures.
Interactive figures and code. Figures compute their results as you change the parameters. Code cells run real Python with NumPy in the page; the first run downloads the interpreter, about 10 MB.
End matter. Each chapter closes with its key points, a few questions that explain their answers, and the original papers and textbook chapters for further reading. Chapters can be marked as read; progress is stored only in your browser.

Where to start

New to programmingBegin with Python for scientific computing, which starts from how the interpreter runs your code, and use the Mathematics book alongside it. A scientist new to machine learningRead Machine learning from the single neuron onwards, then continue to Machine learning for materials. Already running simulationsUse Molecular simulation to check the foundations of what you run, then Enhanced sampling for free energies and rare events. Building an interatomic potentialRead the foundations part of Machine learning for materials, then follow the tutorial that builds and trains a potential from scratch.

Dependency map

Every chapter, grouped by subject, with lines to its prerequisites. Hover a chapter to highlight what it builds on; click to open it.

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Reference tools

Practical pages for day-to-day work on a cluster.

Core textbooks

The books below are the main references across the subjects. Each chapter also lists the papers and book chapters specific to its topic.

C. M. Bishop, Pattern Recognition and Machine Learning, Springer (2006). The standard treatment of probabilistic machine learning.
K. P. Murphy, Probabilistic Machine Learning: An Introduction, MIT Press (2022). Free online; modern and broad.
I. Goodfellow, Y. Bengio and A. Courville, Deep Learning, MIT Press (2016). Free online.
T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning, 2nd ed., Springer (2009). Free online.
C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning, MIT Press (2006). Free online.
D. Frenkel and B. Smit, Understanding Molecular Simulation, 3rd ed., Academic Press (2023).
M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, 2nd ed., Oxford University Press (2017).
M. E. Tuckerman, Statistical Mechanics: Theory and Molecular Simulation, 2nd ed., Oxford University Press (2023).
M. M. Bronstein, J. Bruna, T. Cohen and P. Veličković, Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges, arXiv:2104.13478 (2021).