Appendix D — Further Reading¶

An annotated bibliography of textbooks, major review articles and open courseware that the editors consider most useful for the reader who has finished this book and wants to go further. The list is organised by topic, following the structure of the book itself.

For each entry we have included a one- to three-sentence note about what the work is best for and at what level. The classification "introductory" means a student who has just finished an undergraduate quantum-mechanics or solid-state course can usefully read it; "intermediate" means a finishing PhD student or postdoc will get the most from it; "advanced" means the work is genuinely a specialist reference.

The full bibliographic details — authors, year, publisher, ISBN where applicable — are in Appendix E.

D.1 Quantum mechanics¶

The foundation of everything else in this book. Three textbooks span the difficulty spectrum from undergraduate to graduate.

Griffiths, Introduction to Quantum Mechanics (3^rd edition, 2018). Introductory. The standard first textbook on the subject. Clear, discursive, deliberately physical in its approach. Read it before attempting any of the others if your quantum mechanics is shaky.

Shankar, Principles of Quantum Mechanics (2^nd edition, 1994). Introductory to intermediate. More formal than Griffiths, with careful treatment of the postulates and the linear-algebra structure. The chapter on path integrals is the best introductory treatment we know.

Cohen-Tannoudji, Diu and Laloë, Quantum Mechanics (Vols. I-III, 2^nd edition, 2019). Intermediate to advanced. The encyclopaedic French school text. Every topic is treated thoroughly, every formula is derived. Useful as a reference even if one does not read it cover to cover. Volume III, on relativistic and many-body methods, is particularly relevant to electronic-structure theory.

Sakurai and Napolitano, Modern Quantum Mechanics (3^rd edition, 2020). Intermediate. The graduate-level companion to Griffiths. The chapters on angular momentum and rotation operators are essential preparation for any work on equivariance.

D.2 Solid-state physics¶

For the language of band structures, Brillouin zones, phonons and all the surrounding crystallography.

Ashcroft and Mermin, Solid State Physics (1976). Intermediate. Despite its age, still the standard reference. The treatments of the free-electron and tight-binding models, of phonons, and of the de Haas–van Alphen effect remain unsurpassed. The notation is slightly old-fashioned but mostly faithful to current usage.

Marder, Condensed Matter Physics (2^nd edition, 2010). Intermediate to advanced. A modernised Ashcroft-and-Mermin with better treatment of magnetism, superconductivity and soft-matter topics. The chapters on electronic structure and on phonons are the natural complement to Chapters 5 and 8 of this book.

Kittel, Introduction to Solid State Physics (8^th edition, 2005). Introductory. Wider in scope, lighter in mathematical detail than Ashcroft-and-Mermin. Useful for orientation when one encounters an unfamiliar phenomenon and needs the standard elementary explanation.

Grosso and Pastori Parravicini, Solid State Physics (2^nd edition, 2014). Intermediate. Italian school text with strong emphasis on electronic structure, semiconductors and group- theoretic methods. The treatment of the k·p method and of spin–orbit coupling is the cleanest we know.

D.3 Density functional theory¶

The methodological core of Chapters 5 and 6.

Martin, Electronic Structure: Basic Theory and Practical Methods (2^nd edition, 2020). Advanced. The standard graduate- level text. Comprehensive, careful, and unafraid to discuss the practical difficulties and limitations of DFT calculations. The companion volume Interacting Electrons (2016, with Reining and Ceperley) covers post-DFT methods (GW, BSE, DMFT) at the same level.

Parr and Yang, Density-Functional Theory of Atoms and Molecules (1989). Intermediate to advanced. The quantum-chemistry view of DFT: rigorous, mathematically dense, and surprisingly relevant to materials practitioners. The chapter on the chemical potential is the most insightful discussion of the conceptual content of DFT we have read.

Sholl and Steckel, Density Functional Theory: A Practical Introduction (2009). Introductory. The right book for a graduate student running their first DFT calculation. Covers VASP-style plane-wave pseudopotential calculations, convergence testing, and the standard pitfalls. Light on theory, heavy on practice.

Burke and Wagner, DFT in a Nutshell (International Journal of Quantum Chemistry, 2013). Introductory review article. Twelve pages, free online, and the best short introduction to the conceptual content of DFT we know. A surprising number of senior researchers have learnt the essentials from this article.

Levy, "Universal variational functionals of electron densities..." (PNAS, 1979), and Lieb, "Density functionals for Coulomb systems" (International Journal of Quantum Chemistry, 1983). Advanced. The two papers that put the Hohenberg–Kohn theorems on a firm footing through the constrained-search formulation. Levy gives the physical construction; Lieb supplies the rigorous mathematical analysis. Read them together once the reader wants to understand why the universal functional is well defined, not merely that it exists.

D.4 Molecular dynamics and statistical mechanics¶

The methodological core of Chapters 7 and 8.

Frenkel and Smit, Understanding Molecular Simulation (3^rd edition, 2023). Intermediate. The standard text on MD and Monte Carlo for materials and chemistry. Every common algorithm — Verlet, Nosé–Hoover, replica exchange, free-energy methods — is presented with both physical motivation and ready-to-implement detail. The chapter on free energy is alone worth the price.

Allen and Tildesley, Computer Simulation of Liquids (2^nd edition, 2017). Intermediate. The classical reference, focused on liquids and soft matter. Particularly strong on transport properties and on the practical details of pair correlation functions and structure factors.

Tuckerman, Statistical Mechanics: Theory and Molecular Simulation (2^nd edition, 2023). Advanced. Combines a careful statistical-mechanics development with the corresponding simulation algorithms. The reader who wants to understand the link between an ensemble in the textbook sense and a thermostatted MD trajectory will find it here.

Chandler, Introduction to Modern Statistical Mechanics (1987). Intermediate. Slim and elegant; the right text for a working researcher who wants to refresh the statistical-mechanical foundations underlying Chapters 8 and 11.

D.5 Machine learning for materials¶

The field that Chapters 9 through 12 inhabit. The literature is too new to have settled textbooks; we recommend a mixture of review articles and pivotal primary papers.

Reviews¶

Schmidt, Marques, Botti and Marques, "Recent advances and applications of machine learning in solid-state materials science" (npj Computational Materials, 2019). Intermediate. The standard mid-decade review of the field. Now somewhat dated on the MLIP side but still the best overview of property-prediction work.

Choudhary et al., "Recent advances and applications of deep learning methods in materials science" (npj Computational Materials, 2022). Intermediate. Updated and broader, covering graph neural networks, generative models and the use of foundation models in their early form.

Friederich et al., "Machine-learned potentials for next- generation matter simulations" (Nature Materials, 2021). Intermediate. A balanced review of MLIPs as they stood just before the foundation-model era.

Behler, "Four generations of high-dimensional neural network potentials" (Chemical Reviews, 2021). Intermediate to advanced. A pedagogical history of MLIP architectures from the perspective of one of the founders of the field. Essential reading for anyone who wants to understand why the modern architectures look the way they do.

Primary papers¶

The following are not pedagogical but are too important to omit. All are technical research papers; we expect the reader to skim and return as needed.

Behler and Parrinello, "Generalized neural-network representation of high-dimensional potential-energy surfaces" (Physical Review Letters, 2007). The paper that launched modern MLIPs.

Bartók, Payne, Kondor and Csányi, "Gaussian approximation potentials" (Physical Review Letters, 2010). The first demonstration of GP regression for atomistic energy surfaces.

Schütt et al., "SchNet: A continuous-filter convolutional neural network for modeling quantum interactions" (NeurIPS, 2017). The first deep MLIP, and the architectural ancestor of much of what followed.

Batzner et al., "E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials" (Nature Communications, 2022). The NequIP paper. The first really convincing demonstration that strict equivariance dramatically improves data efficiency.

Batatia et al., "MACE: Higher order equivariant message passing neural networks for fast and accurate force fields" (NeurIPS, 2022). The MACE paper.

Batatia et al., "A foundation model for atomistic materials chemistry" (arXiv:2401.00096, 2024). The MACE-MP-0 paper.

Zeni et al., "MatterGen: a generative model for inorganic materials design" (Nature, 2025). The MatterGen paper.

Merchant et al., "Scaling deep learning for materials discovery" (Nature, 2023). The GNoME paper. A demonstration that graph networks trained at scale, coupled to active learning and high-throughput DFT, can expand the set of known stable inorganic crystals by an order of magnitude. Worth reading both for the method and for the lively debate it provoked about what "discovery" means.

D.6 Foundational data and infrastructure¶

Three resources that should be in every materials-ML practitioner's toolkit.

The Materials Project (materialsproject.org). Free, open, well documented. The starting point for almost any high-throughput study of inorganic materials. The accompanying paper (Jain et al., APL Materials, 2013) is short and well worth reading.

pymatgen (pymatgen.org). The Python interface to the Materials Project and to a large fraction of the related computational ecosystem. The online documentation is the right entry point.

ASE — Atomic Simulation Environment (wiki.fysik.dtu.dk/ase). The Python framework for atomistic calculations and for integrating different electronic-structure codes. The standard glue between DFT codes and MLIPs.

e3nn (e3nn.org; Geiger and Smidt, arXiv:2207.09453, 2022). The reference library for E(3)-equivariant neural networks, and the foundation on which NequIP and MACE are built. The documentation and accompanying paper are the right entry point for any reader who wants to understand how equivariant tensor operations are implemented in practice rather than only in the abstract.

D.7 Open courseware¶

For readers who learn best through structured lectures.

Marzari, "Computational Methods in Physics, Chemistry and Materials Science" (EPFL). Recorded lecture course, freely available online. The most pedagogical complete introduction to periodic DFT we know.

Cohen-Tannoudji's lectures at the Collège de France. In French, but transcribed lecture notes (in French and partly translated) remain a treasure for advanced topics in quantum mechanics. Search the Collège de France video archive.

MIT OpenCourseWare 3.320, "Atomistic Computer Modeling of Materials". Older but still excellent introduction to DFT and MD oriented to materials applications. Lecture notes and problem sets are available.

The Coursera course "Materials Informatics" (Schmidt et al.). Recent and pitched at the right level for the modern reader. Covers much of the same ground as Chapters 9–12 of this book.

Berkeley's "Hands-on tutorials" at the Molecular Foundry. Short video tutorials on the practical use of VASP, Quantum ESPRESSO and the Materials Project. Hosted on YouTube; search for "Foundry materials tutorial".

D.8 Mathematical and computational background¶

For the reader who wants to fill in the mathematical or computational prerequisites.

Strang, Linear Algebra and Its Applications (4^th edition, 2006), and the corresponding MIT OpenCourseWare 18.06. The standard. The MIT video lectures are exceptional.

Trefethen and Bau, Numerical Linear Algebra (1997). For the reader who wants to understand the numerical methods (Lanczos, Davidson, conjugate gradients) that lie under the hood of every DFT code.

Press, Teukolsky, Vetterling, Flannery, Numerical Recipes (3^rd edition, 2007). A general-purpose reference for the working scientist who needs an algorithm now. Idiosyncratic and sometimes opinionated, but vast in scope.

Bishop, Pattern Recognition and Machine Learning (2006). Intermediate. Still the best introduction to the probabilistic view of machine learning, including Gaussian processes (relevant to GAP) and to the kernel methods underlying much of pre-2020 materials ML.

Goodfellow, Bengio and Courville, Deep Learning (2016). Intermediate. The standard introduction to deep learning. Some chapters (especially on regularisation and on optimisation) are directly useful for the MLIP practitioner.

Murphy, Probabilistic Machine Learning: An Introduction (2022) and Probabilistic Machine Learning: Advanced Topics (2023). Intermediate to advanced. More current than Bishop, and covers modern topics — diffusion models, transformers, normalising flows — absent from older texts. The two volumes together are the closest thing to a modern reference for the field.

D.9 Personal recommendations from the editors¶

If asked to nominate the single best companion to this book for each major topic, our votes are:

Quantum mechanics: Shankar.
Solid-state physics: Ashcroft and Mermin, supplemented by Marder for modern topics.
DFT: Martin's Electronic Structure for theory; Sholl and Steckel for first practice.
MD and statistical mechanics: Frenkel and Smit, with Tuckerman alongside for the foundations.
Machine learning for materials: Schmidt et al. (2019) plus Behler (2021) as a starting bibliography; supplement with the primary papers in Appendix E.
Mathematical background: Strang for linear algebra; Murphy for machine learning.

A reader who has worked through this book and seriously engaged with these six recommendations should be able to read essentially any paper in the modern computational materials literature.