Appendix C — Glossary¶
Alphabetical list of \(\sim 80\) technical terms used in the book. Definitions are deliberately short — two or three sentences, oriented toward operational use rather than mathematical rigour. Where a term has a dedicated derivation elsewhere, the relevant chapter is noted.
Active learning. A machine-learning paradigm in which the model itself selects which training examples are most informative to label next. In materials, the typical loop alternates between training a surrogate on existing data, identifying configurations of high predicted uncertainty, evaluating them with DFT, and adding them to the training set. Chapter 11.
Atomic units (a.u.). A system of units in which \(\hbar = m_e = e = 1/(4\pi\epsilon_0) = 1\). Energies are in Hartree, lengths in bohr, times in \(\hbar/E_h\). The natural system for electronic- structure calculations because the Schrödinger equation takes its simplest form. Appendix A.
Born–Oppenheimer approximation. The decoupling of nuclear and electronic degrees of freedom on the grounds that nuclei move much more slowly than electrons. Justifies treating electrons as moving in a static potential from instantaneous nuclear positions. Almost all of computational materials science relies on it. Chapter 4.
Brillouin zone. The Wigner–Seitz cell of the reciprocal lattice; equivalently, the fundamental domain of crystal momentum for a periodic system. All electronic states can be classified by their crystal momentum \(\mathbf{k}\) within this zone. Chapter 5.
Bulk modulus. The pressure required to produce a unit relative change in volume: \(B = -V (\partial p / \partial V)\). A measure of the stiffness of a material under hydrostatic compression. Chapter 6.
Cohesive energy. The energy required to break a solid apart into isolated atoms, per atom. The basic test of any electronic-structure method's accuracy for a given material. Chapter 5.
Configuration interaction (CI). A post-Hartree–Fock method in which the wavefunction is expanded as a linear combination of Slater determinants generated by exciting electrons from a reference. CISD includes single and double excitations; full CI is exact within the basis. Chapter 4.
Convergence criterion. The threshold below which an iterative calculation is declared converged. For DFT, typical criteria are total-energy changes below \(10^{-6}\) eV between SCF cycles, or maximum force components below \(10^{-2}\) eV/Å for geometry optimisations. Chapter 6.
Coupled cluster (CC). A post-Hartree–Fock method using an exponential ansatz \(|\Psi\rangle = e^{\hat T}|\Phi_0\rangle\). CCSD(T) ("the gold standard") is exact through fourth order in perturbation theory for closed-shell systems. Chapter 4.
Cutoff radius. The distance beyond which interatomic interactions in an MLIP or a classical force field are set to zero. Defines the receptive field of the model. Typical values are \(4\)–\(6\) Å for inorganic materials. Chapter 9.
Density functional theory (DFT). The framework, due to Hohenberg and Kohn (1964) and Kohn and Sham (1965), in which the ground-state energy of a many-electron system is written as a functional of the electron density. The workhorse of modern materials science. Chapter 5.
Diffusion model. A class of generative model that learns to reverse a fixed forward noising process. In materials, used to generate crystal structures conditioned on target properties. Chapter 12.
Ewald summation. A trick, due to Ewald (1921), for computing electrostatic energies in a periodic system. Splits the Coulomb sum into a fast-converging short-range part in real space and a fast- converging long-range part in reciprocal space. Chapter 7.
Exchange–correlation functional. The unknown but universal functional \(E_{\text{xc}}[n]\) in DFT, which absorbs everything not captured by the kinetic energy of non-interacting electrons and the Hartree (classical Coulomb) energy. Approximations include LDA, GGA, meta-GGA and hybrids. Chapter 5.
Force constant matrix. The matrix of second derivatives \(\partial^2 E / \partial r_i \partial r_j\) of the total energy with respect to atomic positions, evaluated at equilibrium. Its eigenvectors are vibrational modes; its eigenvalues are squared frequencies. Chapter 8.
Foundation model. A model that is pre-trained at large scale on broad data and adapted to downstream tasks by fine-tuning or prompting. The MLIPs and generative models of Chapter 12 are the materials-science instantiations.
Free energy. The thermodynamic potential whose minimum characterises equilibrium under given constraints. Helmholtz free energy \(F = U - TS\) for constant \(V, T\); Gibbs free energy \(G = U + pV - TS\) for constant \(p, T\). Chapter 8.
Gaussian approximation potential (GAP). A flavour of machine- learning interatomic potential due to Bartók et al. (2010), using Gaussian process regression on SOAP descriptors. Chapter 9.
Generalised gradient approximation (GGA). A class of exchange- correlation functionals that depend on both the density and its gradient. The most common member is PBE (Perdew–Burke–Ernzerhof, 1996). Chapter 5.
Graph neural network (GNN). A neural network architecture operating on graph-structured data, in which information propagates between nodes along edges through message passing. The dominant architecture for property prediction on crystals. Chapter 10.
Harmonic approximation. The expansion of the potential energy around an equilibrium configuration to second order in atomic displacements. Replaces the system by a collection of independent oscillators (phonons). Chapter 8.
Hartree–Fock (HF). The mean-field theory of a many-electron system in which the wavefunction is a single Slater determinant. The starting point for post-Hartree–Fock methods; also the foundation of the hybrid DFT functionals. Chapter 4.
Hellmann–Feynman theorem. The result that the derivative of an electronic-state energy with respect to a parameter is the expectation value of the derivative of the Hamiltonian. Underlies the calculation of forces in DFT. Chapter 5.
Hessian. The matrix of second derivatives of a scalar function with respect to its arguments. For the total energy of a material as a function of atomic positions, the Hessian is the force-constant matrix. Appendix A.
Hybrid functional. An exchange–correlation functional that mixes a fraction of exact (Hartree–Fock) exchange with semi-local DFT exchange. PBE0 and HSE06 are common examples. Computationally far more expensive than GGA. Chapter 5.
Inverse design. The problem of generating a structure (or composition) given a target property, rather than predicting the property of a given structure. The motivating application of the generative models in Chapter 12.
Irreducible representation. A representation of a group that cannot be decomposed into a direct sum of smaller representations. The labelling of phonon modes and electronic states by irreducible representations of the crystal point group encodes selection rules. Chapter 8.
Jacobian. The matrix of first partial derivatives of a vector- valued function. The Jacobian of the atomic position with respect to reduced coordinates is the lattice matrix \(L\).
Kinetic energy cutoff. The maximum kinetic energy of a plane wave included in the basis set for periodic DFT. Determines the size of the basis and the accuracy of the calculation. Convergence tests against this parameter are mandatory. Chapter 6.
Kohn–Sham orbitals. The single-particle orbitals of the fictitious non-interacting Kohn–Sham system, whose density equals that of the true interacting system. They are not the true many-body orbitals but are routinely used to construct band structures. Chapter 5.
LAMMPS. An open-source classical molecular dynamics code, maintained at Sandia National Laboratories. The de facto standard for large-scale classical MD; also a popular harness for running MLIPs in production. Chapter 7.
Lennard–Jones potential. The simplest pairwise potential for van der Waals interactions: \(V(r) = 4\epsilon[(\sigma/r)^{12} - (\sigma/r)^{6}]\). Chapter 7.
Levenberg–Marquardt. An optimisation algorithm interpolating between gradient descent and Gauss–Newton, used to fit non-linear least-squares problems. Appears in classical force-field fitting.
Local density approximation (LDA). The simplest exchange– correlation functional, in which the exchange–correlation energy density at each point is that of the uniform electron gas at the local density. Chapter 5.
MACE. A machine-learning interatomic potential architecture due to Batatia et al. (2022), based on the Atomic Cluster Expansion with equivariant message passing. The basis of MACE-MP-0. Chapter 9, Chapter 12.
Materials Project. A US-hosted database, founded in 2011, of DFT-computed properties of inorganic compounds. As of 2026 contains \(\sim 1.5 \times 10^5\) relaxed structures with formation energies, band gaps and elastic properties; the relaxation trajectories form the MPtrj dataset used to train universal MLIPs.
Mean-square displacement (MSD). \(\langle |\mathbf{r}(t) - \mathbf{r}(0)|^2 \rangle\), averaged over particles. In an isotropic liquid it grows linearly with time, \(\text{MSD} = 6Dt\), defining the diffusion coefficient \(D\). Chapter 7.
Message-passing neural network (MPNN). A general framework for GNNs in which each node updates its state by aggregating messages from its neighbours. The unifying formalism behind SchNet, MACE, NequIP, MEGNet and ALIGNN. Chapter 10.
Metropolis algorithm. The original Markov-chain Monte Carlo algorithm (Metropolis et al. 1953), generating samples from a Boltzmann distribution by accepting or rejecting proposed moves with probability \(\min(1, e^{-\beta \Delta E})\). Chapter 8.
MLIP. Machine-learning interatomic potential. A neural network or kernel regression model that maps a local atomic environment to an energy contribution. Chapter 9.
Molecular dynamics (MD). The numerical integration of the classical equations of motion for a system of atoms. The forces may come from a classical force field, an MLIP, or an on-the-fly DFT calculation (ab initio MD). Chapter 7.
Monkhorst–Pack grid. A scheme for choosing \(\mathbf{k}\)-points in the Brillouin zone, based on a uniform grid offset from the \(\Gamma\) point. The standard choice for sampling the BZ in periodic DFT. Chapter 6.
MPtrj. The Materials Project relaxation trajectory dataset: \(\sim 1.6 \times 10^6\) configurations from \(\sim 1.5 \times 10^5\) relaxations, with PBE+U energies and forces. The training corpus for MACE-MP-0, CHGNet and M3GNet. Chapter 12.
Nudged elastic band (NEB). A method for finding minimum-energy paths between two configurations, by connecting them with a series of replicas joined by spring forces. Used to locate transition states. Chapter 8.
Norm-conserving pseudopotential. A class of pseudopotential in which the integrated charge density inside the core matches that of the all-electron atom. Less computationally efficient than ultrasoft or PAW but more transferable. Chapter 6.
OMat24. A large dataset, released by Meta in 2024, of \(\sim 10^8\) DFT-labelled out-of-equilibrium configurations. Used to train next-generation universal MLIPs. Chapter 12.
One-body / two-body / many-body. Descriptors of an MLIP by the maximum number of atoms whose joint relative configuration directly enters a single energy term. MACE with body order \(\nu = 3\) includes three-atom (triplet) contributions before message passing. Chapter 9.
Pair correlation function \(g(r)\). The probability of finding an atom at distance \(r\) from another, normalised to unity at large distance. The Fourier transform of the static structure factor. Chapter 7.
Partition function. \(Z = \sum_i e^{-\beta E_i}\), the normalising constant of the Boltzmann distribution. Thermodynamic quantities are derivatives of \(\ln Z\). Chapter 8.
PAW (projector augmented wave). A pseudopotential-like scheme that maintains a transformation between the smooth pseudo-wavefunction used in calculation and the all-electron wavefunction. Standard in VASP. Chapter 6.
PBE. The Perdew–Burke–Ernzerhof (1996) GGA exchange–correlation functional. The most widely used functional in materials calculations. Chapter 5.
Periodic boundary conditions (PBC). The convention that a simulation cell is repeated infinitely in all three directions. The natural boundary condition for crystals; an approximation for liquids that becomes exact in the thermodynamic limit. Chapter 6, Chapter 7.
Phonon. A quantum of lattice vibration; equivalently, a collective excitation of the harmonic Hamiltonian for the atomic displacements. Chapter 8.
Plane-wave basis. An expansion of single-particle orbitals in the Fourier basis adapted to the periodicity of the crystal. The natural basis for solid-state DFT. Chapter 6.
Pseudopotential. An effective potential that replaces the core electrons by a smoother potential acting on the valence electrons only, allowing a smaller basis to be used. Chapter 6.
Quasi-harmonic approximation. An extension of the harmonic approximation in which the phonon frequencies depend explicitly on volume; used to compute thermal expansion. Chapter 8.
Reciprocal lattice. The set of vectors \(\mathbf{G}\) such that \(e^{i\mathbf{G}\cdot\mathbf{R}} = 1\) for every direct lattice vector \(\mathbf{R}\). Chapter 5.
SchNet. A continuous-filter convolutional neural network for molecules, due to Schütt et al. (2017). One of the first deep MLIP architectures. Chapter 9, Chapter 10.
SCF (self-consistent field). The iterative procedure by which DFT or Hartree–Fock equations are solved: guess a density, build the Hamiltonian, diagonalise, compute a new density, repeat until convergence. Chapter 5.
Slater determinant. An antisymmetrised product of single- particle orbitals; the simplest wavefunction that obeys fermion statistics. The HF ansatz. Chapter 4.
SOAP (Smooth Overlap of Atomic Positions). A descriptor for local atomic environments, due to Bartók et al. (2013), based on overlapping atom-centred Gaussians. Underlies GAP. Chapter 9.
Spin–orbit coupling. The relativistic interaction between the electron spin and orbital motion; important for heavy elements and for magnetic anisotropy. Chapter 5.
Stress tensor. The \(3 \times 3\) generalised force-per-area on the simulation cell. Its trace gives the (negative of the) pressure. Differentiating the energy with respect to strain components yields the stress tensor. Chapter 6.
Surface energy. The energy per unit area of a surface relative to the bulk. Determines crystal morphology and surface stability. Chapter 6.
Symmetry function. A scalar function of an atomic environment designed to be invariant under rotations, translations and atom permutations. The Behler–Parrinello symmetry functions (2007) were the first such descriptors used in MLIPs. Chapter 9.
Tensor product layer. A layer in an equivariant neural network that combines two equivariant features (each transforming under a representation of the rotation group) into a new feature, using the Clebsch–Gordan decomposition. The building block of MACE, NequIP and SE(3)-transformers. Chapter 9, Chapter 12.
Thermostat. A modification to molecular dynamics that drives the system toward a target temperature. Common choices include Berendsen, Nosé–Hoover and Langevin. Chapter 7.
Trajectory. The time-ordered sequence of configurations produced by an MD or Monte Carlo simulation. Chapter 7.
Universal MLIP. A single machine-learning interatomic potential that performs reasonably well across the periodic table without system-specific training. MACE-MP-0, CHGNet, M3GNet and SevenNet are the canonical examples. Chapter 12.
Variational principle. The statement that the expectation value of the Hamiltonian in any normalised state is at least as large as the ground-state energy. Underlies Hartree–Fock, DFT and most of the rest of electronic-structure theory. Chapter 4.
VASP. Vienna Ab Initio Simulation Package. Commercial DFT code widely used in materials science. Chapter 6.
Verlet algorithm. A second-order symplectic integrator for Newton's equations of motion. The default choice for MD because of its long-time stability. Chapter 7.
Wannier function. A real-space orthonormal basis obtained from a unitary transformation of Bloch functions, localised within a unit cell. Used to interpret bonding and to construct effective Hamiltonians. Chapter 5.
Zero-point energy. The non-zero ground-state energy of a quantum harmonic oscillator, \(\frac12 \hbar \omega\) per mode. The total zero-point energy of a crystal is the sum over phonon modes; it can amount to \(\sim 0.1\) eV/atom for light elements. Chapter 8.
Zero-shot. The use of a pre-trained model on a new task or domain with no further training. Distinguished from few-shot (a small number of labelled examples) and fine-tuning (gradient-based adaptation). Chapter 12.
This list is not exhaustive. Terms used only locally in a single chapter (e.g., specific names of optimisation algorithms or particular spectroscopic methods) are defined in place. When in doubt, search the chapter index.