Chapters map

The full DFT Notes knowledge graph. Boxes are chapters; arrows are “depends on” (read this after the arrow’s source). Boxes with a dashed border are planned but not yet written.

This is the canonical index of every chapter that exists or will exist in this knowledge base. For a human-readable list, see the chapter index.

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graph TD
  C00["00 · Welcome<br/><small>How to read these notes,<br/>notation, prerequisites</small>"]
  C01["01 · Schrödinger equation<br/><small>Postulates, Hamiltonian,<br/>particle-in-a-box</small>"]
  C02["02 · The many-body problem<br/><small>Slater determinants,<br/>exponential wall, full CI</small>"]
  C03["03 · Hartree–Fock<br/><small>Fock operator, SCF,<br/>Slater–Condon rules</small>"]
  C04["04 · Kohn–Sham DFT<br/><small>Hohenberg–Kohn, KS ansatz,<br/>SCF loop</small>"]
  C05["05 · XC functionals<br/><small>LDA → GGA → hybrid → RS,<br/>Jacob's ladder</small>"]
  C06["06 · Basis sets<br/><small>Gaussians, plane waves,<br/>numerical orbitals</small>"]
  C07["07 · Solids & PBC<br/><small>Brillouin zone, Bloch's theorem,<br/>k-point sampling</small>"]
  C08["08 · Pseudopotentials<br/><small>Norm-conserving, ultrasoft,<br/>PAW</small>"]
  C09["09 · Forces & geometry opt<br/><small>Hellmann–Feynman,<br/>BFGS, LBFGS</small>"]
  C10["10 · Phonons & vibrations<br/><small>Frozen phonon, DFPT,<br/>electron–phonon coupling</small>"]
  C11["11 · Band structures<br/><small>DOS, projected DOS,<br/>Fermi surfaces</small>"]
  C12["12 · TDDFT<br/><small>Runge–Gross theorem,<br/>linear response, excitons</small>"]
  C13["13 · DFT+U & beyond<br/><small>Strong correlation,<br/>hybrid functionals, DMFT</small>"]
  C14["14 · Quantum chemistry beyond DFT<br/><small>MP2, CI, CC, CASSCF,<br/>semiempirical, CBS</small>"]
  C15["15 · Relativistic effects<br/><small>Dirac equation, DKH, ZORA,<br/>spin-orbit coupling</small>"]
  C16["16 · Topological materials<br/><small>Chern numbers, Z₂ invariants,<br/>Weyl/Dirac semimetals</small>"]
  C17["17 · Machine learning for DFT<br/><small>Equivariant NNs, MACE,<br/>delta-learning, foundation models</small>"]

  C00 --> C01
  C01 --> C02
  C02 --> C03
  C03 --> C04
  C04 --> C05
  C05 --> C06
  C06 --> C07
  C06 --> C08
  C04 --> C09
  C07 --> C10
  C07 --> C11
  C04 --> C12
  C04 --> C13
  C03 --> C14
  C08 --> C15
  C15 --> C16
  C04 --> C17
  C16 --> C17

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  click C00 "javascript:void(0)" "How to read these notes, the notation table, and the prerequisites for the DFT Notes series."
  click C01 "javascript:void(0)" "The postulates of quantum mechanics, the electronic Hamiltonian, and a worked particle-in-a-box derivation."
  click C02 "javascript:void(0)" "Why a single Slater determinant is not enough, the exponential wall, and the full-CI limit."
  click C03 "javascript:void(0)" "Mean-field theory, the Fock operator, the SCF loop, and the Slater–Condon rules for matrix elements."
  click C04 "javascript:void(0)" "Hohenberg–Kohn existence theorem, the Kohn–Sham ansatz, and the practical KS SCF loop."
  click C05 "javascript:void(0)" "Jacob's ladder: LDA, GGA, meta-GGA, hybrid, and range-separated functionals."
  click C06 "javascript:void(0)" "Gaussian-type orbitals, plane waves, real-space grids, and the Roothaan–Hall equations."
  click C07 "javascript:void(0)" "Bloch's theorem, the Brillouin zone, plane-wave cutoffs, and Monkhorst–Pack k-point sampling."
  click C08 "javascript:void(0)" "Norm-conserving, ultrasoft, and PAW pseudopotentials — construction, transferability, and validation."
  click C09 "javascript:void(0)" "Hellmann–Feynman theorem, BFGS and LBFGS optimisers, and convergence criteria for forces and displacements."
  click C10 "javascript:void(0)" "Frozen-phonon supercells, density-functional perturbation theory, and electron–phonon coupling."
  click C11 "javascript:void(0)" "Density of states, projected DOS, band-structure plots, and Fermi-surface visualisation."
  click C12 "javascript:void(0)" "Runge–Gross theorem, linear-response TDDFT, the Casida equations, and excitons."
  click C13 "javascript:void(0)" "DFT+U for strong correlation, hybrid functionals revisited, and a DMFT outlook."
  click C14 "javascript:void(0)" "Quantum chemistry methods beyond DFT: MP2, CI, CCSD(T), CASSCF, NEVPT2, and the basis-set extrapolation ladder."
  click C15 "javascript:void(0)" "The Dirac equation, scalar-relativistic corrections, and spin-orbit coupling in modern DFT."
  click C16 "javascript:void(0)" "Topology in band theory: Chern numbers, Z₂ invariants, topological insulators, and Weyl semimetals."
  click C17 "javascript:void(0)" "Machine learning for DFT: interatomic potentials, equivariant networks, delta-learning, and foundation models."

Edges

The arrows in the diagram above encode the prerequisite relation. Reading a chapter assumes the reader is comfortable with the material in every chapter that points into it. The table below lists every edge and the reason it exists.

From To Why the dependency
C00 C01 The how-to-read preamble must come first.
C01 C02 The Schrödinger picture and the Hamiltonian are needed to set up the many-body problem.
C02 C03 The Slater-determinant language is the input to Hartree–Fock.
C03 C04 Hartree–Fock is the reference point against which KS-DFT is derived.
C04 C05 The XC functional is the only piece KS-DFT leaves undetermined.
C05 C06 The choice of basis set is entangled with the choice of functional.
C06 C07 Bloch sums are built from a basis; plane waves are a basis.
C06 C08 Pseudopotentials are defined by projector-augmented basis operators.
C04 C09 Forces come from differentiating the KS total energy.
C07 C10 Phonons need a periodic supercell and k-point sampling.
C07 C11 Band structures are defined on the Brillouin zone of a periodic system.
C04 C12 TDDFT is a time-dependent extension of ground-state KS-DFT.
C04 C13 DFT+U is a correction to the KS exchange–correlation.
C03 C14 Post-HF methods (MP2, CI, CC, CASSCF) are alternatives to KS-DFT; HF from chapter 3 is the starting point.
C08 C15 Relativistic effects enter DFT through the pseudopotential (scalar-relativistic and fully-relativistic PPs).
C15 C16 Topological materials need spin-orbit coupling from chapter 15 (SOC opens band gaps, gives Z₂ invariants).
C04 C17 ML for DFT uses KS-DFT energies, forces, and densities as the ground truth for training.
C16 C17 ML methods are increasingly used to predict topological invariants (Chern numbers, Z₂) from band structures.

Tracks

After chapter 05 the chapters split into three conceptual tracks. Pick the track that matches your problem domain — you do not have to read all of them.

Conventions