DFT Notes

A reader-first knowledge base on Density Functional Theory — the standard workhorse of modern quantum chemistry and computational materials science.

These notes are organised as a progressive walk through DFT. Each chapter builds on the last, but every chapter is also self-contained enough to be read in isolation. Math is rendered inline (MathJax 3, with AMS environments loaded); code samples are Python unless otherwise noted; tables and figures are inline rather than appendix-pinned.

Chapters

#	Chapter	Topic
0	Welcome	How to read these notes, notation, and prerequisites
1	Schrödinger equation	The postulates, the Hamiltonian, and what $\psi$ means
2	The many-body problem	Why a single Slater determinant isn’t enough, and why
3	Hartree–Fock	Mean-field theory, the Fock operator, and the SCF loop
4	Kohn–Sham DFT	Hohenberg–Kohn existence + Kohn–Sham practical map
5	XC functionals	Jacob’s ladder: LDA, GGA, hybrids, range-separated
6	Basis sets	Gaussians, plane waves, real-space grids, Roothaan–Hall
7	Solids & PBC	Bloch’s theorem, Brillouin zone, plane-wave cutoff, k-grids
8	Pseudopotentials	Norm-conserving, ultrasoft, PAW, transferability
9	Forces & geometry opt	Hellmann–Feynman, Pulay, BFGS, LBFGS, stress tensor
10	Phonons & vibrations	Frozen-phonon, DFPT, LO-TO, Eliashberg, superconductivity
11	Band structures	DOS, Fermi surface, effective mass, graphene Dirac cone
12	Time-dependent DFT	Runge–Gross, Casida, real-time TDDFT, UV/vis spectra
13	DFT+U & beyond	Strong correlation, DFT+U, hybrid functionals, DMFT
14	Quantum chemistry beyond DFT	MP2, CI, CCSD(T), CASSCF, NEVPT2, semiempirical, CBS
15	Relativistic effects	Dirac equation, DKH, ZORA, X2C, spin-orbit coupling
16	Topological materials	Chern numbers, Z₂ invariants, Weyl/Dirac semimetals, HOTIs
17	Machine learning for DFT	Behler-Parrinello, MACE, NequIP, delta-learning, foundation models

Status: these are sample notes — real prose, real math, real code, but not yet encyclopaedic. They exist to exercise every rendering feature of the future e-reader (see design.md at the root of the repo).

Where to go next

• The chapters map is a live Mermaid graph of every chapter that exists or will exist — past, present, and planned. It shows the natural reading order and the three tracks (Methods / Solids / Dynamics) you can branch into after chapter 05.
• The Python codes folder has the runnable version of every code sample that appears in a chapter, plus the generated plots.

Reading order

If you’re new to DFT, read top-to-bottom. If you’re returning to brush up on a specific topic, each chapter is internally cross-referenced and the section headers follow the same template:

1. The claim — the result or definition, stated precisely.
2. The proof (or derivation) — short, with the load-bearing steps called out.
3. The code — a minimal implementation that produces the result.
4. The catch — what the formula doesn’t* tell you, and the regime where it breaks.

Notation

A short, opinionated list — see chapter 00 for the full table.

Symbol	Meaning
$\psi$	Many-body wavefunction
$\rho$	One-particle density $\rho(\mathbf r) = N \sum_{\sigma} \int \lvert \Psi \rvert^2\, d\mathbf r’$
$\hat H$	Electronic Hamiltonian
$E_\text{xc}$	Exchange–correlation energy
$v_\text{xc}(\mathbf r)$	Exchange–correlation potential
$\phi_i$	Kohn–Sham orbital
$\varepsilon_i$	Kohn–Sham eigenvalue
$\hat T_s$, $\hat J$, $\hat V_\text{ext}$	KS kinetic, Hartree, external operators