Chapter 00 — Welcome

This chapter has no equations. That’s the only one. Everything after this is math.

These notes are an attempt to write down, in one place, the minimum viable theory of Density Functional Theory — enough to read a modern DFT paper, set up a calculation, and know which approximations you are making and why they might be wrong.

How to read this chapter

Chapter 00 is the only meta chapter in the collection. Every later chapter (01, 02, …) is about DFT; this one is about the document you are reading. If you are in a hurry to get to equations, you can read the Notation and What is not in scope sections and skip straight to chapter 01. If you are reading the notes for the first time and want to know what you are getting into, read this chapter linearly.

The document is uniform on purpose

Every content chapter (chapter 01 onward) is built from the same seven-part template (see The chapter structure template below). That uniformity is the whole point of writing a fixed template: once you have read one chapter, you know the shape of every chapter, and you can navigate directly to the part that interests you. The template is enforced by agent:qa-reviewer and recorded in agents.md.

How to read a chapter

There are three reasonable patterns. Pick whichever fits your goal.

1. Linear, first pass. Read claim → derivation → diagram → worked example → problems, in order. This is the right way to learn a chapter the first time. Expect ~30–90 minutes per chapter depending on your background.
2. Reference, second pass. Jump to the claim (always equation 1 of the chapter) to recall the headline result. Then read the What we left out section to see what the chapter is silent on. Use this when you come back to a chapter months later and need a one-paragraph refresher.
3. Problem-driven. Skim the Problems section first. Each problem is calibrated to require material from a particular part of the chapter; if the problem makes no sense to you, that is a signal to go read the part of the chapter it references. This is the right way to use the notes to self-test.

Tip. If you’re reading on a phone, the code blocks are horizontal-scrollable. If you’re reading on the e-reader surface (see the design spec), use the font slider if the math is too small. Every Mermaid diagram is rendered client-side; if it does not appear, open the browser console and look for a CDN error.

The “no step skipped” promise

The author has been emphatic: no calculation will be omitted, nothing should be kept like “do as I exercise” or “from here on”. Every algebraic step in every chapter appears in the chapter. If a step is purely mechanical (expand, distribute, collect) it is labelled as such in one line, but the result is still written out. This is the only reliable way to be sure the math is correct, and it is the only reliable way for a reader to check the math is correct. If you ever find a derivation in these notes that skips a step in violation of this rule, please open an issue.

Cross-references

Chapters cross-reference each other with two distinct signals:

• A plain Markdown link `Chapter 01’ — used for prerequisite material that the reader is expected to have already read.
• A forward link `chapter 04’ — used to flag material that will be developed later but is needed in the current chapter for context.

In both cases, the link is to a chapter, never to a section within a chapter. This is so the dependency graph stays legible (see Roadmap below). If a section-level link seems essential, that is usually a sign the chapter should be split.

The “catch” is woven through

The chapter template has explicit “diagram”, “worked example”, “problems”, and “What we left out” sections. None of them is labelled “catch”. But the spirit of the catch — what does this formula not tell you, and where does it break? — is woven into every part of the template:

• the derivation has Tip, Note, and Warning callouts that flag common pitfalls;
• the worked example deliberately picks parameter values that exercise a known failure mode (small basis, large grid spacing, hard XC functional);
• the “What we left out” section is the catch, made explicit.

When you read a chapter, do not skip the callouts. They are where the lived experience of the topic lives.

Mathematical prerequisites

DFT, at the level of these notes, draws on roughly five mathematical subjects at the undergraduate / first-year-graduate level. Below, each is broken into the specific pieces the notes will use, with a one-sentence statement of what the reader should be able to do with each piece. Nothing in the list is exotic; everything is taught in standard curricula and is re-stated as it is used.

Linear algebra (used throughout)

• Vector spaces and inner products. You should be comfortable with the inner product $\langle u \rvert v \rangle = \sum_i u_i^* v_i$ in $\mathbb C^n$ and its extension $\int u^*(\mathbf r) v(\mathbf r)\, d\mathbf r$ for functions. Every chapter uses inner products.
• Eigendecomposition. Given a Hermitian matrix $\mathbf A$, you should be able to factor it as $\mathbf A = \mathbf U \boldsymbol\Lambda \mathbf U^\dagger$ with $\mathbf U$ unitary and $\boldsymbol\Lambda$ real diagonal. This is the only diagonalisation the notes use as a primitive. Singular-value decomposition is used once or twice in chapter 06 (density fitting) but is not assumed.
• The spectral theorem. Hermitian operators have real eigenvalues and a complete orthonormal set of eigenvectors. The notes use this in chapter 03 (Hartree–Fock) and chapter 04 (Kohn–Sham) to argue that the Fock and Kohn–Sham operators can be diagonalised.
• Bra-ket notation. You should be able to read $\langle \chi_\mu \rvert \hat F \rvert \chi_\nu \rangle$ as “the matrix element of $\hat F$ between basis functions $\mu$ and $\nu$”. We use bra-ket throughout. The only operations we ever perform on a bra or a ket are: take the inner product, insert a resolution of the identity $\sum_i \rvert \phi_i \rangle \langle \phi_i \rvert = \mathbf 1$, and act with an operator from the left or the right.

Calculus of several variables (used throughout)

• Partial derivatives and the chain rule. Used in chapter 04 to derive the Kohn–Sham potential by differentiating the exchange–correlation energy with respect to the density.
• Gradient, divergence, Laplacian. Used in the kinetic operator $-\tfrac{1}{2} \nabla^2$, in the gradient correction to GGA functionals (chapter 05), and in the stress tensor (chapter 09).
• Line, surface, and volume integrals. Used to define the electron number, the dipole moment, and the total energy.
• Green’s theorem / the divergence theorem. Used once, in chapter 07, to relate the gradient of a Bloch function to its surface integral over a unit cell.

Probability and statistics (used in chapters 03, 04, 05, 09)

• Expectation values. The notes use the quantum-mechanical definition $\langle \hat A \rangle = \langle \Psi \rvert \hat A \rvert \Psi \rangle$. You should also be able to translate this to a classical average $\int a(x)\, p(x)\, dx$ when $\hat A$ is replaced by a function of position.
• Joint and marginal distributions. Used in chapter 05 (one-electron reduced density matrices) and chapter 09 (Boltzmann sampling in finite-temperature DFT, briefly).
• Independent and identically distributed samples. Not heavily used, but appears in the discussion of stochastic errors in Monte Carlo and in sampling the Brillouin zone.

Complex analysis (used in chapter 11, on response functions)

• Analytic continuation. The density–density response function $\chi(\mathbf r, \mathbf r’; \omega)$ is a meromorphic function of complex frequency $\omega$ with poles at the excitation energies. You should know what “meromorphic” means and why the location of the poles is physical.
• Cauchy’s residue theorem. Used to derive the fluctuation–dissipation theorem from the spectral representation of $\chi$.
• Contour integration. Used in chapter 11 to evaluate frequency integrals on a contour that avoids the real axis. If you are not continuing past chapter 10, you can skip this.

Special functions (used in chapters 01, 05, 06)

• Spherical harmonics $Y_\ell^m(\theta, \phi)$. Used in chapter 01 (angular part of the hydrogen atom) and in chapter 06 (atom-centred basis functions with $\ell > 0$).
• Hermite polynomials $H_n(x)$. Used in the analytical Gaussian integrals of chapter 06.
• Laguerre polynomials $L_n(x)$. Used in the radial part of the hydrogen atom and in the STO radial functions.
• The error function $\operatorname{erf}(x)$ and the Boys function $F_0(t)$. Both appear in chapter 06 in the closed-form expressions for the two-electron integrals.

Where to look up what you forgot. The classic reference is Abramowitz and Stegun, Handbook of Mathematical Functions (1964), still in print as the NIST Digital Library of Mathematical Functions (dlmf.nist.gov). For complex analysis, Ahlfors, Complex Analysis (any edition), is the standard.

Quantum mechanics, undergraduate level

• The postulates. State vectors, observables, measurement outcomes, the Schrödinger equation, the Born rule. If you have never seen the Schrödinger equation before, start with chapter 01, which lays the postulates out one by one before any DFT-specific material is introduced.
• The harmonic oscillator. Used as a sanity check in chapter 01 and in the example of the Hückel model in chapter 02.
• The hydrogen atom. The exact solution is the only closed-form reference point in DFT. You should know the energy levels $E_n = -1/(2 n^2)\,E_h$ and the form of the 1s, 2s, 2p orbitals. The 1s density is plotted by the hello world program in the next section.

Programming prerequisites

The notes are accompanied by a parallel set of Python scripts under dft_notes/python_codes/. Every chapter has its own folder; every script in that folder corresponds to one of the chapter’s worked-example snippets. The conventions are spelled out in agents.md; the short version is:

• Python 3.11+ (the version pinned in python_codes/README.md).
• Imports are restricted to numpy, scipy, and matplotlib. Anything else is added to the README on a case-by-case basis.
• Headless rendering. Every script begins with `matplotlib.use(“Agg”)’ so that the script runs on a server with no display.
• The plots/’ subfolder holds the committed PNGs. One script produces one figure; the figure file is named ‘plots/-.png'.
• No os.chdir. Paths are constructed relative to the chapter folder using pathlib.Path(__file__).parent.

What you should be able to do

Below is the working subset of Python the notes assume. If any item is unfamiliar, the official tutorial links in the right-hand column are the place to fill the gap.

Hello world — a chapter’s smallest program

The smallest program that does something chapter-relevant is the plot of the hydrogen 1s density. The 1s wavefunction of a hydrogenic atom with nuclear charge $Z$ is

so the corresponding one-electron density is

The full source below produces a PNG that lives in dft_notes/python_codes/chapter_00/plots/01-hydrogen-1s.png. Every later chapter has scripts of comparable length and shape; once you have read this one, you have read the format of them all.

# dft_notes/python_codes/chapter_00/01-hydrogen-1s.py
import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
from pathlib import Path

def hydrogen_1s_density(r, Z=1):
    """Electron density of the hydrogen 1s orbital, |psi_1s(r)|^2."""
    return (Z ** 3 / np.pi) * np.exp(-2.0 * Z  r)

def main():
    r = np.linspace(0.0, 20.0, 400)
    rho = hydrogen_1s_density(r, Z=1)

    fig, ax = plt.subplots(figsize=(6, 4))
    ax.plot(r, rho, color="#0a4", linewidth=2)
    ax.set_xlabel("r (a0)")
    ax.set_ylabel(r"$\rho_{\rm 1s}(r)$ (a0$^{-3}$)")
    ax.set_title("Hydrogen 1s electron density")

    out = Path(__file__).parent / "plots" / "01-hydrogen-1s.png"
    out.parent.mkdir(parents=True, exist_ok=True)
    fig.tight_layout()
    fig.savefig(out, dpi=120)
    print(f"Wrote {out}")

if __name__ == "__main__":
    main()

A reader fluent in ‘numpy’ and ‘matplotlib’ should be able to read this script in under a minute and predict the shape of the figure before running it: an exponentially decaying curve, equal to $1/\pi \approx 0.318$ at $r = 0$ and falling to $\sim 10^{-9}$ by $r = 10\,a_0$.

Tip. The full directory layout (what `chapter_00/’ looks like after this script has been run) is:

```text dft_notes/python_codes/ ├── README.md └── chapter_00/ ├── 01-hydrogen-1s.py └── plots/ └── 01-hydrogen-1s.png ``’

The two-digit prefix (01-) preserves the order of scripts within a chapter; if a future chapter-00 script gets added it would be ‘02-…py’ and its plot ‘02-….png`. This ordering matters because chapters read top-to-bottom.

The chapter structure template

Every content chapter (chapters 01 and later) is built from the same seven-part template. The template is enforced by agent:qa-reviewer and is reproduced verbatim from agents.md. Below, each part is described in plain language: what it is for, what level of detail to expect*, and *what the reader should take away.

1. The claim

The chapter opens with the headline result — the one equation the reader should remember a year from now. It is always a \(*numbere'd*' equation ('\begin{equation} ... \label{eq:ch-NN-headline} ...\end{equation}`), and the body of the chapter cross-references it\) with \eqref{eq:ch-NN-headline}. In chapter 06 the claim is the Roothaan–Hall equation $\mathbf F \mathbf C = \mathbf S \mathbf C \boldsymbol\varepsilon$; in chapter 04 it is the Kohn–Sham eigenvalue equation. The claim is the destination of the derivation; everything in the chapter is in service of it.

What to expect. One to four numbered equations, stated without proof, followed by a one-paragraph statement of what it means and *why it matters. No algebra here.

What to take away. The reader should be able to write down the claim from memory after closing the chapter.

2. The derivation

The body of the chapter. The derivation takes the claim apart into the smaller facts it is built on, and shows — **step by step, with every algebraic step explicit — how those smaller facts combine to give the claim. The author is bound by the “no step skipped” rule: no “by a straightforward manipulation”, no “after some algebra”, no “it can be shown that”. If a step is purely mechanical (expand, distribute, collect), the chapter says so in one line, but it writes the result.

What to expect. Numbered sub-sections (e.g. “6.2 Why we need a basis”), each with its own numbered equations. Cross-references to equations in other chapters by chapter link, cross-references to equations in this chapter by \eqref{}. Inline Tip, Note, and Warning callouts flag common pitfalls and limits of the derivation.

What to take away. The reader should be able to redo the derivation on a blank page, in the same order, with the same intermediate steps. This is the test of whether the chapter has done its job.

3. The code

A runnable Python snippet that produces the result of the chapter (or a piece of it). The snippet is not written ad hoc; it is copied from a file in dft_notes/python_codes/chapter_NN/, with the chapter-relative path of the original file in the comment header. The convention is two-digit prefix, dash, kebab-case slug, .py. The companion script is run by agent:code-runner' to produce the PNG; the PNG is committed under plots/`.

What to expect. A real Python program, ~10–50 lines, that imports only numpy, scipy, and matplotlib’ (with matplotlib.use("Agg")). No display is required. The snippet is self-contained: copy it into a file, run it, get a PNG.

What to take away. The reader should be able to read the snippet and predict the output before running it, and should be able to modify it (change basis, change grid, change functional) and have the change take effect without restructuring the script.

4. The diagram

A Mermaid diagram that summarises the chapter’s structure. Typical choices: a flowchart of the algorithm, a state diagram of the iteration, a class diagram of the objects. The diagram is rendered client-side (CDN-loaded Mermaid), so it requires JavaScript to appear. The diagram is not a substitute for prose; it is a visual index into the chapter.

What to expect. One to three Mermaid blocks, each with a `%%{init: …}%%’ directive for layout hints. A short caption explaining what the diagram is and what the reader should take from it. Cross-references back to the prose sections the diagram summarises.

What to take away. The reader should be able to draw the diagram from memory after closing the chapter, and should be able to point to each box and say what chapter section it corresponds to.

5. Worked example

A fully worked numerical example that takes the chapter’s general result and applies it to a specific system with specific numbers. Every quantity has a numerical value; every intermediate result is shown; the final answer is stated. The example should be small enough to verify by hand (in principle) and large enough to exercise the typical failure modes of the method.

What to expect. The full numerical trace, often as a text’ code block. A reference back to the code snippet (section 3) that produced the plot. The plot itself, included as a Markdown image tag pointing to dft_notes/python_codes/chapter_NN/plots/NN-slug.png.

What to take away. The reader should be able to reproduce the numbers — and should be encouraged to do so, by running the companion script.

6. Problems

Three problems per chapter, ranging easy → hard, each followed by a fully worked solution. Problems use <details class="problem">' for the question and

What to expect. Three to five problems per chapter. Solutions are written in the same step-by-step style as the derivation, and they end with a boxed final answer.

What to take away. The reader who can solve all three problems unaided has genuinely understood the chapter.

7. What we left out

A honest, explicit list of the topics the chapter does not cover. This is part of being honest about the scope. Two to five bullets is the norm. The list is written in the same tone as the rest of the chapter; it is not an apology, it is a hand-off to the next chapter or to a future note.

What to expect. A bulleted list, with one-line explanations of why the topic is omitted. Cross-references forward to later chapters when the omitted topic is developed there.

What to take away. The reader should know, after reading this section, exactly what the chapter is silent on — and where to look if they need that material.

Roadmap

The notes are organised as a dependency graph, not a strictly linear sequence. The graph is shown in two views: a flat chapter-by-chapter view, and a track view that groups chapters by what they teach.

Chapter-by-chapter dependency graph

Each box is a chapter. An arrow $A \to B$ means “chapter $B$ presumes the reader has read chapter $A$”. The orange-bordered box is the chapter you are reading.

%%{init: {'flowchart': {'htmlLabels': true, 'curve': 'basis'}}}%%
graph TD
  C00["Chapter 00<br/>Welcome"]
  C01["Chapter 01<br/>Schrödinger &<br/>postulates"]
  C02["Chapter 02<br/>Many-body<br/>problem"]
  C03["Chapter 03<br/>Hartree-Fock"]
  C04["Chapter 04<br/>Kohn-Sham"]
  C05["Chapter 05<br/>XC functionals"]
  C06["Chapter 06<br/>Basis sets"]
  C07["Chapter 07<br/>Solids & PBC"]
  C08["Chapter 08<br/>Pseudopotentials"]
  C09["Chapter 09<br/>Forces &<br/>geometry opt"]
  C10["Chapter 10<br/>Vibrations &<br/>phonons"]

  C00 --> C01
  C01 --> C02
  C02 --> C03
  C02 --> C04
  C03 --> C05
  C04 --> C05
  C05 --> C06
  C06 --> C07
  C06 --> C08
  C07 --> C09
  C08 --> C09
  C09 --> C10

  style C00 fill:#fff3d6,stroke:#ff8c00,stroke-width:4px,color:#000

The graph has one source (chapter 00) and a “fan-in” structure in the middle: chapters 03 and 04 both depend on chapter 02 and both feed chapter 05. This is deliberate — Hartree–Fock and Kohn–Sham are alternative answers to the same question (chapter 02’s “what is the best independent-particle approximation?”), and the two paths merge again when we discuss exchange– correlation functionals.

Reading implication. If your interest is in molecular DFT and you do not care about solids or phonons, you can read chapters 00 → 01 → 02 → 04 → 05 → 06 in sequence, then stop. Chapters 03, 07, 08, 09, 10 are not on your critical path.

Track view — theory, methods, applications

The same chapters grouped by what they teach. This view is useful when you already know which kind of question you are asking (“how do I represent a periodic system?” → methods track; “where does the functional come from?” → theory track).

%%{init: {'flowchart': {'htmlLabels': true, 'curve': 'basis'}}}%%
graph LR
  subgraph T ["Theory track"]
    direction TB
    T1["Ch 01<br/>Postulates"]
    T2["Ch 02<br/>Many-body"]
    T3["Ch 03<br/>Hartree-Fock"]
    T4["Ch 04<br/>Kohn-Sham"]
    T5["Ch 05<br/>XC functionals"]
    T1 --> T2 --> T3 --> T4 --> T5
  end
  subgraph M ["Methods track"]
    direction TB
    M1["Ch 06<br/>Basis sets"]
    M2["Ch 07<br/>PBC & Bloch"]
    M3["Ch 08<br/>Pseudopotentials"]
    M4["Ch 09<br/>Forces & opt"]
    M1 --> M2 --> M3 --> M4
  end
  subgraph A ["Applications track"]
    direction TB
    A1["Ch 10<br/>Phonons"]
  end

  T5 --> M1
  M4 --> A1

The track view makes the same information as the dependency graph, but arranges it so that you can see the kinds of chapters. The Theory track is essential for everyone; the Methods track is essential for practitioners; the Applications track is essential for materials work but optional for chemical work.

Worked example — anatomy of a chapter

The fastest way to internalise the template is to **take an existing chapter apart with the template in mind. This section does that for the first half of chapter 06 (basis sets). Read the chapter alongside this example; the two together should make the structure obvious.

Locate the claim

Open chapter 06 and look for the first numbered equation. In chapter 06 it is in section 6.1 and is the Roothaan–Hall matrix equation:

That is the claim of chapter 06: the Fock eigenvalue problem in a finite basis is a generalised matrix eigenproblem. The label `eq:ch-06-roothaan-hall’ in the actual chapter file is what allows the rest of the chapter (and other chapters) to refer to this equation.

Locate the derivation

The derivation is the section that follows the claim. In chapter 06 it is the rest of section 6.1: it expands the orbital $\phi_i$ in the basis ${\chi_\mu}$, multiplies by $\chi_\nu^*$, integrates, and collects the integrals into matrices. Every step is shown. The cross-reference to the claim is at the end of the derivation in the phrase “Collecting all $K$ MOs into one matrix gives \eqref{eq:ch-06-roothaan-hall}.”

Locate the diagram, worked example, and problems

Further down the chapter, you will find:

• a Mermaid diagram in section 6.10 (workflow end-to-end);
• a worked example in section 6.9 (STO-3G H$_2$ with full numbers, including the SCF output and a plot of the MOs);
• three problems in section 6.11 (basis-set counting, the Gaussian product theorem, plane-wave count for water in a 10 Å cell).

Locate the “What we left out”

The final section of chapter 06 (section 6.12) lists four deliberate omissions: BSSE, numerical atom-centred orbitals, density fitting, and relativistic ECPs. The author explains why each was omitted (out of scope, developed in chapter 08, out of scope, etc.). This is the “catch” for the chapter as a whole.

A note on density $\rho$ across the roadmap

To see the cross-chapter version of the same idea, follow the symbol $\rho(\mathbf r)$ — the electron density — from chapter 01 to chapter 06. It is the single most reused symbol in the notes.

Following one symbol across chapters is a good way to verify that you have understood the cross-references.

Notation

We use the following conventions consistently. When a chapter deviates, it says so at the top.

Position, geometry, and units

Atomic units ($\hbar = m_e = e = 1$) are used throughout unless a chapter is explicitly working in SI or eV.

Operators

Wavefunctions, states, and densities

Basis set notation (chapter 06)

Linear algebra and indices

Special symbols

A chapter that needs a symbol not in this table will introduce it with a one-line definition at first use, and will repeat the definition if the symbol is reused after a chapter boundary.

What is not in scope

These notes deliberately stop at the Kohn–Sham equations and the practical zoo of exchange–correlation functionals. The following are either out of scope or sketched only briefly:

• Relativistic DFT. Spin-orbit coupling, the Dirac equation, four- component methods.
• Strong correlation. DFT+U, DMFT, multireference methods, quantum Monte Carlo.
• Time-dependent DFT. The Runge–Gross theorem, the linear-response regime, excitons.
• Embedding. ONIOM, QM/MM, subsystem DFT.
• Materials-specific machinery. Phonons, electron-phonon coupling, Wannier functions, Berry-phase quantities.

Each of those is a separate knowledge base.

A short reading list

Problems

The problems in this chapter are meta: they ask you to use the template, the notation, and the roadmap to navigate the rest of the notes, rather than to do new DFT calculations. In later chapters the problems are DFT problems proper.

What’s next

Chapter 01 — the Schrödinger equation and the postulates we’ll be building on.

Disclaimer. These notes are a personal study aid. They are correct to the best of the author’s knowledge, but they are not a substitute for a textbook. Cite primary sources, not these notes.