# Density functional theory & Kohn–Sham

[Hartree–Fock](hartree-fock.md) gave us a mean-field wavefunction but missed **correlation**. Recovering it
by improving the wavefunction (post-HF methods) is accurate but expensive. **Density functional theory
(DFT)** takes a completely different, and cheaper, route: work not with the $3N$-dimensional wavefunction
but with the **electron density** $\rho(\mathbf r)$ — a function of just **three** variables. This is the
workhorse of modern quantum chemistry, and it is what `mychk.scf(ref="r", xc="...").run()` computes.

## The density is enough: Hohenberg–Kohn

It is not obvious that the simple density $\rho(\mathbf r)$ contains as much information as the full
wavefunction. The Hohenberg–Kohn theorems say it does.

:::{prf:theorem} Hohenberg–Kohn
:label: thm-hk

1. **The ground-state density determines everything.** For a system of electrons in an external potential
   $v_{\text{ext}}$ (the nuclei), $v_{\text{ext}}$ is fixed — up to a constant — by the ground-state density
   $\rho(\mathbf r)$. Hence *every* ground-state property, including the energy, is a **functional of the
   density**: $E = E[\rho]$.
2. **A variational principle for the density.** There is a universal functional such that
   $E[\rho] \ge E_0$ for any trial density, with equality at the true ground-state density $\rho_0$.
:::

In principle this is miraculous: the impossible $3N$-dimensional problem collapses to finding one 3-D
function. The catch is equally severe — **the exact functional $E[\rho]$ is unknown**, and its worst piece
is the kinetic energy, which is very hard to write in terms of $\rho$ alone.

## The Kohn–Sham trick

Kohn and Sham found the escape that makes DFT practical. Reintroduce **orbitals**, but only to handle the
kinetic energy: imagine a *fictitious* system of **non-interacting** electrons that has the **same density**
as the real one. Its kinetic energy $T_s[\rho]$ is easy (it is just the orbital kinetic energy, as in HF),
and we lump *everything we do not know* into one term. The energy splits as

$$
E[\rho] = \underbrace{T_s[\rho]}_{\text{non-interacting kinetic}}
        + \underbrace{\int v_{\text{ext}}(\mathbf r)\,\rho(\mathbf r)\,d\mathbf r}_{\text{nuclear attraction}}
        + \underbrace{J[\rho]}_{\text{Hartree (classical Coulomb)}}
        + \underbrace{E_{\text{xc}}[\rho]}_{\text{exchange–correlation}} ,
$$

where the **exchange–correlation functional** $E_{\text{xc}}[\rho]$ absorbs everything left over — exchange,
correlation, *and* the small kinetic-energy difference between the real and non-interacting systems. It is
the *only* unknown.

Minimizing this energy gives the **Kohn–Sham equations**, one-electron eigenvalue equations that look
exactly like Hartree–Fock:

$$
\left[-\tfrac12\nabla^2 + v_{\text{eff}}(\mathbf r)\right]\psi_i = \varepsilon_i\,\psi_i,
\qquad
v_{\text{eff}} = v_{\text{ext}} + v_{\text{H}} + v_{\text{xc}},
\quad v_{\text{xc}} = \frac{\delta E_{\text{xc}}}{\delta \rho},
$$

with the density rebuilt from the occupied Kohn–Sham orbitals, $\rho(\mathbf r) = \sum_i |\psi_i(\mathbf r)|^2$.
Because $v_{\text{eff}}$ depends on $\rho$, these are solved by the **same self-consistent-field loop** as
Hartree–Fock — the same `run(log=...)` cycle table, the same DIIS. In fact, if you set
$E_{\text{xc}}$ to the exact-exchange-only expression, you *recover* Hartree–Fock. DFT and HF are two points
on one continuum.

## Choosing a functional: Jacob's ladder

Everything hinges on the approximation to $E_{\text{xc}}$. These are ranked on a famous **"Jacob's
ladder"** of increasing sophistication — the choices behind `xc=`:

- **LDA** (local density approximation) — $E_{\text{xc}}$ depends on $\rho$ at each point only. The
  simplest rung.
- **GGA** (generalized gradient approximation, e.g. `pbe`) — also uses the density *gradient* $\nabla\rho$.
- **meta-GGA** (e.g. `scan`) — adds the kinetic-energy density $\tau$ (and sometimes $\nabla^2\rho$).
- **Hybrids** (e.g. `b3lyp`, `pbe0`) — mix in a fraction of **exact (Hartree–Fock) exchange**. This rung
  literally blends the HF of the last chapter with DFT, and is the most popular choice for molecules.

qc-rs passes the functional name straight to **libxc**, so any of its LDA/GGA/meta-GGA/hybrid functionals
works in `xc=`.

## The numbers, and an important caveat

Here is water again (RHF vs three functionals, all cc-pVDZ):

| method | `xc=` | rung | energy / $E_h$ |
|---|---|---|---|
| Hartree–Fock | — | — | −76.026772 |
| LDA | `lda` | local | −75.854689 |
| PBE | `pbe` | GGA | −76.333442 |
| B3LYP | `b3lyp` | hybrid | −76.420369 |

Notice something that trips up beginners: **LDA's energy is *higher* than Hartree–Fock's**. That is not a
bug — it is a fundamental difference from the [variational picture](variational-lcao-basis.md) of chapters
2–3. HF gives a rigorous *upper bound* to the exact energy (it uses the exact Hamiltonian), but a DFT total
energy uses an *approximate* $E_{\text{xc}}$, so it is **not a bound at all**. You therefore *cannot* rank
functionals by their total energy — a "better" functional need not give a lower number. Functionals are
judged by how well they reproduce *reference data* (geometries, reaction energies, experiment), not by the
energy itself. (B3LYP's lower value here does reflect the correlation it captures, but that comparison is
not a rigorous ranking.)

## Strengths, weaknesses, and where to go

DFT's appeal is its **cost–accuracy trade-off**: for roughly the price of Hartree–Fock, a good functional
folds in much of the correlation HF misses — which is why it dominates practical chemistry. Its limits
follow from the same source:

- **No systematic route to the exact answer.** Unlike wavefunction methods (HF → MP2 → coupled cluster →
  …), there is no ladder you can climb to convergence; you rely on functionals validated against data.
- **Known failure modes.** Most functionals miss long-range **dispersion** (fixed by the DFT-D3/D4
  corrections in [Solvation & dispersion](../20-guide/solvation-dispersion.md)), and many suffer
  self-interaction error.

You have now met the two pillars of practical electronic structure: **Hartree–Fock** and **DFT**. With the
theory in hand, the [User guide](../20-guide/molecular-input.md) shows how to *run* them well — choosing
references, functionals, guesses, and convergence — starting from [molecular input](../20-guide/molecular-input.md).