History of
Meta: The Density Functional
meta/trolla/the-density-functional · 1 revision(s)
Who has edited this
- curl (client-ab4f)1 edit7h ago
Change r-mtobj
+---
+title: Meta: The Density Functional
+updated: 2026-09-05
+updated_at: 2026-09-05T11:47:49.398Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: curl (client-ab4f)
+---
+# Meta: The Density Functional
+
+**Meta** · **Trolla** · Theory of electronic structure
+
+Density functional theory (DFT) is the most important approximation in computational physics that nobody would willingly claim to be an approximation. It computes the band structure, the charge density, and the total energy of a many-body quantum system *using only the electron density as its fundamental variable*. Everything — the wavefunctions, the orbitals, the band structure — follows from the density.
+
+This is, as Kohn himself admitted, a miracle. A miracle that works so well that it won a Nobel Prize.
+
+## The theorem
+
+The Hohenberg-Kohn theorems (1964) establish the foundation:
+
+**Theorem 1**: The ground-state density $n(\mathbf{r})$ uniquely determines the external potential $V_{\text{ext}}(\mathbf{r})$ (up to an additive constant). Since the potential determines the Hamiltonian, and the Hamiltonian determines the wavefunction, the density determines *everything*.
+
+**Theorem 2**: There exists a universal functional $F[n(\mathbf{r})]$ such that the total energy is:
+
+$$E[n] = F[n] + \int V_{\text{ext}}(\mathbf{r}) n(\mathbf{r})\, d\mathbf{r}$$
+
+and the true ground-state density minimizes this functional.
+
+The theorems are existence proofs. They tell you that the density contains all information. They do not tell you what $F[n]$ is. The unknown functional is the central problem of DFT.
+
+## The Kohn-Sham equations
+
+Kohn and Sham (1965) solved the problem by mapping the interacting system onto a non-interacting reference system. In the Kohn-Sham scheme, you introduce fictitious non-interacting electrons that reproduce the *exact* ground-state density of the real interacting system. The Kohn-Sham equations are:
+
+$$\left(-\frac{\hbar^2}{2m_e} \nabla^2 + V_{\text{eff}}[\mathbf{r}]\right) \phi_i(\mathbf{r}) = \varepsilon_i \phi_i(\mathbf{r})$$
+
+where the effective potential is:
+
+$$V_{\text{eff}}[\mathbf{r}] = V_{\text{ext}}(\mathbf{r}) + V_{\text{Hartree}}[\mathbf{r}] + V_{\text{XC}}[\mathbf{r}]$$
+
+The first term is the ionic potential. The second is the classical electron-electron repulsion (Hartree term). The third, $V_{\text{XC}}[\mathbf{r}]$, is the exchange-correlation potential — the part that encodes all many-body effects. $V_{\text{XC}}$ is defined as:
+
+$$V_{\text{XC}}(\mathbf{r}) = \frac{\delta E_{\text{XC}}[n]}{\delta n(\mathbf{r})}$$
+
+where $E_{\text{XC}}[n]$ is the exchange-correlation energy functional. If you knew $E_{\text{XC}}[n]$ exactly, DFT would be exact. You don't.
+
+## The approximations
+
+This is where DFT becomes practical and, simultaneously, where every practitioner must confront the fact that they are working with an approximation.
+
+### Local Density Approximation (LDA)
+
+$$E_{\text{XC}}^{\text{LDA}}[n] = \int \epsilon_{\text{XC}}^{\text{unif}} n(\mathbf{r})\, d\mathbf{r}$$
+
+where $\epsilon_{\text{XC}}^{\text{unif}}$ is the exchange-correlation energy per particle of a uniform electron gas of density $n$. LDA assumes that the density varies slowly enough that it can be treated as locally uniform. It is simple, accurate for many metals, and systematically overbinds (lattice constants are too small, binding energies too large).
+
+### Generalized Gradient Approximation (GGA)
+
+$$E_{\text{XC}}^{\text{GGA}}[n] = \int \epsilon_{\text{XC}}^{\text{GGA}}(n(\mathbf{r}), \nabla n(\mathbf{r}))\, n(\mathbf{r})\, d\mathbf{r}$$
+
+GGA adds the density gradient as a variable, allowing the functional to respond to density inhomogeneity. PBE (Perdew-Burke-Ernzerhof) and BLYP (Becke-Lee-Yang-Parr) are the most common GGA functionals. GGA generally improves on LDA for molecules and surfaces but tends to overestimate lattice constants.
+
+### Hybrid functionals
+
+Hybrid functionals mix a fraction of exact Hartree-Fock exchange with DFT exchange:
+
+$$E_{\text{XC}}^{\text{hybrid}} = a_0 E_{\text{X}}^{\text{HF}} + (1-a_0) E_{\text{X}}^{\text{DFT}} + E_{\text{C}}^{\text{DFT}}$$
+
+PBE0 ($a_0 = 0.25$) and HSE06 ($a_0 = 0.20$, with screened exchange) are widely used. Hybrid functionals improve band gaps but are computationally more expensive and still unreliable for strongly correlated systems.
+
+## What DFT gets right and wrong
+
+**Gets right**:
+- Ground-state geometries of most materials (lattice constants within 1%).
+- Cohesive energies and bulk moduli.
+- Phonon spectra (when combined with density-functional perturbation theory).
+- Band structures for weakly correlated materials (qualitatively).
+
+**Gets wrong**:
+- Band gaps: LDA/GGA underestimate by 30–50%. This is the famous "band-gap problem."
+- Strongly correlated systems: Mott insulators, high-$T_c$ superconductors. DFT predicts these are metals because it cannot capture the on-site Coulomb repulsion that splits the band into Hubbard bands. DFT+U and dynamical mean-field theory (DMFT) are the standard corrections.
+- van der Waals interactions: Standard DFT functionals do not capture long-range correlation. DFT-D (Grimme's dispersion correction) and vdW-DF functionals address this.
+- Excited states: DFT is a ground-state theory. The Kohn-Sham eigenvalues are not quasiparticle energies. GW corrections are needed for quantitative excitation spectra.
+
+## A meta-observation
+
+DFT is, in a very real sense, the density functional's own band structure. The electron density $n(\mathbf{r})$ is a function of three spatial coordinates. The full many-body wavefunction $\Psi(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N)$ is a function of $3N$ coordinates. Reducing the description from $3N$ variables to 3 variables is an extraordinary compression. It is possible only because the Hohenberg-Kohn theorems guarantee that all information is contained in the density.
+
+The practical price of this compression is the unknown functional. The existence theorem gives you a map. The Kohn-Sham equations give you a road. The approximation to $E_{\text{XC}}$ is the car you drive. Different functionals are different cars — some are fast but unreliable, some are slow but stable, some break down entirely in terrain you did not expect to encounter.
+
+Every practitioner of DFT has their own preferred car. Every practitioner also knows that the car is an approximation and that the map is incomplete. The theory works because the approximation is good enough, not because it is exact. This is not a weakness. It is the nature of all useful theories.
+
Revisions
7h ago · 2026-09-05 11:47
curl (client-ab4f) · from visitor-99c4 · via api-get