Density Functional Theory in Condensed Matter

Core Idea

Density functional theory (DFT) replaces the intractable many-electron Schrodinger equation with an equivalent problem involving only the electron density n(r). The Hohenberg-Kohn theorems (1964) prove that the ground state energy is a unique functional of n(r) and that minimizing this functional yields the exact ground state density. The Kohn-Sham scheme (1965) maps the interacting problem onto non-interacting electrons moving in an effective potential that includes exchange and correlation effects. DFT with approximate exchange-correlation functionals (LDA, GGA, hybrid) has become the standard method for calculating band structures, crystal structures, lattice constants, elastic properties, and phase stability of real materials from first principles.

Explainer

The fundamental challenge of condensed matter theory is the many-body problem: N interacting electrons in N^ion ions, governed by the Schrodinger equation for a wavefunction Psi(r_1, ..., r_N) that depends on 3N coordinates. For a macroscopic solid, N ~ 10²³, and direct solution is utterly impossible. Density functional theory circumvents this by proving that the ground state energy is determined entirely by the electron density n(r) — a function of just three coordinates, regardless of N.

The Hohenberg-Kohn theorems (1964) established two results. First, the external potential V_ext(r) (and hence all properties) is a unique functional of the ground state density n(r) — there is a one-to-one mapping. Second, the true ground state density minimizes the energy functional E[n]. These theorems are exact but not directly useful because the kinetic energy functional T[n] and the exchange-correlation functional E_xc[n] are unknown. The breakthrough came with the Kohn-Sham scheme (1965), which maps the interacting problem onto a system of non-interacting electrons moving in an effective potential V_eff = V_ext + V_Hartree + V_xc. The non-interacting system is chosen to reproduce the exact ground state density, and its kinetic energy is computed exactly from single-particle orbitals. All the many-body complexity is isolated in E_xc[n], which is typically small and can be approximated.

The most common approximations for E_xc are the local density approximation (LDA), which uses the exchange-correlation energy of a uniform electron gas at the local density, and the generalized gradient approximation (GGA, e.g., PBE), which also includes density gradients. These approximations work remarkably well for ground state properties: lattice constants are predicted to ~1%, bulk moduli to ~5-10%, and crystal structure predictions are usually correct. The Kohn-Sham equations are solved self-consistently by iterating until the input and output densities agree, using plane-wave basis sets with pseudopotentials or augmented wave methods.

DFT's limitations are well understood. The Kohn-Sham eigenvalues are not quasiparticle energies, so band gaps are systematically underestimated (~30-50% in LDA/GGA). Strongly correlated systems (Mott insulators, heavy fermions) are poorly described because the exchange-correlation functional cannot capture the physics of strong on-site correlations. Van der Waals interactions are absent in standard LDA/GGA. These limitations have driven the development of extensions: hybrid functionals (B3LYP, HSE) for better gaps, DFT+U for correlated systems, RPA and GW for accurate excitation spectra, and DFT-D for dispersion corrections. Despite these caveats, DFT is the default first-principles method in condensed matter, chemistry, and materials science — it is arguably the most impactful computational method in all of physical science.