Density Functional Theory (DFT)

Quantum mechanics tells us: a system of N atoms is fully described by the wave function Ψ(r₁,r₂,...,rₙ). But for N electrons this is a 3N-dimensional object — practically impossible to compute for N>10. DFT idea: instead of the wave function, use the electron density ρ(r) — only 3 dimensions! Walter Kohn proved in 1964 that all properties of a system are determined by ρ(r). This is the foundation of DFT.

1927 — Thomas-Fermi model: first attempt to use electron density — simple but inaccurate. 1964 — Pierre Hohenberg and Walter Kohn: "Hohenberg-Kohn theorems" — mathematical foundation of DFT (Physical Review, 1964). 1965 — Walter Kohn and Lu Jeu Sham: "Kohn-Sham equations" — practical DFT computation scheme. 1998 — Walter Kohn received the Nobel Prize in Chemistry for DFT. Today DFT is the most widely used computational method in chemistry and materials science.

First theorem: the external potential V_ext(r) is uniquely determined by the electron density ρ(r). As a result — the total energy is a functional of ρ(r). Second theorem: the ground state energy is minimal — variational principle. Total energy functional:

T[ρ] — kinetic energy, V_ext[ρ] — external potential (nuclei), V_ee[ρ] — electron-electron interaction (Coulomb + exchange-correlation).

Kohn-Sham (1965): the real interacting electron system is replaced by a non-interacting system with an effective potential V_eff(r). This gives single-electron Schrödinger equations:

V_eff(r) = V_ext(r) + V_Hartree(r) + V_xc(r). V_xc — exchange-correlation potential — the only "unknown" in DFT, approximated by various functionals (LDA, GGA, hybrid).

Electron density is computed from occupied orbitals:

ρ(r) — probability density of finding an electron. ∫ρ(r)dr = N (total number of electrons). Mulliken population analysis distributes ρ among atoms.

HOMO (Highest Occupied Molecular Orbital) and LUMO (Lowest Unoccupied Molecular Orbital). The HOMO-LUMO gap = molecular "electronic Band Gap":

DFT → Crystal Structure: DFT always starts with atomic coordinates — defining the unit cell. DFT → Band Structure: DFT gives accurate band structure — far more precise than tight-binding. DFT → NEGF: NEGF quantum transport uses the DFT Hamiltonian — this is the DFT+NEGF approach for nanodevices.

Pharmaceuticals: design of new drug molecules — HOMO-LUMO gap determines chemical reactivity. Catalysis: calculating adsorption energies on catalyst surfaces. Batteries: DFT modeling of Li-ion battery electrode materials. Semiconductors: accurate Band Gap calculation for Si, GaAs device design.

Small HOMO-LUMO gap → chemically reactive molecule. Large gap → stable, less reactive. H₂ has large gap — very stable. Benzene has small gap — electrons are delocalized.