Density-functional theory (DFT) reduces the daunting many-electron Schrödinger problem by choosing the electron density ρ(r) instead of the full wavefunction as the basic variable. The first Hohenberg–Kohn theorem states that the external potential is uniquely determined by the ground-state density, while the second establishes a variational principle for an energy functional. Practical exchange-correlation approximations such as LDA and GGA followed, allowing accurate calculations of solid-state band structures and reaction energies on catalytic surfaces. Today DFT is indispensable for designing semiconductor chips, rechargeable batteries, and green catalysts.

DFT has become the de facto standard for first-principles electronic-structure calculations, with typical O(N^3) scaling compared to the O(N^4) cost of Hartree–Fock. In the Kohn–Sham scheme, non-interacting reference electrons are introduced and all many-body effects are captured by an exchange-correlation functional Exc[ρ]. Approximations range from the Local Density Approximation (LDA) based on the homogeneous electron gas, through Gradient-Corrected (GGA) and hybrid functionals such as B3LYP or PBE0. Extensions like time-dependent DFT (TD-DFT) for excited states, DFT+U for strongly correlated systems, and van der Waals-corrected DFT for weak interactions widen the scope. These tools quantify Li diffusion barriers in batteries, optical spectra of photocatalysts, and phase diagrams of Earth’s deep-mantle minerals.

Beyond the foundational Hohenberg–Kohn theorems, Kohn’s key contribution was the introduction of the Kohn–Sham reference system, which enabled practical implementations using plane-waves, Gaussians, real-space grids, and full-potential LAPW bases. Linear-response formulations allow phonon dispersion calculations, and hybrid workflows with GW or Bethe–Salpeter equations have become standard. Modern developments include meta-GGA functionals such as SCAN/r2SCAN and emerging machine-learning XC models. Persistent challenges—self-interaction error, band-gap underestimation, strong correlation—are being addressed via constrained DFT, exact-exchange mixing, and beyond-DFT methods. Kohn’s framework remains the backbone of ab-initio simulations, powering materials informatics and high-throughput discovery pipelines.