Self-Consistent Equations Including Exchange and Correlation Effects¶
Why this mattered¶
Kohn and Sham turned the Hohenberg–Kohn existence theorem for density functional theory into a practical computational framework. The key move was to replace the many-electron interacting problem with self-consistent one-electron equations for a fictitious noninteracting system having the same ground-state density. All difficult many-body effects were gathered into an exchange-correlation potential, approximated in the paper using properties of the uniform electron gas. This made density, rather than the many-electron wavefunction, the central object of electronic-structure calculation.
The paradigm shift was practical as much as conceptual: after this paper, electronic structure could be computed for real solids, molecules, surfaces, and later nanostructures without explicitly tracking the exponentially complex many-electron wavefunction. The resulting Kohn–Sham equations gave researchers a tractable route between simple Hartree-like mean-field models and full many-body theory, enabling calculations accurate enough for chemistry and materials science while remaining scalable enough for broad use.
Subsequent breakthroughs in local, gradient-corrected, hybrid, and more specialized exchange-correlation functionals all grew from the structure this paper defined. Modern computational materials discovery, quantum chemistry workflows, surface catalysis modeling, semiconductor and metal physics, and much of first-principles condensed-matter simulation rely on the Kohn–Sham formulation. Its enormous citation count reflects that it did not merely improve an existing method; it established the standard language in which interacting-electron ground-state problems are still most often made computable.
Abstract¶
From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of $\frac{2}{3}$.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.
Related¶
- enables → A new mixing of Hartree–Fock and local density-functional theories — Kohn-Sham density functional theory provided the orbital-based framework in which B3LYP mixes exact Hartree-Fock exchange with density-functional exchange-correlation.
- enables → Efficient pseudopotentials for plane-wave calculations — Kohn-Sham self-consistent density-functional equations provided the electronic-structure framework in which plane-wave pseudopotentials are constructed and tested.
- enables → Generalized Gradient Approximation Made Simple — Kohn-Sham density-functional theory supplied the exchange-correlation functional framework that PBE simplifies as a generalized-gradient approximation.
- enables → Density-functional approximation for the correlation energy of the inhomogeneous electron gas — Kohn-Sham self-consistent density-functional equations supplied the exchange-correlation framework that the 1986 correlation-energy approximation improves.
- enables → Commentary: The Materials Project: A materials genome approach to accelerating materials innovation — Kohn-Sham density functional theory supplied the first-principles electronic-structure engine used for high-throughput Materials Project calculations.
- enables → Unified Approach for Molecular Dynamics and Density-Functional Theory — Kohn-Sham self-consistent density functional theory supplied the electronic-energy formulation embedded in Car-Parrinello molecular dynamics.
- enables → Projector augmented-wave method — The 1965 Kohn-Sham equations supplied the density-functional framework whose all-electron reconstruction problem the PAW method addresses.
- enables → Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation — The Kohn-Sham equations made practical density-functional calculations possible, enabling later GGA exchange-correlation functionals for atoms, molecules, solids, and surfaces.
- cite ← A new mixing of Hartree–Fock and local density-functional theories — B3LYP builds on Kohn-Sham density-functional theory, using the 1965 self-consistent exchange-correlation framework as the DFT basis for hybridization.
- cite ← Efficient pseudopotentials for plane-wave calculations — Troullier and Martins rely on the Kohn-Sham self-consistent density-functional equations as the electronic-structure framework for their pseudopotentials.
- cite ← Generalized Gradient Approximation Made Simple — PBE is a generalized-gradient exchange-correlation functional built within the Kohn-Sham density-functional theory framework.
- cite ← Density-functional approximation for the correlation energy of the inhomogeneous electron gas — The 1986 correlation-energy approximation builds on the Kohn-Sham framework by supplying a practical exchange-correlation functional for the self-consistent density-functional equations.
- cite ← Commentary: The Materials Project: A materials genome approach to accelerating materials innovation — The Materials Project relies on density functional theory calculations grounded in the Kohn-Sham self-consistent equations for exchange and correlation.
- cite ← Unified Approach for Molecular Dynamics and Density-Functional Theory — Car-Parrinello molecular dynamics builds its electronic-structure forces on Kohn-Sham density-functional theory with exchange-correlation effects.
- cite ← Projector augmented-wave method — The PAW method relies on the Kohn-Sham self-consistent density-functional equations as its underlying electronic-structure framework.
- cite ← Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation — Perdew et al. ground their exchange-correlation approximations in the Kohn-Sham density-functional formalism.