This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| Kohn–Sham equations | |
|---|---|
| Name | Kohn–Sham equations |
| Field | Quantum chemistry; Condensed matter physics |
| Introduced | 1965 |
| Developers | Walter Kohn; Lu Jeu Sham |
| Related | Density functional theory; Hartree–Fock method; Born–Oppenheimer approximation |
Kohn–Sham equations The Kohn–Sham equations are a set of self-consistent single-particle equations central to modern electronic-structure theory, originally formulated by Walter Kohn and Lu Jeu Sham in 1965; they provide a practical scheme within density functional theory to compute ground-state properties of many-electron systems. The approach maps an interacting electron problem onto a noninteracting reference with the same electron density, enabling computations used across chemistry and condensed-matter physics by groups and institutions worldwide.
The Kohn–Sham formalism emerged from efforts that included work by Walter Kohn and Lu Jeu Sham and built on earlier contributions from Paul Dirac, John Slater, and Douglas Hartree, situating itself in the lineage of methods developed at institutions such as the University of California, Santa Barbara, the University of Toronto, and institutions affiliated with the Nobel Prize in Chemistry and Physics. The formulation is foundational for computational packages developed by teams at IBM, Bell Labs, and the Massachusetts Institute of Technology, and underpins studies ranging from materials investigated at Oak Ridge National Laboratory to molecules characterized at the Max Planck Institute. Practitioners often combine the Kohn–Sham scheme with ideas from the Born–Oppenheimer approximation and techniques influenced by the work of Lev Landau, Richard Feynman, and P. W. Anderson.
The Kohn–Sham framework is rooted in the Hohenberg–Kohn theorems, which assert a one-to-one correspondence between ground-state electron density and external potential, a principle that connects to the conceptual developments of Erwin Schrödinger and Paul Dirac; it formalizes how a noninteracting reference system reproduces the exact interacting density using an effective potential. The effective potential contains contributions analogous to the Hartree potential and an exchange–correlation term, concepts that reflect historical threads from the Hartree–Fock method developed by Douglas Hartree and Vladimir Fock as well as exchange ideas linked to John Slater and J. C. Slater. The rigorous underpinnings relate to variational principles articulated by Enrico Fermi and to mathematical foundations explored by Norbert Wiener and John von Neumann.
The Kohn–Sham equations are a set of coupled single-particle Schrödinger-like equations solved self-consistently for Kohn–Sham orbitals, with the total energy functional partitioned into kinetic, external, Hartree, and exchange–correlation contributions, a decomposition that echoes frameworks used by Walter Kohn, Lu Jeu Sham, and contemporaries. Implementation involves diagonalizing effective Hamiltonians in bases such as plane waves popularized by Steven Louie and Marvin Cohen, localized atomic-like orbitals used by Roald Hoffmann and Linus Pauling, or Gaussian-type orbitals promoted by John Pople and Walter Kohn collaborators. The self-consistent field cycle uses techniques from numerical analysis advanced by Alan Turing and John von Neumann, and computational acceleration methods developed in contexts including Bell Labs and CERN.
Practical Kohn–Sham calculations require approximations to the exchange–correlation functional; common classes include the local density approximation (LDA) linked to the work of John Perdew and David Langreth, the generalized gradient approximation (GGA) associated with Perdew, Burke, and Ernzerhof, and hybrid functionals combining GGA with Hartree–Fock exchange as popularized by Axel Becke and John Pople. Advanced approximations such as meta-GGA, range-separated hybrids, and functionals informed by many-body perturbation theory connect to methods developed by Lars Hedin, Giovanni Onida, and Lev Landau, while nonlocal correlation functionals for dispersion trace lineage to work by Stefan Grimme and Alfredo Truhlar. Benchmarking and functional development often occur at laboratories and universities including Lawrence Berkeley National Laboratory, the University of Cambridge, and ETH Zurich.
Kohn–Sham equations are implemented in numerous software packages and codes maintained by communities associated with IBM Research, Argonne National Laboratory, the Quantum ESPRESSO consortium, and Gaussian, with algorithmic contributions from developers at Rice University, Princeton University, and Stanford University. Numerical strategies include plane-wave pseudopotential methods influenced by David Vanderbilt, projector-augmented wave methods introduced by Peter Blöchl, and all-electron approaches such as the linearized augmented plane wave method used by teams at the Fritz Haber Institute and the Paul Scherrer Institute. Parallelization and high-performance computing practices enabling large-scale Kohn–Sham simulations are driven by supercomputing centers like Oak Ridge and Los Alamos and by collaborations with vendors such as Intel and NVIDIA.
The Kohn–Sham scheme underlies studies of molecular spectroscopy in groups led by Ahmed Zewail and Rudolph Marcus, catalysis research at institutions including the California Institute of Technology and the Weizmann Institute, and materials design efforts exemplified by work at the National Renewable Energy Laboratory and Toyota’s research centers. It is applied to band-structure predictions in solids investigated by theorists such as Philip Anderson and Marvin Cohen, surface science problems explored at IBM and Bell Labs, and nanoscience studies involving researchers at Cornell University and the Massachusetts Institute of Technology. Case studies include computations for transition-metal complexes studied by Roald Hoffmann, organic photovoltaics advanced by researchers at MIT, and battery materials evaluated at Lawrence Livermore National Laboratory.
Despite widespread success, Kohn–Sham methods face limitations in describing strongly correlated systems studied by groups around Nobel laureates such as Klaus von Klitzing and the Hubbard-model community, in capturing van der Waals interactions without special corrections pioneered by Stefan Grimme, and in treating excited states where time-dependent density functional theory developed by E. Runge and E. K. U. Gross or many-body GW methods from Lars Hedin are often preferred. Extensions include DFT+U approaches used in transition-metal oxide research at the Max Planck Society, dynamical mean-field theory collaborations involving Antoine Georges, and embedding methods pursued at institutions such as Harvard University and Imperial College London. Ongoing research efforts span national laboratories, universities, and industry partners including IBM, Toyota, and BASF.