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| Kohn-Sham | |
|---|---|
| Name | Kohn–Sham formalism |
| Field | Theoretical chemistry; Condensed matter physics |
| Introduced | 1965 |
| Founder | Walter Kohn; Lu Jeu Sham |
| Notable works | "Self-Consistent Equations Including Exchange and Correlation Effects" |
Kohn-Sham
Kohn–Sham is a widely used computational framework in theoretical chemistry, condensed matter physics, and materials science that transforms the many-body problem of interacting electrons into an auxiliary noninteracting problem, enabling practical calculations of electronic structure. The approach underlies numerous methods and software packages employed at institutions such as Bell Labs, University of California, Santa Barbara, Harvard University, and Max Planck Society. It provides the basis for predicting properties relevant to experiments at facilities like CERN and Brookhaven National Laboratory and for interpreting results from techniques associated with Nobel Prize–winning discoveries in quantum chemistry.
The Kohn–Sham formalism builds on foundational concepts developed by Walter Kohn and Lu Jeu Sham, bridging ideas from Thomas–Fermi model, Hartree–Fock method, and density-based approaches used across Princeton University, University of California, Berkeley, and Cambridge University. By representing the electronic ground state via an electron density that is computationally tractable, it complements experimental probes carried out at laboratories such as Los Alamos National Laboratory and Argonne National Laboratory. The formalism facilitates connection between first-principles theory and applied research pursued by organizations like IBM and Bell Labs.
The central Kohn–Sham equations are a set of self-consistent single-particle equations whose solutions yield orbitals reproducing the true interacting electron density. Mathematically similar structures appear in methods developed at Massachusetts Institute of Technology, Stanford University, and ETH Zurich, and they are implemented within major electronic-structure codes originating from groups at Oak Ridge National Laboratory and Lawrence Berkeley National Laboratory. The equations contain kinetic, external potential, Hartree, and exchange–correlation terms; solving them iteratively often involves algorithms pioneered by researchers affiliated with Microsoft Research, Bell Labs, and Los Alamos National Laboratory.
Exchange–correlation functionals encapsulate the many-body effects beyond the classical Hartree term and are the primary source of approximation in the Kohn–Sham framework. Families of functionals—local density approximation (LDA), generalized gradient approximation (GGA), hybrid functionals, and meta-GGA—were developed through collaborations and research programs at IBM Research, University of Cambridge, University of Oxford, and University of California, Irvine. Notable functionals and parameterizations trace back to contributions by researchers associated with Bell Labs, Argonne National Laboratory, Rutgers University, and University of Florida, and they continue to be benchmarked against experiments performed at National Institute of Standards and Technology and Lawrence Livermore National Laboratory.
Implementations of the Kohn–Sham equations appear in a wide array of software packages developed by teams at MIT, Stanford University, Columbia University, University of Vienna, and Cambridge University and in community projects supported by European Research Council and National Science Foundation grants. Numerical techniques such as plane-wave basis sets, localized Gaussian bases, real-space grids, and projector-augmented wave methods were advanced by researchers at ETH Zurich, Université Paris-Saclay, Tokyo Institute of Technology, and Seoul National University. Linear-scaling algorithms, density-matrix purification, and fast-Fourier-transform optimizations draw on computational strategies from Lawrence Berkeley National Laboratory and Argonne National Laboratory and are integral to high-performance computing efforts on systems at Oak Ridge National Laboratory and Sandia National Laboratories.
The Kohn–Sham approach enables prediction of molecular geometries, reaction energetics, electronic band structures, defect energetics, and optical spectra—capabilities exploited in research at DuPont, Toyota, BASF, and academic centers including Caltech, Imperial College London, and University of Chicago. It supports design efforts for photovoltaic materials studied at National Renewable Energy Laboratory and for catalysts investigated at Brookhaven National Laboratory and SLAC National Accelerator Laboratory. Studies of superconductivity, magnetism, and topological phases use Kohn–Sham–based calculations carried out by groups at Rutgers University, University of Illinois Urbana–Champaign, and Northwestern University to interpret measurements from facilities like Diamond Light Source and Advanced Photon Source.
Limitations of the Kohn–Sham approach include the approximate nature of exchange–correlation functionals, challenges in describing strong correlation, van der Waals interactions, excited states, and charge-transfer processes—issues addressed by extensions developed at Argonne National Laboratory, Lawrence Berkeley National Laboratory, and Max Planck Society. Methods that build on or go beyond Kohn–Sham include time-dependent density functional theory (TDDFT), DFT+U, dynamical mean-field theory (DMFT), and quantum Monte Carlo approaches; these were advanced by researchers at Princeton University, Columbia University, University of Cambridge, and University of Munich. Multiscale frameworks and machine-learning-enhanced functionals are active research areas supported by institutions such as Google DeepMind, ETH Zurich, and University of Toronto.
The Kohn–Sham formalism was introduced in a seminal paper by Walter Kohn and Lu Jeu Sham and subsequently refined by a global community including contributors at Bell Labs, Harvard University, University of Oxford, and Stanford University. Key developments in functional design, numerical algorithms, and software engineering came from teams at IBM Research, Argonne National Laboratory, Lawrence Berkeley National Laboratory, and Max Planck Institute for Solid State Research. The framework’s adoption and evolution involved collaborations across universities, national laboratories, and industrial research centers, culminating in widespread integration into curricula at Massachusetts Institute of Technology, University of Cambridge, and University of California, Berkeley.