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| Lieb–Oxford inequality | |
|---|---|
| Name | Lieb–Oxford inequality |
| Field | Mathematical physics; Quantum chemistry |
| Introduced | 1981 |
| Discoverer | Elliott H. Lieb; Stephen Oxford |
Lieb–Oxford inequality
The Lieb–Oxford inequality is a mathematical bound arising in Lieb and Oxford's work that places a lower bound on the indirect part of the Coulomb energy in many-body quantum mechanics and quantum chemistry models, with crucial impact on density functional theory and rigorous estimates in Thomas–Fermi theory. It connects rigorous analysis by scholars associated with institutions such as Princeton University, Harvard University, and University of California, Berkeley to computational practice in groups at Max Planck Society and industrial research labs like Bell Labs.
The inequality was introduced in a 1981 paper by Lieb and Oxford and subsequently refined by researchers at places including Rutgers University and University of Cambridge. It provides a universal constant that bounds the exchange–correlation or indirect Coulomb energy in terms of the one-particle density used in density functional theory implementations by teams at IBM Research and academic groups at ETH Zurich and MIT. The result has influenced developments related to the Hohenberg–Kohn theorem, the Kohn–Sham equations, and comparisons with Hartree–Fock method estimates.
In its basic form the inequality gives, for an N-particle wavefunction associated with electrons in the Coulomb field studied by researchers at Los Alamos National Laboratory and Lawrence Berkeley National Laboratory, a lower bound on the indirect part of the Coulomb interaction in terms of an integral of the electron density, a key object in Kohn–Sham theory and analyses by groups at University of Oxford and University of Tokyo. The bound uses constants whose optimal values were later studied by mathematicians connected to Courant Institute, University of Chicago, and Princeton Plasma Physics Laboratory. The statement is often presented alongside comparison inequalities such as those by Hardy, Rellich, and Sobolev that are staples in analysis programs at Steklov Institute and CNRS.
Proof techniques draw on rearrangement inequalities familiar from work by Hardy, Littlewood, and Pólya, operator methods developed by Reed and Simon, and semiclassical analysis associated with Onsager and Newton-era analogues. Approaches leveraging screening and local density estimates were advanced by teams at University of California, Santa Barbara and Imperial College London, while alternative derivations use convexity arguments encountered in research at University of Bonn and University of Paris-Saclay.
The original constant obtained by Lieb and Oxford has been the subject of numerous improvements by mathematicians affiliated with University of Utrecht, University of Toronto, McMaster University, and University of Cambridge. Subsequent work by researchers connected to Scuola Normale Superiore and École Polytechnique tightened bounds using numerical and analytical tools; debates over sharpness invoke methods from scholars at Princeton University and Stanford University. Proposed optimal constants play a role in constraints used by developers at Gaussian (software), NWChem, and quantum chemistry groups at Los Alamos.
The inequality underpins rigorous justifications for exchange–correlation functionals utilized by practitioners in density functional theory across institutions such as Argonne National Laboratory, Sandia National Laboratories, and university groups at University of Illinois Urbana-Champaign. It constrains approximate functionals employed in the Kohn–Sham framework and informs error estimates in hybrid methods blending Hartree–Fock method with local or semilocal functionals, techniques used by teams at BASF and Bayer. The Lieb–Oxford bound appears in analytical studies of molecular stability investigated at Royal Society-supported projects and in mathematically rigorous treatments by researchers at Institute for Advanced Study.
Extensions include versions for different spatial dimensions studied by groups at Columbia University and Yale University, magnetic-field variants considered by researchers at Max Planck Institute for Physics and University of Geneva, and spin-dependent or current-dependent formulations explored by scholars at University of Sydney and University of Melbourne. Related inequalities have been developed in contexts linked to Thomas–Fermi theory and semiclassical limits investigated at Princeton University and Caltech.
Explicit model systems examined by research groups at Argonne National Laboratory and Oak Ridge National Laboratory—including uniform electron gas settings, atoms, and simple molecules—illustrate how the bound controls exchange–correlation energy compared with results from quantum Monte Carlo computations performed by teams at University of Illinois and ETH Zurich. Numerical counterexamples used to test sharpness have been produced by computational groups associated with University of Cambridge and Weizmann Institute of Science, guiding further refinements by mathematicians at Université de Montréal and Imperial College London.