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| Hartree approximation | |
|---|---|
| Name | Hartree approximation |
| Field | Quantum mechanics |
| Introduced | 1927 |
| Inventor | Douglas Hartree |
Hartree approximation
The Hartree approximation is a self-consistent, single-particle method in quantum mechanics for approximating many-body wavefunctions and energies. It reduces an N-body problem to a set of coupled one-body equations by representing the many-body wavefunction as a product of single-particle orbitals and replacing interparticle interactions with average fields. Originating in the interwar period, the method influenced atomic, molecular, and condensed matter theory and provided a stepping stone toward later methods developed by physicists and chemists.
The Hartree approximation was developed in the context of atomic theory by Douglas Hartree and contemporaries such as Erwin Schrödinger, Paul Dirac, Wolfgang Pauli, and Niels Bohr. It grew from attempts to describe multi-electron atoms like Helium, Lithium, and Neon using tractable equations influenced by earlier work on the Schrödinger equation and the Thomas–Fermi model. The approach frames the N-electron problem in a basis where each electron moves in an effective potential arising from all other electrons and fixed nuclei such as in models of the Hydrogen atom and heavier elements studied at institutions including the University of Cambridge and the University of Manchester.
In the Hartree formalism the N-electron wavefunction is approximated by a product of single-electron orbitals, often called Hartree orbitals, obeying one-particle equations. Starting from the non-relativistic many-electron Hamiltonian used in treatments of Atomic spectroscopy and Molecular orbital theory, the electron–electron Coulomb interaction is replaced by a mean potential constructed from the electron density. The resulting equations resemble the one-body Schrödinger equation with an additional Hartree potential term similar to potentials used in early calculations at institutions such as Imperial College London and facilities like the Cavendish Laboratory.
The self-consistent Hartree equations are derived by variational minimization of the total energy subject to orbital normalization constraints, invoking techniques familiar to researchers at places like Princeton University and Caltech. Each orbital satisfies a one-body differential or integral equation containing the external nuclear potential and a Coulomb mean-field term computed from the remaining orbitals' densities. The self-consistent field (SCF) procedure iterates orbitals and potentials until convergence, a concept also central to methods developed at Harvard University and in computational projects at labs such as Los Alamos National Laboratory.
The Hartree approximation is a precursor to the Hartree–Fock method formulated with contributions from Vladimir Fock, John C. Slater, and others at institutions like Bell Labs. Hartree–Fock improves on Hartree by enforcing antisymmetry via a Slater determinant, introducing exchange integrals absent in Hartree. Both Hartree and Hartree–Fock fall within the broader class of mean-field methods used across domains influenced by figures such as Lev Landau and John Bardeen; these methods include density functional approaches advanced by researchers associated with the Kohn–Sham equations and connections to techniques developed at research centers like the Max Planck Society.
The Hartree approximation has been used in early atomic structure calculations for elements examined at observatories like Royal Greenwich Observatory and in molecular studies relevant to work at the Royal Society. It served in modeling electron distributions in simple atoms and ions for spectroscopy experiments tied to laboratories such as the National Institute of Standards and Technology. The approach also informed modeling in solid-state contexts researched at institutions including IBM Research and supported pedagogical expositions at universities like Yale University and Columbia University.
Hartree neglects antisymmetry and exchange effects required by the Pauli principle emphasized by Wolfgang Pauli and later formalized in the work of Eugene Wigner and John von Neumann. This omission leads to quantitative errors in total energies, ionization potentials, and spectroscopic properties investigated in collaborations at centers like Lawrence Berkeley National Laboratory. Corrections include Hartree–Fock exchange, post-Hartree–Fock correlation methods developed by scientists at institutions such as University of Oxford, and density functional theory contributions originating from Pierre Hohenberg and Walter Kohn.
Practical Hartree calculations employ basis sets and discretization schemes used in computational packages originating from collaborations including Argonne National Laboratory and academic software projects at Massachusetts Institute of Technology. Techniques include finite-difference grids, spline bases, and Gaussian-type orbitals influenced by basis development by researchers at ETH Zurich and the University of Bristol. Convergence acceleration methods such as direct inversion in the iterative subspace (DIIS) were pioneered in computational chemistry groups at DuPont and later incorporated in codes used at national supercomputing centers like Oak Ridge National Laboratory.