A variational approximation restricting a many-electron wavefunction to one Slater determinant. Minimizing its energy gives self-consistent one-electron equations with Coulomb and exchange operators. It includes exchange from antisymmetry but omits correlation beyond the determinant.
The self-consistent one-electron operator obtained by varying a Slater determinant. It contains a local hartree potential and a nonlocal exchange action. Its occupied eigenfunctions define a stationary determinant; its eigenvalues are Lagrange multipliers and are not by themselves the total many-electron energy.
A variational single-Slater-determinant treatment of interacting electrons and a neutralizing positive background. For uniform plane-wave occupations the direct background and electron terms cancel, leaving kinetic and same-spin exchange energy. Allowing spin polarization or spatially modulated orbitals enlarges the variational class.
In a uniform three-dimensional Hartree-Fock electron gas with repulsive contact coupling , a spin-independent neutralizing background can cancel the Hartree energy. The remaining same-spin exchange term is negative and quadratic in each spin density. Integrating the filled single-spin Fermi sphere gives . At fixed total density the exchange term favors spin polarization, whereas the kinetic term opposes it.
Let and . In units the spin-dependent energy is . Comparing only and gives the endpoint crossing . Expanding near gives a quadratic coefficient , so the unpolarized state first loses local stability at . The endpoint crossing is therefore not the actual continuous onset; intermediate polarizations matter.
The uniform paramagnetic three-dimensional electron-gas determinant is unstable within Hartree-Fock theory to a spin-density-wave deformation, even at high density. Comparing only spatially uniform polarized and unpolarized determinants therefore does not find the unrestricted Hartree-Fock ground state.[ref]
In a three-dimensional unpolarized gas, exchange energy per particle is , with Coulomb convention . It scales as minus the inverse dimensionless spacing radius, while kinetic energy scales as its inverse square. Exchange acts only between equal-spin occupied orbitals.
For number densities , . Uniform spin polarization raises kinetic energy by concentrating occupation in one Fermi sphere, while lowering exchange energy.
For real spatial orbitals, in atomic units. It enters the energy of a Slater determinant through antisymmetry and becomes small for well-separated localized orbitals.
The electrostatic repulsion integral in atomic units. For a normalized hydrogenic 1s orbital of exponent , the same-orbital value is . It is a two-electron integral, unlike a diagonal Hückel one-electron parameter.
In frozen-orbital Hartree-Fock theory, removing an electron from occupied orbital changes the energy by approximately . Orbital relaxation and correlation are omitted, so a measured ionization energy is not an exact orbital-energy measurement.
A Hartree-Fock ansatz allowing different spatial orbitals for alpha and beta spin. This can lower the energy of a stretched bond, but the determinant need not be an eigenstate of total spin squared.
Articles by others on the same topic
The Hartree-Fock (HF) method is a fundamental approach in quantum chemistry and computational physics used to approximate the electronic structure of many-electron atoms and molecules. It simplifies the complex problem of interacting electrons in a field created by themselves and their nuclei by making several key approximations. ### Key Features of the Hartree-Fock Method: 1. **Mean-Field Theory**: HF is based on the assumption that each electron moves in an average field created by the other electrons and the nuclei.
