HartreeFock

Hartree-Fock is an approximate method in quantum chemistry for determining the wavefunction and energy of a many-electron system in a stationary state. It represents the N-electron wavefunction as a single Slater determinant of spin-orbitals, enforcing antisymmetry so that electrons move in an average field produced by the others.

Historically, the method stems from Hartree’s self-consistent field approach and was extended to include antisymmetry by Fock. The practical formulation became widely used after Roothaan introduced a matrix-based way to solve the equations for molecular systems in the early 1950s. The result is a self-consistent field procedure that yields the best single-determinant description within a chosen basis set.

In the Hartree-Fock procedure, the Fock operator is built from one-electron integrals and two-electron Coulomb and exchange terms. Solving the resulting Roothaan equations iteratively produces molecular orbitals that minimize the energy of the single determinant. Variants include restricted HF for closed-shell systems, unrestricted HF for open-shell systems, and restricted open-shell HF for certain cases. The method is typically implemented with a basis set of atomic orbitals, and the computed energy is often used as a reference point for more advanced methods.

Limitations include neglect of dynamic electron correlation beyond exchange, which can lead to inaccurate absolute energies and properties. HF results depend on the basis set and can suffer from spin contamination in unrestricted formulations. Despite these limits, Hartree-Fock remains a foundational starting point for many ab initio techniques, such as configuration interaction and coupled-cluster theory, and it is widely used to obtain molecular geometries and qualitative trends.