Editing Hartree–Fock method (section)

=== Variational optimization of orbitals ===
[[File:Hartree-Fock.png|thumb|right|325px|Algorithmic flowchart illustrating the Hartree–Fock method]]
The [[Variational method (quantum mechanics)|variational theorem]] states that for a time-independent Hamiltonian operator, any trial wave function will have an energy [[expectation value]] that is greater than or equal to the true [[ground-state]] wave function corresponding to the given Hamiltonian.  Because of this, the Hartree–Fock energy is an upper bound to the true ground-state energy of a given molecule. In the context of the Hartree–Fock method, the best possible solution is at the ''Hartree–Fock limit''; i.e., the limit of the Hartree–Fock energy as the basis set approaches [[Orthonormal basis#Incomplete orthogonal sets|completeness]]. (The other is the ''[[Configuration interaction|full-CI limit]]'', where the last two approximations of the Hartree–Fock theory as described above are completely undone. It is only when both limits are attained that the exact solution, up to the Born–Oppenheimer approximation, is obtained.) The Hartree–Fock energy is the minimal energy for a single Slater determinant.

The starting point for the Hartree–Fock method is a set of approximate one-electron wave functions known as ''[[spin-orbital]]s''. For an [[atomic orbital]] calculation, these are typically the orbitals for a [[hydrogen-like atom]] (an atom with only one electron, but the appropriate nuclear charge).  For a [[molecular orbital]] or crystalline calculation, the initial approximate one-electron wave functions are typically a [[linear combination of atomic orbitals]] (LCAO).

The orbitals above only account for the presence of other electrons in an average manner. In the Hartree–Fock method, the effect of other electrons are accounted for in a [[mean-field theory]] context. The orbitals are optimized by requiring them to minimize the energy of the respective Slater determinant. The resultant variational conditions on the orbitals lead to a new one-electron operator, the [[Fock operator]]. At the minimum, the occupied orbitals are eigensolutions to the Fock operator via a [[unitary transformation]] between themselves. The Fock operator is an effective one-electron Hamiltonian operator being the sum of two terms. The first is a sum of kinetic-energy operators for each electron, the internuclear repulsion energy, and a sum of nuclear–electronic [[Coulomb's law|Coulombic]] attraction terms. The second are Coulombic repulsion terms between electrons in a mean-field theory description; a net repulsion energy for each electron in the system, which is calculated by treating all of the other electrons within the molecule as a smooth distribution of negative charge.  This is the major simplification inherent in the Hartree–Fock method and is equivalent to the fifth simplification in the above list.

Since the Fock operator depends on the orbitals used to construct the corresponding [[Fock matrix]], the eigenfunctions of the Fock operator are in turn new orbitals, which can be used to construct a new Fock operator. In this way, the Hartree–Fock orbitals are optimized iteratively until the change in total electronic energy falls below a predefined threshold. In this way, a set of self-consistent one-electron orbitals is calculated.  The Hartree–Fock electronic wave function is then the Slater determinant constructed from these orbitals. Following the basic postulates of quantum mechanics, the Hartree–Fock wave function can then be used to compute any desired chemical or physical property within the framework of the Hartree–Fock method and the approximations employed.