Hartree-Fock

In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative variational procedure to calculate the Slater determinant (or the molecular orbitals which it is made of) for which the expectation value of the electronic molecular Hamiltonian is minimum. Whilst it calculates the exchange energy exactly, it does not include the effect of electron correlation. The procedure is named after Douglas Hartree, who devised the self-consistent field method, and V. A. Fock, who demonstrated the rigour of Hartree's method and reformulated it into the matrix form used today. Expressed in a Slater-type or Gaussian-type basis set, the Hartree-Fock equation can be transformed into matrix form called Roothaan equations.

Hartree-Fock algorithm

The Hartree-Fock method is typically used to solve the time-independent Schrödinger equation for a multi-electron atom or molecule described in the fixed-nuclei approximation by the electronic molecular Hamiltonian. Because of the complexity of the differential equations for any but the smallest systems, (see Hydrogen atom), the problem is usually impossible to solve analytically, and so the numerical technique of iteration is used. The method makes four major simplifications in order to deal with this task:

Related Topics:
Schrödinger equation - Electronic molecular Hamiltonian - Hydrogen atom - Iteration

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• The Born-Oppenheimer approximation is inherently assumed. The true wavefunction is actually a function of the coordinates of each of the nuclei, in addition to those of the electrons.
• Typically, relativistic effects are completely neglected. The momentum operator is assumed to be completely non-relativistic.
• The basis set is composed of a finite number of orthogonal functions. The true wavefunction is a linear combination of functions from a complete basis set.
• The energy eigenfunctions are assumed to be anti-symmetrized linear combinations of products of one-electron wavefunctions. The effects of electron correlation, beyond that of exchange energy resulting from the anti-symmetrization of the wavefunction, are completely neglected.

The variational theorem states that, for a time-independent Hamiltonian operator, any trial wavefunction will have an energy expectation value that is greater than or equal to the true ground state wavefunction 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. The limit of the Hartree-Fock energy as the basis set becomes infinite is called the Hartree-Fock limit. It is a unique set of one-electron orbitals, and their eigenvalues.

Related Topics:
Variational theorem - Expectation value - Ground state

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The starting point for the Hartree-Fock method is a set of approximate one-electron orbitals. For an atomic calculation, these are typically the orbitals for a hydrogenic atom (an atom with only one electron, but the appropriate nuclear charge). For a molecular or crystalline calculation, the initial approximate one-electron wavefunctions are typically a linear combination of atomic orbitals. This gives a collection of one electron orbitals that, due to the fermionic nature of electrons, must be anti-symmetric. This antisymmetry is achieved through the use of a Slater determinant.

Related Topics:
Atomic - Molecular - Fermionic - Slater determinant

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

At this point, a new approximate Hamiltonian operator, called the Fock operator, is constructed. The first terms in this Hamiltonian are a sum of kinetic energy operators for each electron, the internuclear repulsion energy, and a sum of nuclear-electronic coulombic attraction terms. The final set of terms models the electronic coulombic repulsion terms between each electron with a sum. The sum is composed of 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 fourth simplification in the above list, (see post-Hartree-Fock).

Related Topics:
Fock operator - Post-Hartree-Fock

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The newly constructed Fock operator is then used as the Hamiltonian in the time-independent Schroedinger Equation. Solving the equation yields a new set of approximate one-electron orbitals. This new set of orbitals is then used to construct a new Fock operator, as in the preceding paragraph, beginning the cycle again. The procedure is stopped when the change in total electronic energy is negligible between two iterations. In this way, a set of so-called "self-consistent" one-electron orbitals are calculated. The Hartree-Fock electronic wavefunction is then equal to the Slater determinant of these approximate one-electron wavefunctions. From the Hartree-Fock wavefunction, any chemical property of the system in question can be calculated in an approximate manner.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~