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.

Mathematical formulation

The Fock operator

Because the electron-electron repulsion term of the electronic molecular Hamiltonian involves the coordinates of two different electrons, it is necessary to reformulate it in an approximate way. Under this approximation, (outlined under Hartree-Fock algorithm), all of the terms of the exact Hamiltonian except the nuclear-nuclear repulsion term are re-expressed as the sum of one-electron Fock operators. Each one-electron Fock operator is in turn composed of a sum of terms:

Related Topics:
Electronic molecular Hamiltonian - Hartree-Fock algorithm - Fock operators

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:hat F(1) = hat H^{core}(1)+sum_{j=1}^{n/2}

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where:

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:hat F(n)

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is the Fock operator for the n-th electron in the system,

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:hat H^{core}(n)

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is the core Hamiltonian for the n-th electron,

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:hat J_j(n)

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is the Coulomb operator, defining the repulsive force between the j-th and n-th electrons in the system,

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:hat K_j(n)

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is the exchange operator, defining the effect of exchanging two electrons.

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The total Fock operator for the system is then calculated as the sum of all of the individual one-electron Fock operators.

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Linear combination of atomic orbitals

:Main article: basis set

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Typically, in modern Hartree-Fock calculations, the one-electron wavefunctions are approximated by a Linear combination of atomic orbitals. These atomic orbitals are called Slater-type orbitals. Furthermore, it is very common for the "atomic orbitals" in use to actually be composed of a linear combination of one or more Gaussian-type orbitals, rather than Slater-type orbitals, in the interests of saving large amounts of computation time.

Related Topics:
Linear combination of atomic orbitals - Slater-type orbitals - Gaussian-type orbitals - Slater-type orbitals

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Various basis sets are used in practice, most of which are composed of Gaussian functions. Typically, an orthogonalization method such as the Gram-Schmidt process is performed in order to produce a set of orthogonal basis functions. This can save considerable computational time when the computer is solving the Roothaan equations by converting the overlap matrix effectively to a unit matrix, thereby removing it from the calculations.

Related Topics:
Basis sets - Gram-Schmidt process - Roothaan equations - Overlap matrix - Unit matrix

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