Hartree-Fock

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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 improved the rigour of Hartree's method to make it consistent with the Pauli principle 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.

The discussion here is only for the Restricted Hartree-Fock method, where the atom or molecule is a closed-shell system with all orbitals (atomic or molecular) are doubly occupied. Open shell systems, where some of the electrons are not paired, can be dealt with by one of two Hartree-Fock methods:-

Restricted Open-shell Hartree-Fock (ROHF)
Unrestricted Hartree-Fock (UHF)

1 Hartree-Fock algorithm
2 Mathematical formulation
- 2.1 The Fock operator
- 2.2 Linear combination of atomic orbitals
3 Numerical stability
4 Weaknesses, extensions, and alternatives
5 Software packages
6 See also
7 External links

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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:

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.

Image:Hartree-Fock.png 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.

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.

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).

The newly constructed Fock operator is then used as the Hamiltonian in the time-independent Schrödinger 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.

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Mathematical formulation

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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 operators outlined below. The "(1)" following each operator symbol simply indicates that the operator is 1-electron in nature.

where:

is the one-electron Fock operator,

<math>\hat H^{core}(1)=-\frac{1}{2}\nabla^2_1 - \sum_{\alpha} \frac{Z_\alpha}{r_{1\alpha}}</math>

is the one-electron core Hamiltonian,

is the Coulomb operator, defining the electron-electron repulsion energy due to the j-th electron,

is the exchange operator, defining the electron exchange energy. Finding the Hartree-Fock one-electron wavefunctions is now equivalent to solving the eigenfunction equation:

<math>\hat F(1)\phi_i(1)=\epsilon_i \phi_i(1)</math>

where <math>\phi_i(1)</math> are a set of one-electron wavefunctions, called the Hartree-Fock Molecular Orbitals.

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Linear combination of atomic orbitals

Main article: basis set

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.

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.

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Numerical stability

Numerical stability can be a problem with this procedure- there are various ways of combating this instability. One of the most basic and generally applicable is called F-mixing. With F-mixing, once a single electron wavefunction is calculated it is not used directly. Instead, some combination of that calculated wavefunction and the previous wavefunctions for that electron is used - the most common being a simple linear combination of the calculated and immediately preceding wavefunction. A clever dodge, employed by Hartree, for atomic calculations was to increase the nuclear charge, thus pulling all the electrons closer together. As the system stabilised, this was gradually reduced to the correct charge.

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Weaknesses, extensions, and alternatives

Of the four simplifications outlined under Hartree-Fock algorithm, the fourth is typically the most important. Neglecting electron correlation can lead to large deviations from experimental results. A number of approaches to this weakness, collectively called post-Hartree-Fock methods, have been devised to include electron correlation to the multi-electron wave function. One of these approaches, Moller-Plesset perturbation theory, treats correlation as a perturbation of the Fock operator. Others expand the true multi-electron wavefunction in terms of a linear combination of Slater determinants - such as Multi-configurational self-consistent field, Configuration interaction, Quadratic configuration interaction, Complete active space SCF.

An alternative to Hartree-Fock calculations used in some cases is density functional theory, which gives approximate solutions to both exchange and correlation energies, but is not a purely ab initio method in practice. Indeed, it is common to use calculations that are a hybrid of the two methods - the popular B3LYP schema is one such method.

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