Hartree-Fock
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In computational physics, the Hartree-Fock calculation scheme is a self-consistent iterative procedure to calculate the so-called "best possible" single determinant solution to the time-independent Schrödinger equation of a many-electron system in a Coulombic potential of fixed nuclei. As a consequence to this, whilst it calculates the exchange energy exactly, it does not calculate the effect of electron correlation at all. Because the nuclei are modeled as fixed point charges, it is only applicable after the Born-Oppenheimer approximation has been made. The name is for 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.
The starting point for the Hartree-Fock method is a set of approximate 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 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.
The many-body Hamiltonian is used to model the interaction between the electrons and the nucleus,
<math> \hat H=\sum_{i}{-\frac{1}{2}\nabla_i^2}-\sum_{i}{\sum_{a}{\frac{Z_a}{\left |\mathbf{r_i}-\mathbf{d_a}\right |}}}+\frac{1}{2}\sum_{i}{\sum_{j\ne i}{\frac{1}{\left |\mathbf{r_i}-\mathbf{r_j}\right |}}}+\frac{1}{2}\sum_{a}{\sum_{b\ne a}{\frac{Z_a Z_b}{\left |\mathbf{d_a}-\mathbf{d_b}\right |}}} <math>
where
- <math>\mathbf{r_i}<math> is the vector position of electron
- <math>i<math> with vector components in Bohr radii,
- <math>Z_a<math> is the charge of fixed nucleus a in units of the elementary charge,
- <math>\mathbf{d_a}<math> is the vector position of nucleus
- <math>a<math> with vector components in Bohr radii.
The first term in the Hamiltonian is the sum of the kinetic energy operators for each electron in the system. The second term is the sum of the electron-nucleus Coulombic attractions. The third term is the sum of the electron-electron Coulombic repulsions. The final term is the sum of the nucleus-nucleus Coulombic repulsions, also known as the nuclear repulsion energy. Because the Born-Oppenheimer approximation has been made, and the nuclear repulsion operator does not depend upon the electron positions, it can be calculated once at the beginning of the Hartree-Fock procedure and subsequently treated as a constant dependent only upon the nuclear positions.
Typically, in modern Hartree-Fock calculations, the wavefunction is approximated as a product of one-electron wavefunctions, which are in turn approximated by a Linear combination of atomic 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 functions, rather than Slater-type orbitals, in the interests of saving computation time. Approximating the wavefunction in this way can be achieved using the Roothaan equations, which approximate the true Hamiltonian of the system.
Once an initial wavefunction is constructed, an electron is selected. The effect of all the other electrons is summed up, and used to generate a potential. (This is why the procedure is sometimes called a mean-field procedure.) This gives a single electron in a defined potential, for which the Schrödinger equation can be solved, giving a slightly different wavefunction for that electron. This process is then repeated for each of the other electrons, which completes one step of the procedure. The whole procedure is then repeated, until the change from one step to the next is sufficiently small.
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.
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 based on a purely quantum mechanical solution. Indeed, it is common to use calculations that are a hybrid of the two methods - the popular B3LYP schema is one such. Hartree-Fock calculations can be used as the starting point for more sophisticated methods, such as many-body perturbation theory.