Information AboutHartree-fock Theory |
| CATEGORIES ABOUT HARTREE-FOCK | |
| computational physics | |
| atomic physics | |
| computational chemistry | |
| quantum chemistry | |
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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:- 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 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. 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 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, : 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: where are a set of one-electron wavefunctions, called the ''Hartree-Fock Molecular Orbitals.'' Linear combination of atomic orbitals Main article: 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. 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. 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. SOFTWARE PACKAGES For a list of software packages known to handle Hartree-Fock calculations, see the Software Packages section of Computational Chemistry . SEE ALSO Related fields Concepts People EXTERNAL LINKS |
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