Hartree Fock theory developed for solving equations electronic Scrhodinger

Hartree Fock theory developed to solve equations electronic Scrhodinger. Simple method to solve many-particle system.

Many-electron wave function as a product of one-electron wave functions:

Wave function for the molecule N-shell electrons closed where the electrons occupy molecular orbitals 1 Ψa with spin α, 2 occupied molecular orbital electrons with spin β Ψa and so on. Thus, the wave function must satisfy pauli principle and change the sign under permutations of each pair of electrons. To achieve this, we write the wave function as the sum of all possible permutations with appropriate sign:

where N is the number of electrons, the entire amount can be written as a determinant:

An initial factor that ensures that the wave function dinormalisai, if the normalized molecular orbitals. The wave function can be written:

When the determinant wave function combined with the variation principle to obtain optimal wave function, in the sense in accordance with the total lowest energy, must meet the Hartree-Fock equation:

where σ is α or β. Fock operator f1 is:

three terms in this expression is a core Hamiltonian:

Coulomb operator J, which:

Exchange operator, K (exchange operators):

Although the Hartree-Fock equation looks simple, the Fock operator looks like Hamilton, we see from this definition are factually dependent on the wave functions of all the electrons. To continue, we have to guess the initial shape of the wave function, used in the definition of Coulomb and exchange operators, and solving the Hartree-Fock equations. The process was then followed by using the newly discovered wave function to each calculation cycle of energy and wave function left unchanged until the selected criteria. This is the origin of the term self-consistent field (SCF) for this type of procedure.

The difficulty in this procedure is the solution of the Hartree-Fock equation. To make progress, we must express the wave function as a linear combination of atomic orbitals M Xi, and write:

The use of such a linear combination led to a set of equations that can be expressed in the form of a matrix known as Roothaan equation:

FC = SCε

where F is a matrix formed from Fock operator:

and S is the overlaping for integral matrix:

Hartree-Fock method can be divided into two: Restricted Hartree-Fock (RHF) and Unrestricted Hartree-Fock (UHF). Restricted Hartree-Fock (RHF) is a method that is applied to the notion that electrons with opposite spins form pairs and fill the same spatial orbitals. the advantages of the method RHF is bersesuain with spin magnetic moment of electrons dispensed in exact for unpaired electrons in the same orbital. RHF method is almost always used for closed cell system (Closed Shell). Unrestricted Hartree-Fock (UHF) is a molecular orbital method most common to the open cell system (open Shell), where the number of electrons is not the same for each round. While theory, RHF using molecular orbital single twice, once multiplied by a function α and β multiplied by the function. UHF using different molecular orbitals for electrons α and β, is called the orbital method is different for different spin / different orbitals for different spins (Dods).

Base Set

Base set is a collection of mathematical functions used to compile the orbit of an orbital cluster. Existing set of mathematical functions are arranged in a linear combination with meyertakan coefficient value in it. Commonly used functions are clusters of atoms making up the molecule orbit. Chemical calculations are generally carried out in a set calculation basis consisting of the wave functions are arranged in a linear fashion. The process of collection of atomic orbital calculations compiled following the rules of orbital Slater Slater called. Broadly speaking, Slater orbital arch-shaped exponential down that generally approximated by a linear combination of Gaussian functions.

The set of base are classified according to the number of primitive functions and how the formation of a gaussian function, Extended. The set contains a minimal basis basis functions to represent each (skin and skin valence) atomic orbitals. Further development of the base set is done by replacing each function basis of a minimum set of base with several different functions base on orbital exponent, ζ, which became known as the set of an expanded base, which is included in the set of base expanded, among others, double zeta ( DZ) which contains twice the base functionality of the minimal base and triple zeta (TZ) which contains three times the base functionality of the minimal base. The set of split valence basis (SV -Split Valence-) is a combination of a minimum base set and the set of base expanded. There is some basis functions for each atomic orbitals in the valence shell electrons but only one base function to describe atomic orbitals on the skin.