Linear combination of atomic orbitals

The LCAO method (of English. Linear combination of atomic orbitals linear combination of atomic orbitals ') is a quantum superposition of atomic orbitals, and a method for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as a wave function in terms of hydrogen in the Schrödinger equation. In a chemical reaction, the orbital wave-functions are modified, i.e., the cloud of electrons is changed depending on the participating atoms, a chemical bond.

The LCAO method was published in 1929 by John Lennard -Jones with the description of binding diatomic molecules of the second period, but was previously used by Linus Pauling for H2 .

Principle

A basic assumption is that the number of molecular orbitals of the number of atomic orbitals is similar in the linear expansion. The n atomic orbitals are combined into n molecular orbitals, which are numbered with an index from 1 to n and the need not all be the same. The term of linear expansion for the i-th molecular orbital is:

Or

Wherein ( phi) is a molecular orbital, represented as a sum of n atomic orbitals (CHI), each multiplied by a corresponding coefficient, and r ( numbered from 1 to n) is the combined atomic orbital in the term. The coefficients are the proportions of the contribution of the atomic orbitals on the molecular orbital. Hartree -Fock method is used to determine the coefficient of expansion.

The orbitals are expressed as linear combinations of the basis functions which are centered as one-electron functions on the nuclei of the atoms involved in the molecule. The atomic orbitals used are usually from hydrogen ( eg Slater type orbital ), as these are known analytically, but others may also be selected, such as the Gaussian orbitals in the standard basis set.

By minimizing the total energy of the system, a matching set is determined on coefficients of the linear combination. This quantitative approach is now known as the Hartree- Fock method. Since the introduction of molecular modeling, the LCAO method is used, however, less to optimize a wave function rather than to a qualitative rating, which is helpful in predicting and in explaining the results obtained by modern methods. In this case, the shape of the molecular orbitals and their respective energies approximated by comparing the energies of the individual atomic orbitals (or molecular fragments ) are derived and the level repulsion or applied like. The graphs generated for clarification are called correlation diagrams ( engl. correlation diagrams ). The energies of the atomic orbitals needed come from calculations or can be determined via the Koopmans theorem experimentally.

The first step consists of assigning an item group to the molecule. A common example is water, which has a C2v symmetry. The following are the reducible representation of the binding is listed in the water:

Each operation in the point group is carried out with respect to the molecule. The number of bonds unchanged is the character of an operation. This reducible representation is decomposed into the sum of irreducible representations. The irreducible representations correspond to the symmetry of the orbitals involved.

MO diagrams provide a simple qualitative treatment of the LCAO approximation.

Quantitative theories are the Hückel approximation, the extended Hückel method and the Pariser-Parr - Pople method.

Considering a system with a plurality of elements (e.g., atoms), centered on, it can be seen that the wave function of the electron describes when the element is isolated. The wave function that describes the electron in the entire system can be approximated by a linear combination of wave functions:

Derivation

Describes the wave function of an electron, if the element is isolated.

Under the hypothesis that the order

Is not significative except for the potential modification by an element of the wave function is not so important.

Each solution of the equation of the total system

Can be approximated by a linear combination of the isolated wave functions: