LCAO method

The LCAO method ( 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 relation to hydrogen in the Schrödinger equation . In a chemical reaction the orbital wave functions are modified, i. H. the electron cloud is changed depending on the atoms participating in a chemical bond .

The LCAO method was published in 1929 by John Lennard-Jones with the description of the binding of diatomic molecules of the second period , but was previously used by Linus Pauling for H ₂ ⁺ .

principle

A basic assumption is that the number of molecular orbitals equals the number of atomic orbitals 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 which need not all be the same. The term of linear expansion for the i th molecular orbital is:

${\displaystyle \ \phi _{i}=c_{1i}\chi _{1}+c_{2i}\chi _{2}+c_{3i}\chi _{3}+\cdots +c_{ni}\chi _{n}}$

or

${\displaystyle \ \phi _{i}=\sum _{r}c_{ri}\chi _{r}}$

where ( phi ) is a molecular orbital , represented as the sum of n atomic orbitals ( chi ), each multiplied by a corresponding coefficient , and r (numbered from 1 to n) represents the atomic orbital combined in the term. The coefficients are the proportions of the contribution of the atomic orbitals to the molecular orbital. The Hartree-Fock method is used to find the coefficients of expansion. ${\displaystyle \ \phi _{i}}$ ${\displaystyle \ \chi _{r}}$ ${\displaystyle \ c_{ri}}$

The orbitals are expressed as linear combinations of the basis functions , which are one- electron functions that are centered on the atomic nuclei of the atoms involved in the molecule. The atomic orbitals used are usually of the hydrogen (e.g. Slater Type Orbitals ) as these are analytically known, but others can also be selected, e.g. B. the Gauss orbitals in the standard basis set.

By minimizing the total energy of the system, an appropriate set of coefficients of the linear combination is determined. This quantitative approach is now known as the Hartree-Fock method . Since the introduction of molecular modeling , however, the LCAO method has been used less for optimizing a wave function than for a qualitative assessment, which is helpful in predicting and explaining the results obtained with modern methods. In this case, the shape of the molecular orbitals and their respective energies are derived approximately from the comparison of the energies of the individual atomic orbitals (or molecular fragments) and the level repulsion or the like is applied. The generated for clarification graphs are referred to as correlation diagrams (engl. Correlation diagrams ), respectively. The required energies of the atomic orbitals come from calculations or can be determined experimentally using Koopmans theorem .

The first step is to assign a point group to the molecule. A common example is water , which has a C _2v symmetry. The reducible representation of the bond in water is listed below:

Every operation in the point group is performed on the molecule. The number of unaltered bindings is the nature of an operation. This reducible representation is broken down into the sum of the 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 .

If one looks at a system with several elements (e.g. atoms), centered on , one finds that the wave function of the electron describes when the element is isolated. The wave function that the electron describes in the entire system can be approximated by a linear combination of wave functions : ${\displaystyle i}$ ${\displaystyle {\vec {r}}_{i}}$ ${\displaystyle |\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }$ ${\displaystyle i}$ ${\displaystyle |\Psi ({\vec {r}})\rangle }$ ${\displaystyle |\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }$

${\displaystyle |\Psi ({\vec {r}})\rangle \simeq \sum _{i}\alpha _{i}|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }$

Derivation

The wave function describes an electron when the element is isolated. ${\displaystyle |\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }$ ${\displaystyle i}$

${\displaystyle E_{i}|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle =\left({-\hbar ^{2} \over {2m_{e}}}\nabla ^{2}+V_{i}({\vec {r}}-{\vec {r}}_{i})\right)|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }$

Under the hypothesis that the order of magnitude

${\displaystyle \langle \Psi _{i}({\vec {r}}-{\vec {r}}_{i})\mid V_{j}({\vec {r}}-{\vec {r}}_{j})\mid \Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle =\int _{\Omega }\Psi _{i}^{*}({\vec {r}}-{\vec {r}}_{i})V({\vec {r}}-{\vec {r}}_{j})\Psi _{i}({\vec {r}}-{\vec {r}}_{i})}$

is not significant except for , the potential modification by an element is not so important for the wave function . ${\displaystyle j=i}$ ${\displaystyle j\neq i}$ ${\displaystyle |\Psi _{i}(r-R_{i})\rangle }$

Any solution to the equation of the total system

${\displaystyle E|\Psi \rangle =\left(-{\hbar ^{2} \over {2m_{e}}}{\nabla ^{2}}+\sum _{i}V_{i}(r-R_{i})\right)|\Psi \rangle }$

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

${\displaystyle |\Psi \rangle =\sum _{i}\alpha _{i}|\Psi _{i}(r-R_{i})\rangle }$

See also

• Tight binding method
• Holstein-Herring method

Web links

• LCAO @ chemistry.umeche.maine.edu Link

Individual evidence

1. ↑ Huheey, James. Inorganic Chemistry: Principles of Structure and Reactivity .
2. ^ Werner Kutzelnigg: Friedrich Hund and Chemistry. In: Angewandte Chemie International Edition in English. 35, 1996, pp. 572-586, doi : 10.1002 / anie.199605721 .
3. ^ Robert S. Mulliken: Spectroscopy, molecular orbitals, and chemical bonding. Nobel Lecture .