- Linear combination of atomic orbitals molecular orbital method
A linear combination of atomic orbitals or LCAO is a
quantum superpositionof atomic orbitalsand a technique for calculating molecular orbitals in quantum chemistry[Huheey, James. "Inorganic Chemistry:Principles of Structure and Reactivity"] . In quantum mechanics, electron configurationsof atoms are described as wavefunctions. In mathematical sense, these wave functions are the basis set of functions, the basis functions, which describe the electrons of a given atom. In chemical reactions, orbital wavefunctions are modified, i.e. the electron cloudshape is changed, according to the type of atoms participating in the chemical bond.
It was introduced in 1929 by
Sir John Lennard-Joneswith the description of bonding in the diatomic molecules of the first main row of the periodic table, but had been used earlier by Linus Paulingfor H2+. [ Friedrich Hundand Chemistry, Werner Kutzelnigg, on the occasion of Hund's 100th birthday, Angewandte Chemie, 35, 573 - 586, (1996) ] [ Robert S. Mulliken's Nobel Lecture, Science, 157, no. 3785, 13 - 24, (1967) ]
A mathematical description is
where (phi) is a
molecular orbitalrepresented as the sumof n atomic orbitals (chi), each multiplied by a corresponding coefficient . The coefficients are the weights of the contributions of the n atomic orbitals to the molecular orbital. The Hartree-Fockprocedure is used to obtain the coefficients of the expansion from the Hartree-Fock procedure.
The orbitals are thus expressed as
linear combinations of basis functions, and the basis functions are one- electronfunctions centered on nuclei of the component atoms of the molecule. The atomic orbitals used are typically those of hydrogen-like atoms since these are known analytically i.e. Slater-type orbitals but other choices are possible like Gaussian functions from standard basis sets.
By minimizing the total
energyof the system, an appropriate set of coefficients of the linear combinations is determined. This quantitative approach is now known as the Hartree-Fockmethod. However, since the development of computational chemistry, the LCAO method often refers not to an actual optimization of the wave function but to a qualitative discussion which is very useful for predicting and rationalizing results obtained via more modern methods. In this case, the shape of the molecular orbitals and their respective energies are deduced approximately from comparing the energies of the atomic orbitals of the individual atoms (or molecular fragments) and applying some recipes known as level repulsionand the like. The graphs that are plotted to make this discussion clearer are called correlation diagrams. The required atomic orbital energies can come from calculations or directly from experiment via Koopmans' theorem.
This is done by using the symmetry of the molecules and orbitals involved in bonding. The first step in this process is assigning a
point groupto the molecule. A common example is water, which is of C2v symmetry. Then a reducible representationof the bonding is determined demonstrated below for water:
Each operation in the point group is performed upon the molecule. The number of bonds that are unmoved is the character of that operation. This reducible representation is decomposed into the sum of irreducible representations. These irreducible representations correspond to the symmetry of the orbitals involved.
MO diagrams provide simple qualitative LCAO treatment.
Quantitative theories are the
Huckel method, the extended Huckel methodand the Pariser–Parr–Pople method.
Quantum chemistry computer programs
* LCAO @ chemistry.umeche.maine.edu [http://chemistry.umeche.maine.edu/Modeling/lcao.html Link]
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