Alinear combination of atomic orbitals orLCAO is aquantum superposition ofatomic orbitals and a technique for calculatingmolecular orbitals inquantum chemistry.[1] In quantum mechanics,electron configurations of atoms are described aswavefunctions. In a mathematical sense, these wave functions are thebasis set of functions, the basis functions, which describe the electrons of a given atom. Inchemical reactions, orbital wavefunctions are modified, i.e. theelectron cloud shape is changed, according to the type of atoms participating in thechemical bond.
It was introduced in 1929 by SirJohn Lennard-Jones with the description of bonding in the diatomic molecules of the first main row of theperiodic table, but had been used earlier byLinus Pauling for H2+.[2][3]
An initial assumption is that the number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion. In a sense,n atomic orbitals combine to formn molecular orbitals, which can be numberedi = 1 ton and which may not all be the same. The expression (linear expansion) for thei th molecular orbital would be:
or
where is a molecular orbital represented as the sum ofnatomic orbitals, each multiplied by a corresponding coefficient, andr (numbered 1 ton) represents which atomic orbital is combined in the term. The coefficients are the weights of the contributions of the n atomic orbitals to the molecular orbital. TheHartree–Fock method is used to obtain the coefficients of the expansion.The orbitals are thus expressed aslinear combinations ofbasis functions, and the basis functions are single-electron functions which may or may not be centered on thenuclei of the componentatoms of themolecule. In either case the basis functions are usually also referred to as atomic orbitals (even though only in the former case this name seems to be adequate). The atomic orbitals used are typically those ofhydrogen-like atoms since these are known analytically i.e.Slater-type orbitals but other choices are possible such as theGaussian functions fromstandard basis sets or the pseudo-atomic orbitals from plane-wave pseudopotentials.
By minimizing the totalenergy of the system, an appropriate set ofcoefficients of the linear combinations is determined. This quantitative approach is now known as the Hartree–Fock method. However, since the development ofcomputational 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 aslevel repulsion and 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 viaKoopmans' theorem.
This is done by using the symmetry of the molecules and orbitals involved in bonding, and thus is sometimes calledsymmetry adapted linear combination (SALC). The first step in this process is assigning apoint group to the molecule. 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.
Molecular orbital diagrams provide simple qualitative LCAO treatment. TheHückel method, theextended Hückel method and thePariser–Parr–Pople method, provide some quantitative theories.