Linear combinations of atomic orbitals

A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of 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 cloud shape is changed, according to the type of atoms participating in the chemical bond.

It was introduced in 1929 by Sir John Lennard-Jones with the description of bonding in the diatomic molecules of the first main row of the periodic table, but had been used earlier by Linus Pauling for H₂⁺.

A mathematical description follows.

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 form n molecular orbitals, which can be numbered i = 1 to n and which may not all be the same. The expression (linear expansion) for the i th molecular orbital would be:

or

where ${\displaystyle \ \phi _{i}}$ (phi) is a molecular orbital represented as the sum of n atomic orbitals ${\displaystyle \ \chi _{r}}$ (chi), each multiplied by a corresponding coefficient ${\displaystyle \ c_{ri}}$, and r (numbered 1 to n) 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. The Hartree–Fock procedure is used to obtain the coefficients of the expansion.

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