Linear Combination of Atomic Orbitals (LCAO)

Molecular orbital wavefunctions can, in principle, be obtained by solving the Schrödinger wave equation for the whole molecule -- but this is far too mathematically complex to do exactly for any but the very simplest systems, so approximation methods are used instead. The most widely-used such method is the linear combination of atomic orbitals (LCAO).

Atomic orbitals are represented by a wave function, ψ. Consider two atomic orbitals, represented by wave functions ψA and ψB, of comparable energy, combining to form two molecular orbitals -- one bonding (ψ_bonding) and one antibonding (ψ_antibonding). Under the LCAO method, the wave functions of these two resulting molecular orbitals are obtained simply by the linear (additive/subtractive) combination of the two atomic-orbital wave functions:

The formation of the bonding molecular orbital can be pictured as the result of CONSTRUCTIVE interference between the two atomic orbitals (the two waves are 'in phase', carrying the same sign, so they reinforce and ADD); the formation of the antibonding molecular orbital can be pictured as the result of DESTRUCTIVE interference (the two waves are 'out of phase', carrying opposite signs, so they partially cancel). …