Principles of Linear Combination of Atomic Orbitals (LCAO)

Principles of Linear Combination of Atomic Orbitals (LCAO)

The wave functions for molecular orbitals can be obtained by solving the Schrödinger wave equation for the molecule. However, solving the Schrödinger wave equation directly for complex molecules is mathematically difficult. Therefore, the Linear Combination of Atomic Orbitals (LCAO) approximation method is used to obtain the wave functions for molecular orbitals.

Atomic orbitals are represented by wave functions ψ. Let us consider two atomic orbitals represented by the wave functions ψ_A and ψ_B of atoms A and B respectively, possessing comparable energy. When they combine, they form two molecular orbitals: a bonding molecular orbital (ψ_bonding) and an anti-bonding molecular orbital (ψ_anti-bonding).

The formation of a bonding molecular orbital is the result of addition (i.e., constructive interference) of the two atomic orbitals. Conversely, the formation of an anti-bonding molecular orbital is the result of subtraction (i.e., destructive interference) of the two atomic orbitals.

ψ_MO = ψ_A ± ψ_B
ψ_BMO = ψ_A + ψ_B
ψ_ABMO = ψ_A − ψ_B

Explanation: In a homonuclear or heteronuclear bimolecular system, an electron near one nucleus belongs predominantly to the wave function of that specific nucleus at any particular moment. But when the electron is distributed between two nuclei, it is governed by their combined wave function. This configuration is the core basis of the LCAO principle.

If ψ_A and ψ_B represent the baseline atomic wave functions of the interacting atoms, the structural outcome behaves as follows:

Hence, according to the LCAO principle, we mathematically define the state transformations as:

ψ_MO = ψ⁽¹⁾_A ± ψ⁽¹⁾_B

ψ_BMO = ψ⁽¹⁾_A + ψ⁽¹⁾_B
ψ_ABMO = ψ⁽¹⁾_A − ψ⁽¹⁾_B

Related Topics

Rules or Conditions for Linear Combination of Atomic Orbitals Molecular Orbital Wave Function for H₂ and H₂ ion