Principles of Linear Combination

Principles of Linear Combination

The wave functions for the molecular orbitals can be obtained by solving Schrodinger wave equation for the molecule. Solving the Schrodinger wave equation is too complex. So, linear combination of atomic orbitals is used to obtain the wave function 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 atom A and B respectively with comparable energy that combines to form two molecular orbitals. One is bonding molecular orbitai (ψ_bonding) and the other is anti-bonding molecular orbital (ψ_anti-bonding). Formation of bonding molecular orbital is result of addition (i.e. constructive interference) of two atomic orbitals however, Formation of antibonding molecular orbital is result of subtraction (i.e. destructive interference) of two atomic orbitals. The wave function for molecular orbitals, ψ_A and ψ_B can be obtained by the LCAO as shown below-
ψ_MO = ψ_A ± ψ_B.
ψ_BMO = ψ_A + ψ _B.
ψ_ABMO = ψ_A − ψ_B. Explanation: In a bimolecular system, an electron near to one nucleus belongs to the wave function of that nucleus at a particular moment. But when the electron is between two nuclei, then it belongs to its combined wave function. This is called LCAO principle.
If ψ_A and ψ_B be the wave functions of two atoms, then-

Hence according to LCAO principle, we have-
ψ_MO = ψ⁽¹⁾_A ± ψ⁽¹⁾_B.
Thus, when two atomic orbitals of two different atoms undergo LCAO, we get two molecular orbitals ψ_BMO and ψ_ABMO and electron moves in these molecular orbitals-
ψ_BMO = ψ⁽¹⁾_A + ψ⁽¹⁾_B.
ψ_ABMO = ψ⁽¹⁾_A − ψ⁽¹⁾_B.

---

Rules or Conditions for Linear Combination of Atomic Orbitals

Molecular Orbital Wave Function for H₂ and H₂ ion

 Share