Principles of Linear Combination

Principles of Linear Combination

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 ψA and ψB of atom A and B respectively with comparable energy that combines to form two molecular orbitals. One is bonding molecular orbitai (ψbonding) and the other is anti-bonding molecular orbital (ψanti-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, ψA and ψB can be obtained by the LCAO as shown below-
ψMO = ψA ± ψB.
ψBMO = ψA + ψ B.
ψABMO = ψA − ψB. 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 ψA and ψB be the wave functions of two atoms, then-
Principle of Linear Combination of Atomic Orbitals
Hence according to LCAO principle, we have-
ψMO = ψ(1)A ± ψ(1)B.
Thus, when two atomic orbitals of two different atoms undergo LCAO, we get two molecular orbitals ψBMO and ψABMO and electron moves in these molecular orbitals-
ψBMO = ψ(1)A + ψ(1)B.
ψABMO = ψ(1)A − ψ(1)B.

Rules or Conditions for Linear Combination of Atomic Orbitals

Molecular Orbital Wave Function for H2 and H2 ion


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