# 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 H_{2} and H_{2} ion

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