A nontrivial solution exists only if

det(H - ES) = 0,

i.e.,

    

 

This determinant is known as the SECULAR DETERMINANT.

 

Expanding the determinant, we get a quadratic equation for E:

 

(H11 - ES11)(H22 - ES22) - (H12 - ES12)2 = 0

 

·       This means that there will be two values of E which satisfy this equation. 

·       These two values  are the energies of the two molecular orbitals and  the index “i” has been used to label them. 

·       We denote the two molecular orbital energies as E1 and E2. 

·       In our example, one will be the bonding, the other the antibonding p-orbital formed by the combination of the two C 2p orbitals.

 

The original atomic orbital energies, e1 and e2, are:

        and                                          (3)

 

The MOs must be normalized...