Bond vs. Molecular Orbital Theory

In 1927, Heitler and London carried out a calculation for the hydrogen molecule using what has become known as valence bond theory.12 Each electron of the pair could be assigned to nuclei corresponding to wavefunctions of the type 

In the first, electron 1 is in the Is atomic orbital a and electron 2 is in b. In the second, the electrons are reversed. The two wave function when considered separately lead to the same energy. If we wish to form the hydrogen molecule, it is necessary to take linear combinations of the above, and the two new wavefunctions become (neglecting the normalization constant)  

In molecular orbital theory, molecular orbitals are formed by linear combinations of atomic orbitals, and the bonding molecular orbital is a + j b.14 The molecular wave function is the product of the wave functions for the two electrons or [ a(l) + b(l)][ J a(2) + ( b(2)]. When the energy is calculated using this wavefunction, it is found to be 80.0 kcal/mol with a length of 0.732A. In this case, the valence bond result is slightly more satisfactory than the molecular orbital result, but neither is really satisfactory. 

It can be seen that the first two terms are the same as the valence bond wavefunction, and there are an additional two terms. The first two are commonly called the covalent terms because each has the electrons associated with both centers. The final two are known as ionic terms because each places two electrons at one center (i.e. H+ H and H H ). Valence bond theory generally neglects ionic terms of this type, whereas in MO theory the covalent and ionic terms are treated equally. 

A better result can be obtained if the proportion of covalent and ionic terms can be adjusted. This can be done by mixing the doubly excited state with the ground state MO wavefunction. The new wavefunction is then 

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