Pauling actually begins his discussion with the wavefunction for the molecule A­A:

Ψ_AA = aΨ_A­A + bΨ_A⁺A¯ + bΨ_A¯A⁺

The contribution of bΨ_A⁺A¯ and bΨ_A¯A⁺ to the overall bond energy would be small. Without saying so, Pauling goes on to ignore this portion of the wavefunction.

This leaves: Ψ_AA = aΨ_A­A

A similiar wavefunction can be written for the molecule B­B.

Here is what he then says:

Now let us consider a molecule A­B, involving a single bond between two unlike atoms. If the atoms were closely similar in character, the bond in this molecule could be represented by a wave function such as 3-1 [the first one above], an average of those for the symmetric molecules A­A and B­B. Let us describe such a bond as a normal covalent bond. (p. 80, italics his)

The key point is that the wave function for A­B can be considered as an average of those for A­A and B­B.

So, if you knew the value for Ψ_AA and for Ψ_BB, you could average those two values to get one for aΨ_A­B.

Now, be real careful. That last value is on the right-hand side (the first value) of the wavefunction at the top of the file you came from. (I've put it just below also.) IT IS NOT Ψ_AB. This is an experimentally determined value.

Let's do that again because it's important. Here the wavefunction for AB:

Ψ_AB = aΨ_A­B + bΨ_A⁺B¯ + dΨ_A¯B⁺

The "normal covalent bond" is the part I've put in boldface. It is obtained by averaging two experimental values (Ψ_AA and Ψ_BB). The actual value for the bond energy is Ψ_AB, which is only obtained by experiment.

Pauling will discover a difference in the values for Ψ_AB and Ψ_A­B. From that difference, he will draw the meaning of electronegativity.