Pauling actually begins his discussion with the wavefunction for the molecule AA:
ΨAA = aΨAA + bΨA+A¯ + bΨA¯A+
The contribution of bΨA+A¯ and bΨA¯A+ to the overall bond energy would be small. Without saying so, Pauling goes on to ignore this portion of the wavefunction.
This leaves: ΨAA = aΨAA
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 ΨAA and for ΨBB, you could average those two values to get one for aΨAB.
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 ΨAB. This is an experimentally determined value.
Let's do that again because it's important. Here the wavefunction for AB:
ΨAB = aΨAB + bΨA+B¯ + dΨA¯B+
The "normal covalent bond" is the part I've put in boldface. It is obtained by averaging two experimental values (ΨAA and ΨBB). The actual value for the bond energy is ΨAB, which is only obtained by experiment.
Pauling will discover a difference in the values for ΨAB and ΨAB. From that difference, he will draw the meaning of electronegativity.