It is possible that the population of A may include some of the population of atom B on its right. This is seen in the intersection space between them. The projector P AB is made from the defining functions of both atoms A and B . This finding will not be unexpected by chemists, even if they know nothing of projection mathematics. It has long been accepted that atoms may share electrons, and in so doing form bonds .
Bonds are about as poorly defined quantum mechanically as atoms.
Since sharing can now be defined between atoms, bonding was the next logical thing. Following the usual practice of quantum mechanics, Gould evaluates the eigenvalues and eigenfunctions of these operators, and shows that they occur in pairs. Even more interestingly, he shows that these can be compared with the bonding and anti-bonding orbital pairs found in more conventional chemical electronic theory. Each eigenfunction in used as a defining space for its own projection operator, which is used to determine the population of the eigenfunction. Herzberg's rule for the bond index is used with the eigenfunctions of each operator to get bond indices for covalency and ioncity.