An atomic property expressed as a sum of bond contributions

We shall be interested in observables that have non-vanishing commutators with the Hamiltonian H. Suppose the commutator of H and G is related to the observable A by the relation 

In this case A and G shall be referred to as associated operators. The averaging of this commutator and its complex conjugate over the atom f2, as indicated in eqn (6.2), and multiplication by JV/2 then yield the average value of A for an atom in a molecule, the quantity /4(H). In such a case, the commutator average equals the change in the value of the property A when a free atom H combines to form a molecule, since this average for the isolated atom vanishes (eqn (6.4)). If one denotes the change in the property A by A/4(H), then from eqn (6.2) one obtains 

Thus the change in the value of the property A that occurs as a result of an atom entering into chemical combination is given by the flux in the vector current density of its associated generator G through the atomic surface. 

An atomic surface S(H, r) is, in general, composed of a number of interatomic surfaces, there being one such surface for each atom O linked to H by an atomic interaction line or, in the case of a bound system, by a bond path. That is. 

In this manner the atomic average of the observable satisfying eqn (6.98) for the associated generator G is expressible as a sum of contributions where each contribution is determined by an integral over the surface which the atom shares with each of its bonded neighbours. This atomic average is necessarily the change in the value of the property relative to its value in the isolated atom. An example is provided by eqn (6.18), which shows that the 

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