ABSTRACT
The inequality (2.7) indicates that the use of atomic projection operators is
not very well suited to identify an atom in the case when interatomic overlap
is present: one cannot present an arbitrary function as the sum of its projec-
tion into the different atomic subspaces. This is a peculiarity of the basis sets
used in quantum chemistry-mathematicians usually use only orthonormal-
ized basis sets. It would hardly be the best choice to follow their approach in
quantum chemistry, because of the chemical significance of the interatomic
overlap. Therefore in a quantum chemical framework orthonormalized basis
sets usually appear only as auxiliary entities used in the derivations or numer-
ical calculations. (It is the usual practice to use basis sets in which even the
basis functions centered on the same atom are non-orthogonal, but this is only
because of computational convenience and has no conceptual significance-
see, however, also Section 4.3.)
For an orthonormalized basis the sum of the projection operators on the in-
dividual basis functions represents a resolution of identity, and of course that
remains valid even if one at first groups the functions into subsets. That means
that if an orthonormalized basis set were used, the sum of atomic projectors
would be the identity operator for the given basis set. In order to facilitate
our analysis, we shall introduce a generalization of this idea: the “atomic de-
composition of identity” [20]. This permits to treat different decomposition
problems (population analysis, energy components, etc.) in a common, ab-
stract framework and the results of different Hilbert-space or 3D schemes of
analysis can be obtained by a simple substitution.
