Relation between the Hilbert space and “fuzzy atoms” analyses

The “fuzzy atoms” formalism of analysing the wave function in terms of the three-dimensional (3D) space permits one to obtain “effective atomic orbitals” (3D variants of natural hybrids) from an SCF calculation. It is demonstrated here that a “Hilbert-space” analysis performed in the basis of these “effective AO-s” exactly reproduces the results of the “fuzzy atoms” 3D analysis: the Mulliken net and overlap populations calculated in that special basis coincide with the “fuzzy atoms” net and overlap populations, and the same holds for the bond orders calculated in that basis and in the “fuzzy atoms” framework.