A general hierarchy of the coarsed-grained electron probability distributions induced by exhaustive partitions of the physical space is presented. It is argued that when the space is partitioned into atomic regions the consideration of these distributions may provide a first step toward an orbital invariant treatment of resonant structures. We also show that, in this case, the total molecular energy and its components may be partitioned into structure contributions, providing a fruitful extension of the recently developed interacting quantum atoms approach (J. Chem. Theory Comput. 2005, 1, 1096). The above ideas are explored in the hydrogen molecule, where a complete statistical and energetic decomposition into covalent and ionic terms is presented.