Atomic energies are used to describe local stability in eight low-lying water hexamers: prism, cage, boat 1, boat 2, bag, chair, book 1, and book 2. The energies are evaluated using the quantum theory of atoms in molecules (QTAIM) at MP2/aug-cc-pVTZ geometries. It is found that the simple, stabilizing cooperativity observed in linear hydrogen-bonded water systems is diminished as clusters move from nearly planar to three-dimensional structures. The prism, cage, and bag clusters can have local water stabilities differing up to 5 kcal mol(-1) as a result of mixed cooperative and anticooperative interactions. At the atomic level, in many cases a water may have a largely stabilized oxygen atom but the net water stability will be diminished due to strong destabilization of the water's hydrogen atoms. Analysis of bond critical point (BCP) electron densities shows that the reduced cooperativity results in a decrease in hydrogen bond strength and an increase in covalent bond strength, most evident in the prism. The chair, with the greatest cooperativity, has the largest average electron density at the BCP per hydrogen bond, whereas the cage has the largest total value for BCP density at all hydrogen bonds. The cage also has the second largest value (after the prism) for covalent bond critical point densities and an oxygen-oxygen BCP which may factor into the experimentally observed stability of the structure.