Configurations of hydrogen bonded water can be classified into hydrogen bonding families, or configurations that vary only by combinations of reversals of cycles. These hydrogen bonding families were shown to having similar energies. Progressively lower energies may therefore be arrived at by iteratively establishing hydrogen bonding families and finding configurations that are not members of these families; the latter invariably implies the reorientation of hydrogen bonding. In this study, nonmembers were produced by the optimization of prospective minima. In general, the breaking of the hydrogen bonding network to produce new ones produce other families with their own range of energies.