One key problem in parameterizing water models is that there is precisely zero phase space overlap between simulations carried out with rigid water models with different geometry parameters. Take, as an example, the TIP3P and SPC models which have different rigid bond lengths for the OH and HH bonds; bond lengths corresponding to one water model can never be sampled during the simulation of the other water model. We have shown one can efficiently capture the free energy differences between water models by creating phase space overlap between these states by mapping between configurations, and including the Jacobians of the these mapping transformations in the free energy calculations. We have laid out specifically how to carry out this process in the case of mapping between rigid water models in our previous work, which we also review in the Methods section here. Using this mapping, we showed how only a few hundreds of samples are required to calculate accurate and precise expectation values and free energy estimates for parameter perturbations in water models. By implementing a combination of reweighting and the mapping techniques, it therefore should be possible to get substantial reductions in computational costs involved in a full parameterization based on free energy estimates and other thermodynamic observables. In this paper, we demonstrate the applicability of the parameterization scheme by optimizing simple rigid water models to capture the behavior of liquid water over a large range of temperatures and pressures. To fully parameterize even a rigid water model, we need to explore a multidimensional water model parameter space. The six parameters in the three and four site rigid water model as shown in Figure are: charge q O , Lennard-Jones σ and of oxygen, the OH bond length, HOH bond angle, and in four-site water models, the OM distance between the virtual site (carrying the negative charge of the water model) and oxygen atom. The charge on both the hydrogens is necessarily +q O /2. As is typical, the LJ σ and of both hydrogens are chosen in this family of models to be zero. For three site models OM is also set to be zero.