Exact quantum mechanical calculations on the excess electronic states of the electron-water molecule system have been performed in the static-exchange approximation. The computational model includes a steep, but smooth confining potential which keeps the excess electron in the vicinity of the neutral molecule. Elimination of the core states of the water molecule by the application of the Phillips-Kleinman repulsion operator, and the removal of the large core oscillations of the wave function of the excess electron by the linear combination of the core states and the valence state result in a smooth pseudo-wave function. The pseudo-wave function has proper asymptotic behavior with the correct eigenvalue, and, thus, can serve as a model for comparisons to test the validity of various approximations employed in electron-molecule pseudopotential theory. From the comparisons we conclude that of the most commonly used approximations for the repulsion and the exchange operators only the combination of the local repulsion (LR) approximation and the semiclassical exchange (SCE) works partly satisfactorily. This particular combination reproduces the exact eigenvalue reasonably well, whereas the fit of the electron density is moderate. Although the calculated local potential, based on the LR-SCE approximation, is similar in its most characteristic features to those employed earlier for hydrated electron calculations, we propose this potential to be considered as a reasonable starting point for further work. Since the other examined approximations fail seriously, we find them inappropriate to use in the development of a new effective pair potential.