A simplified version of the so-called Yukagua model of water, proposed and studied via Monte Carlo simulations elsewhere by two of the authors [Physica A 265, 396 (1999)], is considered. The model includes dipolar, quadrupolar, and octupolar short-range interactions of tetrahedral symmetry that mimic hydrogen bonds, but it differs from the original Yukagua model in that dipolar and quadrupolar terms are not coupled. For this model we solve, using Baxter-Wertheim factorization technique, the Ornstein-Zernike relation for the molecular pair correlation functions closed by generalized mean spherical closures. From the molecular correlations we derive the atom-atom (oxygen-oxygen, oxygen-hydrogen, and hydrogen-hydrogen) pair correlations. We compare our results with those obtained from diffraction experiments and also from other theoretical treatments available in the recent literature including a modified version of the hypernetted chain integral equation for the SPC/E water model and also ab initio simulations of Car-Parrinello type.