The Poincaré-Hopf relation is studied for molecular electrostatic potentials (MESPs) of a few test systems such as cyclopropane, cyclobutane, pyridine, and benzene. Appropriate spheres centered at various points, including the center of mass of the system under study, are constructed and the MESP gradient is evaluated on the corresponding spherical grid. The change in directional nature of MESP gradient on the surface of these spheres gives indication of the critical points of the function. This is used for developing a method for locating the critical points of MESP. The strategy also enables a general definition of the Euler characteristic (EC) of the molecule, independent of any region or space. Further, the effect of basis set and level of theory on the EC is discussed.