A procedure for directly fitting the potential energy curve of a diatomic molecule has been applied to the X1sigma+ ground states of LiCl, TlCl, RbF and CsF. Extensive, high-precision infrared and pure-rotational data sets for all isotopomers of the aforementioned molecules have been employed in direct least-squares fits of a radially dependent Hamiltonian operator. The Born-Oppenheimer potentials are represented by a modified Lennard-Jones function that is shown to be flexible and provide the proper behavior in the long-range region of the potential. While the potential fits of LiCl and TlCl required the inclusion of Born-Oppenheimer breakdown functions, no such functions were necessary for either RbF or CsF.