  The energy levels of the Schr\"odinger equation under the Eckart-Hellmann
potential (EHP) energy function are studied by the Nikiforov-Uvarov-Functional
Analysis (NUFA) method. We obtained the analytic solution of the energy spectra
and the wave function in closed form with the help of Greene-Aldrich
approximation. The numerical bound states energy for various screening
parameters at different quantum states and vibrational energies of EHP for
CuLi, TiH, VH, and TiC diatomic molecules were computed. Four particular cases
of this potential were achieved. To test the accuracy of our results, we
computed the bound states energy eigenvalues of Hellmann potential which are in
excellent agreement with the report of other researchers.
