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By employing the dissociation energy and the equilibrium bond length for a diatomic molecule as explicit parameters, we generate improved expressions for the well-known Rosen-Morse, Manning-Rosen, Tietz, and Frost-Musulin potential energy functions. It is found that the well-known Tietz potential function that is conventionally defined in terms of five parameters [T. Tietz, J. Chem. Phys. 38, 3036 (1963)] actually only has four independent parameters. It is shown exactly that the Wei [Phys. Rev. A 42, 2524 (1990)] and the well-known Tietz potential functions are the same solvable empirical function. When the parameter h in the Tietz potential function has the values 0, +1, and -1, the Tietz potential becomes the standard Morse, Rosen-Morse, and Manning-Rosen potentials, respectively.

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By employing an improved new approximation scheme to deal with the centrifugal term, we solve approximately the Klein-Gordon equation with equal scalar and vector generalized Pöschl-Teller potentials for the arbitrary orbital angular momentum number l. The bound state energy equation and the unnormalized radial wave functions have been approximately obtained by using the supersymmetric shape invariance approach and the function analysis method. It is found that the present approximate analytical results are in better agreement with those obtained by using the numerical integration approach for small values of α than the approximate results obtained by using the conventional approximation to the centrifugal term.