A method for constructing analytical diatomic molecular potential energy functions is proposed.The potential energy function is examined with 36 diatomic molecules or ions-homonuclear ground-state of neutral diatomic molecules Li₂-X¹∑_g⁺,Na₂-X¹∑_g⁺,heternuclear ground-state of charged diatomic molecular ion (BC)^--X³∏,heternuclear excitation-state neutral diatomic molecules PbS-A¹∑⁺,BaO-A¹∑,heternuclear excitation-state of charged diatomic molecular ion (CS)⁺-A²∏,homonuclear excitation-state of neutral diatomic molecule K₂-B¹∏_u,homonuclear excitation-state for charged diatomic molecule N₂⁺-B²∑_u⁺ etc.Vibrational energy levels of molecules calculated are consistent with RKR data (Rydberg-Klein-Rees) and experimental data in high-precision.It shows that the potential energy function is superior to Murrell-Sorbie potential in calculating vibrational energy levels of molecules.