Two new representations of diatomichyphen;molecule vibrational potentials are presented. One of these includes most of the previously employed series approximations as special cases. The new representations are tested against older ones by using Peekrsquo;s rsquo;rsquo;exactrsquo;rsquo; numerical Bornndash;Oppenheimer potentials for the 1ssgr;g, 2ppgr;u, and 3dsgr;gstates of H2+as test problems. Accuracy comparisons are made with Dunham, Thakkar, Ogilviendash;Tipping, Coulombhyphen;subtracted Ogilviendash;Tipping, and Padeacute; representations. A central idea of the new treatment is that it isnotnecessary to use a global representation of the potential over the region 0quest;R/Reinfin;. Rather one can choose separate representations suitable for the subregionsR/Request;1 andR/Request;1 and then match them smoothly to each other and to the Dunham expansion atR/Re=1. Attention is focused on finding improved approximations forR/Regsim;1, since this region exerts strong control on the vibrational eigenvalue spectrum and it is here, perhaps, where the older techniques are at their weakest. Known properties of the exact potential such as theRrarr;infin; behavior and the dissociation energy can be built into the new formsapriori. If the dissociation energy is not known, the new methods allow it to be estimated with better accuracy then could be done previously. If one knows only Dunham coefficients, the Coulombhyphen;subtracted Ogilviendash;Tipping series is the superior representation. If one knows, in addition, the dissociation energy, the new representations are superior and give more accurate results on the intervalR/Request;1.
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