Diatomic molecule vibrational potentials: Accuracy of representations

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Abstract
A method is presented for increasing the radius of convergence of certain representations of diatomic molecule vibrational potentials. The method relies on using knowledge of the analytic structure of such potentials to the maximum when attempting to approximate them. The known singular point (due to the centrifugal and/or Coulomb potentials) at zero internuclear separation should be included in its exact form in an approximate representation. The efficacy of this idea is tested [using Peek’s ’’exact’’ numerical Born–Oppenheimer potential for the (1sσg)2Σ+g state of H+2 as a test problem] when the representational form is the series of (1) Dunham, (2) Simons, Parr, and Finlan, (3) Thakkar, and (4) Ogilvie–Tipping, and also (5) when the form is a [2, 2] or a [3, 3] Padé approximant. Significant improvements in accuracy are obtained in some of these cases, particularly on the inner wall of the potential. A comparison of the effectiveness of the five methods is made both with and without the origin behavior being included exactly. This is useful in itself as no comprehensive accuracy comparison of the standard representations seems to have appeared in the literature. The Ogilvie–Tipping series, corrected at the origin for singular behavior, is the best representation presently available for states analogous to the (1sσg)2Σ+g state of H+2.