On the Determination of Molecular Potential Curves from Spectroscopic Data

The problem of determining accurately a molecular potential curve from spectroscopic data on the system is analyzed, and the difficulties are discussed in detail. In general the principal difficulty arises from the importance in the energy expression of terms in high powers in the quantum numbers; to deal with this, recourse may be taken to a method employing graphical integration. Formulas based on the first order of the W.B.K. approximation are usually but not always adequate. One can obtain markedly better results in determining potential curves by dealing directly with the energy levels which it is desired to reproduce, rather than by the usual method of making the curve reproduce a limited number of spectroscopic constants. A method of successive approximations has proved to be particularly effective in giving accurate results. Formulas are given for the convenient manipulation of potential curves of the types suggested by Morse, P\"oschl and Teller, Hylleraas, Dunham, and also a generalization of the Morse curve. This last curve has proved to be most satisfactory, combining great flexibility with relative ease of manipulation; explicit solutions for vibrational wave functions are not obtainable for it, however. The discussion is illustrated by extensive computations on the lowest ³Σg state of H₂, for which a potential curve very accurate in the range of nuclear separations 1.3<~r<~2.9aH is obtained.