Dissociation Energy and Long-Range Potential of Diatomic Molecules from Vibrational Spacings of Higher Levels

Abstract

An expression is derived which relates the distribution of vibrational levels near the dissociation limit D of a given diatomic species to the nature of the long-range interatomic potential, in the region where the latter may be approximated by D − Cn / Rn. Fitting experimental energies directly to this relationship yields values of D, n, and Cn. This procedure requires a knowledge of the relative energies and relative vibrational numbering for at least four rotationless levels lying near the dissociation limit. However, it requires no information on the rotational constants or on the number and energies of the deeply bound levels. D can be evaluated with a much smaller uncertainty than heretofore obtainable from Birge–Sponer extrapolations. The formula predicts the energies of all vibrational levels lying above the highest one measured, with uncertainties no larger than that of the binding energy of the highest level. The validity of the method is tested with model potentials, and its usefulness is demonstrated by application to the precise data of Douglas, Mo/ller, and Stoicheff for the B 3Π0u+ state of Cl2.

Quantization rule for highly excited vibrational states of H

“…Replacing (18) by the renormalized singlepower tail (20) only marginally affects this result. The further entries in the last column in Table 3 show that the values predicted for the scattering length consistently lie in the range �30.5 to �31 au, regardless of which reference potential is chosen for V tail (R).…”

“…With the quantization function based on the single-power tail (18), this yields v D ¼ 1.02676, and the value of the scattering length that follows via (11) is a ¼ þ762.82 au. On the basis of the renormalized single-power tail (20), we obtain v D ¼ 1.02675 and a ¼ þ762.81 au. This can be compared to the full-scale ab initio calculations of [27,28], which yield a ¼ þ750�5 au and a ¼ þ725.5 au respectively, for the p-H scattering length in ungerade symmetry.…”

“…The well-known semiclassical quantization rule of LeRoy and Bernstein [20,21] fails towards threshold, because it ignores the crucial energy dependence of the outer reflection phase in this anticlassical, extreme quantum limit. The fact that conventional WKB quantization fails near threshold has actually been known for a long time, see e.g.…”

“…The plots are shown in Figure 2, and the numerical values on which they are based are listed in Table 2. The column headedṼ ð1Þ tail ðRÞ lists the numbers obtained on the basis of the single-power tail (20), in which nonadiabatic effects due to the reducedmass renormalization in the hydrogen atom are taken into account. The small differences to the results obtained with the potential tail (18) are not discernible in the figures.…”

“…The straight dashed green and solid red lines are fitted according to (10) (18), (20), (21), (22) and (23) (10), together with the tail parameters � a, b and 0 . The last column shows the value obtained for the scattering length according to (11).…”