Spectroscopic determination of Fisher information on vibrational states of diatomic molecules using the example of the X 1I£ g + state of a Li2 molecule

An expression governing Fisher information of vibrational states of a diatomic molecule is obtained in the first-order Wentzel-Kramers-Brillouin (WKB) approximation, and a quantitative criterion of its applicability is proposed. The expression is determined only by the dependence of energy of vibrational levels on quantum number v and the reduced mass of the molecule; it is not related to any additional model assumptions and does not require knowledge of vibrational wavefunction. It is established that the dependence of Fisher information on v attains maximum at certain value v (0). The efficiency of the obtained expression is demonstrated by calculating the Fisher information of vibrational levels v = 0-40 of the X I-1 pound (g) (+) state of the Li-7(2) molecule. It is established that v (0) = 23 for this electronic state. Calculation of the Fisher information of levels v = 0-38 of this state with the help of the obtained relation is in excellent (within mean-square deviation sigma I" = 0.01%) agreement with the results of calculation based on numerical solution of the Schrodinger equation by using the earlier-published semi-empirical potential curve of this state. At the same time, the expressions available in the literature based on the Wilson-Sommerfeld quantization rule describe the Fisher information of these levels only qualitatively (sigma I" = 18%). It is established that the reason for deviation for levels v = 39 (sigma = 0.3%) and v = 40 (sigma = 1.9%) is not related to using the WKB approximation but is due to the numerical differentiation procedure of strongly nonlinear (in the vicinity of the upper bound of v) dependence of the energy of vibrational terms on v appearing in the obtained expression.