Correlations and accuracy of estimation of spectroscopic constants and term values

Abstract

A comparison and evaluation of the various methods for reducing spectroscopic data to spectroscopic constants or term values is made with the aid of an analysis of a number of “synthetic” Σ-Σ bands generated from fixed sets of constants with random noise superimposed on the line positions. It is shown that the strong correlations that exist between the upper-state constants B′, D′ and the lower-state constants B 0, D 0 can be effectively broken up by using the difference constants Δ B = B′ - B 0 and Δ D = D′ - D 0, along with ν 0, for representing the upper states. The lower state constants B 0 and D 0 and their standard errors calculated from the combination differences Δ 2 F″( J) are shown to be as good as those obtained from direct polynomial fits. If data for a number of bands originating in the same lower state are available, a considerable increase in accuracy of estimating the lower-state constants can be attained by analyzing the bands simultaneously, e.g., using combination differences, provided the data are free from systematic errors. The dependence of the accuracy of determining the constants B 0, D 0, ν 0, Δ B, and Δ D on the extent of the band analyzed was investigated by varying the minimum and maximum J-values. The plots showing this dependence for both the actual errors and standard errors can be used e.g., to assess the band size necessary to attain a desired accuracy for a given constant. Åslund's term-value method is cast in a form which permits simple derivation of explicit formulas for the correlation coefficients connecting all the upper- and lower-state term values, and of explicit relations between the term values and the combination differences. Modifications of the term-value method suitable for the case where one or more transitions originate in an unperturbed state are explored. It is shown that the “difference term values” referred to the lower (unperturbed) state with the same value of J, T i( J)- T 0( J), are essentially uncorrelated to the lower state constants B 0 and D 0. Since these quantities can be expressed directly as the eigenvalues of the energy matrix for a given J in the presence of perturbations, their use for representing the energies of perturbed rovibronic states is recommended.