Knowledge Bank

Within individual high-J ($J\simeq 50$)P-branch and R-branch manifolds in the highly resolved $\nu 3$ band of $SF6$, slight deviations of observed line positions from positions prescribed by the diagonal $F(4)$-coefficients of $Moret-Bailly1$ have been explained successfully by Galbraith, McDowell, and $Cantrell2$ (GMC). They diagonalized Hecht’s $Hamiltonian3$ applied to octahedral $XY6$ molecules, including terms of higher-order fourth-rank tensor operators that are off-diagonal in the quantum number R, in the basis of $(6J+3)$ states with the same value of J. The present calculations employ the ``second-order perturbation” formulas of Moret-$Bailly1$ and reproduce the results of GMC with excellent agreement. When the off-diagonal fourth-rank corrections to the line positions are plotted against the $F(4)$ positions within the manifold, they trace out parabolas that become smoother and better-fitted as J increases. The present calculations show that, ultimately, the mathematical parabolas are properties of the high-J off-diagonal $F(4)$-coefficients.