Variance and shift of transition arrays for electric and magnetic multipole transitions

Generalized analytical expressions for the two-electron relativistic Unresolved-Transition-Array (UTA) energy variance and shift for electric and magnetic transitions of general multipole order are presented. The revised expressions are shown to agree with the exact moments calculated directly from the energy levels of two-electron configurations. We show that for electric transitions of even multipole order and for magnetic transitions, the available expressions in the literature, which are implemented in widely used atomic codes, are incorrect. We suggest an alternative method for the calculation of the UTA energy variance and shift by using the analytical expressions for the two-electron energy levels and line-strengths. The method is much more efficient and simple than the use of the traditional lengthy analytic expressions. Finally, the effect of UTA widths on Super-Transition-Array (STA) spectral opacity is shown for several examples.