Corrections to radiative rates between atomic configurations

The computation of radiative opacity or emissivity of hot dense matter is a challenging task. It requires accounting for an immense number of energy levels and lines across various excitation and ionization states. Whether in local thermodynamic equilibrium (LTE) or non-LTE plasmas, statistical methods provide significant assistance. Many computational codes are based on the Detailed Configuration Accounting approximation, which involves averaged rates between configurations. In that approach, only the mean energies of the configurations are considered, and the effects of the energy distributions of the levels within the initial and final configurations are typically neglected. A long time ago, Klapisch proposed a method to correct the rates. The corresponding formalism includes the energy shift and variance of the Unresolved Transition Array, as well as the average energies of the configurations. We extend this formalism and investigate its impact on opacity calculations in two specific cases: first, the iron experiment conducted at Sandia National Laboratories under conditions similar to those at the base of the Sun's convective zone, dominated by L-shell 2p-$n$d transitions, and second, laser experiments--still for iron--at much lower temperature. The latter measurements shed light on our understanding of the envelopes of $\beta$-Cephei-type stars, where the relevant transitions are intra-M-shell $\Delta n=0$ (3-3) transitions, specifically 3s-3p and 3p-3d, in the XUV range. The issue of ensuring the validity of Kirchhoff's law when plasmas approach LTE is also addressed, and a prescription is proposed, applying both to the standard configuration-to-configuration case and to the aforementioned corrections, which account for the energy distribution of the levels within a configuration.