Energies, Auger width and branching ratios of some core-excited states for a beryllium-like system

The saddle-point variational method with multiconfiguration interaction wave function is used to obtain the energies of the core-excited triplet 1$s$2$p^{3} \quad ^{3}P^{o}$ and $^{3}D^{o}$ states for the beryllium isoelectronic sequence ($Z$=4-10), including the mass polarization and relativistic corrections. Restricted variational method is carried out to extrapolate a better energy. The saddle-point complex-rotation method is used to study the Auger width and branching ratios in the Auger decay of the beryllium-like core-excited 1$s$2$p^{3} \quad ^{3}P^{o}$ and $^{3}D^{o}$ states and double core-excited 2$s$2$p^{3} \quad ^{1}D^{o}$, $^{3}S^{o}$, $^{3}P^{o}$, $^{3}D^{o}$ and $^{5}S^{o}$ resonances. The relative branching ratios of the main decay channels of these resonance states are explained satisfactorily using the spin-alignment-dependent theory. The oscillator strengths and transition rates are also calculated. Our results are compared with the available experimental and other theoretical results in the literature.