Astronomy & Astrophysics (A&A) is an international journal which publishes papers on all aspects of astronomy and astrophysics

Individual orbital optimization of wave functions for the initial and final states produces the most accurate wave functions for given expansions, but complicates the calculation of transition-matrix elements since the two sets of orbitals will be nonorthogonal. The orbital sets can be transformed to become biorthonormal, in which case the evaluation of any matrix element can proceed as in the orthonormal case. The transformation of the wave-function expansion to the new basis imposes certain requirements on the wave function, depending on the type of transformation. An efficient and general method was found a few years ago for expansions in determinants, spin-coupled configurations, or configuration state functions for molecules belonging to the ${\mathit{D}}_{2\mathit{h}}$ point group or its subgroups. The method requires only that the expansions are closed under deexcitation and thus applies to restricted active space wave functions. This type of expansion is efficient for correlation studies and includes many types of expansions as special cases. The above technique has been generalized to the atomic, symmetry adapted case requiring the treatment of degenerate shells ${\mathit{nl}}^{\mathit{N}}$, with arbitrary occupation numbers 0≤N≤4l+2. A computer implementation of the algorithm in the multiconfiguration Hartree-Fock atomic-structure package for atoms allows the calculation of transition moments for individually optimized states. An application is presented for the B i 1${\mathit{s}}^2$2${\mathit{s}}^2$2p ${}_{}{}^2{}_{}{}^{}$ ${\mathit{P}}^{\mathit{o}}$→1${\mathit{s}}^2$2s2${\mathit{p}}^2$ ${}_{}{}^2{}_{}{}^{}$D electric dipole transition probability, which is highly sensitive to core-polarization effects.

We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for all multipartite symmetric states in the general $N$-qubit case. For this purpose, we introduce 2 parameters playing a crucial role, namely, the diversity degree and the degeneracy configuration of a symmetric state. Those parameters give rise to a simple method of identifying operational families of SLOCC entanglement classes of all symmetric $N$-qubit states, where the number of families grows as the partition function of the number of qubits.