Coupled tensorial forms of the second-order effective Hamiltonian for open-subshell atoms in jj-coupling

Symbolic programming packageN Coperatorswith applications to theoretical atomic spectroscopy

“…Whilst the Racah package [11][12][13], based on Maple, is known to handle several similar procedures such as the calculation of transformation matrices and recoupling coefficients, evaluation of many-particle matrix elements etc., the advantages and distinctive features of NCoperators are that (i) it manages the reduction of the product of more than two SU(2)-irreducible representations therefore giving an opportunity to calculate the matrix elements on the basis of many-open-shell wave functions (eg. six open shells appear even in the second-order Rayleigh-Schrödinger perturbation theory based on effective operator approach [14]); (ii) it is based on the algebraic manipulations of the quantities considered neither the diagrammatic representations such as the Jucys graphs exploited in the angular momentum theory [15], though the diagrammatic visualization, to some extent, is implemented as well; (iii) the output of expressions, including Clebsch-Gordan coefficients and 3nj-symbols, obtained within Mathematica interface fits the standard text-based form; (iv) Racah refers to Varshalovich et al [16], while NCoperators implements the sum rules given by Jucys et al [15; 17]; in many particular cases, the difference is in the phase factor and different notions (eg. the ClebschGordan coefficient versus the Wigner 3j-symbol); (v) at last, but not least, the package NCoperators allows one to generate expansion terms of the Rayleigh-Schrödinger perturbation theory up to the third-order [18], and afterward express them in the irreducible tensor form.…”