Coupled tensorial forms of atomic two-particle operator

Full text: For many-electron atoms and ions the ability to present a two-electron operator and its matrix elements in an optimal form may be decisive for successful calculation solutions of many theoretical spectroscopy problems. Here a two-particle operator is expressed in Second Quantization Representation (SQR). Special attention is paid to the approach when the expressions for the operator considered are given in terms of submatrix elements of the coupled two-electron wave function. This gives more freedom in choosing a convenient way for the calculations of matrix elements for open-shell atoms. In a SQR two approaches were considered to express an arbitrary two-electron operator in a coupled tensorial form for multi-shell atoms. The expressions of both topologically different approaches are applicable to the study of the operators representing atomic interactions as well as the operators describing some effective interactions appearing, for instance, in an atomic Many Body Perturbation Theory (MBPT) or a coupled cluster method. The first approach is more suitable when one seeks effectively to calculate the matrix elements of (one) particular operator because the internal ranks of operator are involved in the calculation of many-electron angular part. The second approach is superior for the problems where several operators with different tensorial structure are considered, for example, the formation of energy matrix of atomic Hamiltonian in Breit-Pauli approximation. In this case, only the resulting ranks of operators enter in the expressions for the submatrix elements of many-electron angular part and the computer codes used for such calculations could be more efficient if they are based on the second approach. The final result of the study is a large set of a two-particle operators acting in the space of the states of one-, two-, three- and four-shells. The expressions can be used for both nonrelativistic (LS coupling) and relativistic (jj coupling) approximations. (author)