Triple excitations in the relativistic coupled-cluster formalism and calculation of Na properties

A practical high-accuracy relativistic method of atomic structure calculations for univalent atoms is presented. The method is rooted in the coupled-cluster formalism and includes nonperturbative treatment of single and double excitations from the core and single, double, and triple excitations involving valence electron. Triple excitations of core electrons are included in the fourth order of many-body perturbation theory. In addition, contributions from the disconnected excitations are incorporated. Evaluation of matrix elements includes all-order dressing of lines and vertices of the diagrams. The resulting formalism for matrix elements is complete through the fourth order and sums certain chains of diagrams to all orders. With the developed method we compute removal energies, magnetic-dipole hyperfine-structure constants $A$, and electric-dipole amplitudes. We find that the removal energies are reproduced within 0.01–0.03 % and the hyperfine constants of the $3s_{1∕2}$ and $3p_{1∕2}$ states with a better than 0.1% accuracy. The computed dipole amplitudes for the principal $3s_{1∕2}\text{−}3p_{1∕2;3∕2}$ transitions are in an agreement with 0.05%-accurate experimental data.

Figure 1

Diagrammatic representation of valence triple excitations. The double-headed arrow represents the valence state.

Figure 2

Representative contributions to the rhs of the valence triples equation. The horizontal dashed line denotes Coulomb interaction and the solid lines denote cluster amplitudes.

Figure 3

Sample contributions of triples and disconnected excitations to the valence singles equation.

Figure 4

Effects of disconnected and valence triple excitations on valence doubles.